Research ArticleOdd Jacobi Manifolds and Loday-Poisson Brackets

Andrew James Bruce

Institute of Mathematics Polish Academy of Sciences Sniadeckich 8 PO Box 21 00-956 Warszawa Poland

Correspondence should be addressed to Andrew James Bruce andrewjamesbrucegooglemailcom

Received 24 December 2013 Accepted 18 February 2014 Published 7 April 2014

Academic Editor Luis J Alias

Copyright copy 2014 Andrew James BruceThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifoldWe refer to such Poisson-like brackets as Loday-Poisson brackets We examine the relations between the Hamiltonian vector fieldswith respect to both the odd Jacobi structure and the Loday-Poisson structure Furthermore we show that the Loday-Poissonbracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field

In memory of Jean-Louis Loday (1946ndash2012)

1 Introduction

In this paper we construct a Loday-Poisson bracket on thealgebra of smooth functions on an odd Jacobi supermanifold(119872 119878 119876) see [1] The key observation here is that thehomological vector field 119876 isin Vect(119872) is a Jacobi vector fieldwith respect to the odd Jacobi structure Due to this onecan directly employ the notion of derived brackets as definedby Kosmann-Schwarzbach [2 3] We will draw heavily fromthese works (the notion of a derived bracket can be tracedback to J L Koszul (1990) in unpublished notes The notionwas also put forward byThTh Voronov (1993) but remainsunpublished)

On any differential super Lie algebra (119860 [ ] 119863) withbracket of even or odd parity one can construct the ldquoderivedbracketrdquo

(119886 119887)119863= (minus1)

119886+1[119863 (119886) 119887] (1)

This bracket is not in general a Lie bracket (or a Lieantibracket) due to the lack of skew symmetry Howeverthe bracket does satisfy a form of the Jacobi identity SuchLie algebra mod the skew symmetry was first introducedby Loday under the name ldquoLeibniz algebrasrdquo [4 5] we willshortly be a little more precise here

On an odd Jacobi manifold we have a differential Lieantialgebra that is an odd Lie bracket provided by the odd

Jacobi bracket and a differential provided by the homologicalvector field

In essence on an odd Jacobi manifold one also has ldquoaPoisson bracket mod the skew symmetryrdquo We make theprevious statement more precise and then investigate therelations between the Hamiltonian vector fields given by theodd Jacobi and the Loday-Poisson structure Furthermorewe study the notion of an odd form of noncommutativemultiplication on the smooth functions on an odd Jacobimanifold generated by the homological vector field It isshown that the Loday-Poisson bracket satisfies the rightLeibniz rule over both the standard product of smoothfunctions and the odd derived product

Remark 1 Grabowski and Marmo in [6 7] prove that on aclassical (purely even) manifold 119873 the Jacobi-Loday identitytogether with the (generalized) Leibniz rule for a binarybracket on119862infin(119873) implies the skew symmetry there can onlybe Poisson or even Jacobi brackets on classical manifolds Infact they show that similar statements also hold for Nambu-Poisson brackets and for generalizations of Lie algebroidbrackets However here we consider supermanifolds mean-ing that we have a Z

2-grading and more importantly non-

trivial nilpotent functions which invalidate the conditions oftheir theorem

Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 630749 10 pageshttpdxdoiorg1011552014630749

2 Journal of Mathematics

This paper is arranged as follows In Section 2 werecall the definition of an odd Jacobi manifold and presentsome results needed throughout the rest of this paper InSection 3 we construct the Loday-Poisson bracket on an oddJacobi manifold and in Section 4 we present some standardexamples The Hamiltonian vector fields with respect to theodd Jacobi brackets and the Loday-Poisson brackets are thesubject of Section 5 The derived product generated by ahomological vector field is defined and studied in Section 6Here it is shown that the Loday-Poisson bracket satisfies anappropriate Leibniz rule over the noncommutative derivedproduct In Section 7 the previous constructions are appliedto the specific case of Jacobi algebroids We conclude thispaper with some final remarks in Section 8 The remainderof the introduction is devoted to some preliminary notionsand nomenclature

11 Preliminaries All vector spaces and algebras will beZ2-graded We will generally omit the prefix super By

manifold we will mean a smooth supermanifold We denotethe Grassmann parity of an object by tilde 119860 isin Z

2 By even

or oddwewill be referring explicitly to theGrassmann parityA Poisson (120576 = 0) or Schouten (120576 = 1) algebra is

understood as a vector space 119860 with a bilinear associativemultiplication and a bilinear operation sdot sdot

120576 119860 otimes 119860 rarr 119860

such that

Grading 119886 119887120576= 119886 + + 120576

Skew symmetry 119886 119887120576= minus(minus1)

(119886+120576)(+120576)119887 119886

120576

Jacobi identity sumcyclic(119886119887119888) (minus1)(119886+120576)(119888+120576)

119886 119887 119888120576120576

=

0

Leibniz rule 119886 119887119888120576= 119886 119887

120576119888 + (minus1)

(119886+120576)119887119886 119888

120576

For all homogenous elements 119886 119887 and 119888 isin 119860 Extension toinhomogeneous elements is via linearity If the Leibniz ruledoes not hold identically but is modified as

119886 119887119888120576= 119886 119887

120576119888 + (minus1)

(119886+120576)119887119886 119888

120576minus 119886 1

120576119887119888 (2)

then we have even (120598 = 0) or odd (120598 = 1) Jacobi algebrasA manifold 119872 such that 119862infin(119872) is a PoissonSchouten

algebra is known as a PoissonSchouten manifold In partic-ular the cotangent of a manifold comes equipped with acanonical Poisson structure

Let us employ natural local coordinates (119909119860 119901119860) on119879lowast119872

with 119909119860 = 119860 and 119901119860

= 119860 Local diffeomorphisms on 119872

induce vector bundle automorphism on 119879lowast119872 of the form

1199091198601015840

= 1199091198601015840

(119909) 1199011198601015840 = (

120597119909119861

1205971199091198601015840)119901

119861 (3)

We will in effect use the local description as a natural vec-tor bundle to define the cotangent bundle of a supermanifoldThe canonical Poisson bracket on the cotangent is given by

119865 119866 = (minus1)119860119865+119860 120597119865

120597119901119860

120597119866

120597119909119860minus (minus1)

119860119865 120597119865

120597119909119860120597119866

120597119901119860

(4)

A manifold equipped with an odd vector field 119876 suchthat the nontrivial condition 119876

2 = (12)[119876 119876] = 0 holdsis known as a Q-manifold The vector field 119876 is known as ahomological vector field for obvious reasons

The notion of a Loday-Poisson bracket is important inthis paper We define such a bracket as ldquoa Poisson bracketbut without the skewsymmetryrdquo Note that the Jacobi identityas presented above for an even bracket (with or without aLeibniz rule) implies that the adjoint endomorphism 119891 999492999484

ad119891= 119891 sdot is a derivation over the bracket itself That is

ad119891119892 ℎ = ad

119891(119892) ℎ + (minus1)

119891119892119892 ad

119891 (ℎ) (5)

Note that the above has direct meaning even if the skewsymmetry of the bracket is weakened Thus the most usefuldefinition of the Jacobi identity for an even bracket when thesymmetry is lost is

119891 119892 ℎ = 119891 119892 ℎ + (minus1)119891119892

119892 119891 ℎ (6)

We refer to this form of the Jacobi identity as the (right)Jacobi-Loday identity [4] If we have an even bracket thatsatisfies the Jacobi-Loday identity and the right Leibniz rulebut not necessarily the skew symmetry then we call such abracket as a right Loday-Poisson bracket Left Loday-Poissonbrackets can be similarly defined as can Loday-Poissonbrackets that satisfy the Leibniz rule in both directions Forexample classical Poisson brackets are specific examples ofLoday-Poisson bracket albeit skew symmetricWewill exam-ine only right Loday-Poisson brackets but we will typicallydrop explicit reference to the word right To the authorrsquosknowledge (noncommutative) Loday-Poisson algebra wasfirst discussed by Casas and Datuashvili [8]

Remark 2 Much of this paper will generalize directly to QS-manifolds in the sense of Voronov [9] A primary example ofa QS-manifold is the antitangent bundle Π119879119872 of a Poissonmanifold 119872 The Schouten structure is supplied by theKoszul-Schouten bracket and the homological vector field bythe de Rham differential [10] The associated Loday-Poissonbracket is a natural extension of the Poisson bracket on119862infin(119872) to differential forms over119872 but note this extension

is as a Loday bracket only [2] The constructions here willnot directly generalize to even Jacobi supermanifolds dueto incompatibility of the Grassmann parities There is nocanonical choice of a homological Jacobi vector field on aneven Jacobi supermanifold

2 Odd Jacobi Manifolds

Lichnerowicz [11 12] introduced the notion of a Poissonman-ifold as well as a Jacobi manifold Such manifolds have foundapplications in classical mechanics and play an importantrole in quantisation In this section we recall some of thebasic notions as pertaining to odd Jacobi (super) manifoldsNo proofs are given here and can be found in [1] or followdirectly from the definitions For a modern discussion ofJacobi structures including Z-graded versions see [13ndash15]

Journal of Mathematics 3

Definition 3 An odd Jacobi structure (119878 119876) on a manifold119872

consists of

(i) an odd function 119878 isin 119862infin(119879lowast119872) of degree two in fibrecoordinates

(ii) an odd vector field 119876 isin Vect(119872)

such that the following conditions hold

(1) the homological condition 1198762 = (12)[119876 119876] = 0(2) the invariance condition 119871

119876119878 = 0

(3) the compatibility condition 119878 119878 = minus2Q119878

Here Q isin 119862infin(119879lowast119872) is the principal symbol or ldquoHamil-tonianrdquo of the vector field119876The brackets sdot sdot are the canoni-cal Poisson brackets on the cotangent bundle of themanifold

A manifold equipped with an odd Jacobi structure (119878 119876)

is known as an odd Jacobi manifold

Remark 4 At first glance the above definition seems dif-ferent from the classical notion of a Jacobi manifold Thecompatibility condition involves the principal symbol of thehomological vector field rather than the vector field itselfHowever all we are really doing is thinking of the vector fieldas a linear function on the cotangent bundle

Definition 5 The odd Jacobi bracket on 119862infin(119872) is defined as

[119891 119892]119869= (minus1)

119891+1119878 119891 119892 minus (minus1)

119891+1Q 119891119892

= (minus1)(119861+1)119891+1

119878119861119860 120597119891

120597119909119860120597119892

120597119909119861

+ (minus1)119891(119876

119860 120597119891

120597119909119860)119892 + 119891(119876

119860 120597119892

120597119909119860)

(7)

with 119891 119892 isin 119862infin(119872)

The odd Jacobi bracket makes the algebra of smoothfunctions on119872 into an odd Jacobi algebra

Remark 6 The definition of an odd Jacobi manifold givenhere is not quite the most general and one can include anodd function in the construction of an odd Jacobi structuresee Grabowski and Marmo [14] for details Note that thedefinition given by Grabowski andMarmo coincide with thatgiven here (up to conventions) up on setting the odd functionto zero Thus there are examples of odd Jacobi brackets onsupermanifolds not covered by the constructions here

Definition 7 Given a function 119891 isin 119862infin(119872) the associated

Hamiltonian vector field is given by

119891 999492999484 119883119891isin Vect (119872)

119883119891(119892) = (minus1)

119891[[119891 119892]]

119869minus 119876 (119891) 119892

(8)

Note that the homological vector field 119876 is itself Hamil-tonian with respect to the unit constant function 119876 = 1198831 =

[[1 sdot]]119869

Definition 8 Avector field119883 isin Vect(119872) is said to be a Jacobivector field if and only if

119871119883119878 = 120594 119878 = 0 119871

119883119876 = 120594Q = 0 (9)

where 120594 isin 119862infin(119879lowast119872) is the symbol or ldquoHamiltonianrdquo of thevector field 119883 Note that the homological vector field 119876 is aJacobi vector field

Proposition 9 Let 119885 isin Vect(119872) be a vector field on an oddJacobi manifold Then the following are equivalent

(1) 119885 is a Jacobi vector field(2) 119885 is a derivation over the odd Jacobi bracket

119885([119891 119892]119869) = [119885 (119891) 119892]

119869+ (minus1)

119885(119891+1)[119891 119885 (119892)]

119869(10)

(3) [119885119883119891] = (minus1)

119885119885119883(119891)

for all Hamiltonian vectorfields119883

119891

Proposition 10 AHamiltonian vector field119883119891isin Vect(119872) is

a Jacobi vector field if and only if 119891 isin 119862infin(119872) is Q-closed

Proposition 11 The assignment 119891 999492999484 119883119891is a morphism

between the odd Lie algebra on 119862infin(119872) provided by the oddJacobi brackets and the Lie algebra of vector fields Specificallythe following holds

[119883119891 119883

119892] = minus119883

[119891119892]119869

(11)

for all 119891 119892 isin 119862infin(119872)

Proposition 12 On an odd Jacobi manifold the followingidentity holds

119883119891119892

= (minus1)119891119891119883

119892+ (minus1)

119892(119891+1)119892119883

119891+ (minus1)

119891+119892+1119891119892119876 (12)

Remark 13 An even or odd Lie bracket defined on sectionsof an even line bundle over a manifold which is a first-orderdifferential operator with respect to each argument is knownas a Kirillov bracket [16] Jacobi brackets are Kirillov bracketson trivial line bundles In this respect Jacobi brackets serveas a local description of Kirillov brackets but should be seenas a secondary notion We will not pursue the more generalsituation of Kirillov brackets here and focus only on Jacobistructures

3 Loday-Poisson Brackets

We are now in a position to state the main theorem of thispaper We draw heavily on the notion of derived brackets inthe sense of Kosmann-Schwarzbach see [2 3] for details

Theorem 14 Let (119872 119878 119876) be an odd Jacobi manifold then119862infin(119872) comes equipped with a canonical Loday-Poisson

bracket

Proof We prove the theorem by direct construction of theLoday-Poisson bracket The bracket is canonical up to minor

4 Journal of Mathematics

issues of conventions Following Kosmann-Schwarzbach [2]we define the Loday-Poisson bracket as

119891 119892119869= (minus1)

119891+1[119876(119891) 119892]

119869 (13)

The vector field 119876 is homological and Jacobi This facttogether with the Jacobi identity for the odd Jacobi bracketimplies that the Loday-Poisson is even and satisfies theJacobi-Loday identity For completeness we outline the stepshere and urge the reader to consult [2]

From the definitions and the Jacobi identity for the oddJacobi bracket we have

119891 119892 ℎ119869119869= (minus1)

119891+119892[119876 (119891) [119876 (119892) ℎ]

119869]119869

= (minus1)119891+119892

([[[[119876(119891) 119876(119892)]]119869 ℎ]]

119869

+ (minus1)119891119892[[119876(119892) [[119876(119891) ℎ]]

119869]]119869)

(14)

Using the fact that the homological vector field is a Jacobivector field we have

119891 119892 ℎ119869119869

= (minus1)119892+119891+1

[(minus1)119891+1

119876([119876 (119891) 119892]119869) ℎ]

119869

+ (minus1)119891119892+119891+119892

[119876 (119892) [119876 (119891) ℎ]119869]119869

minus (minus1)119892[[119876

2(119891) 119892]

119869 ℎ]

119869

(15)

As 1198762 = 0 the above gives

119891 119892 ℎ119869119869= 119891 119892

119869 ℎ

119869+ (minus1)

119891119892119892 119891 ℎ

119869119869 (16)

Note however that the bracket is not automatically skewsymmetric The Leibniz rule follows from the definitionsdirectly

119891 119892ℎ119869= (minus1)

119891+1[119876(119891) 119892ℎ]

119869

= (minus1)119891+1

[119876(119891) 119892]119869ℎ

+ (minus1)119891+1+119891119892

119892[119876(119891) ℎ]119869

minus [119876 (119891) 1]119869119892ℎ

= 119891 119892119869ℎ + (minus1)

119891119892119892119891 ℎ

119869

minus [119876(119891) 1]119869119892ℎ

(17)

which follows from the modified Leibniz rule for the oddJacobi bracket and the definition of the Loday-Poissonbracket Then as [119891 1]

119869= (minus1)

119891119876(119891) we have [119876(119891) 1]

119869=

plusmn119876(119876(119891)) = 0 as 119876 is homological This could also be shownvia direct application of the Jacobi identity for the odd Jacobibracket Then

119891 119892ℎ119869= 119891 119892

119869ℎ + (minus1)

119891119892119892119891 ℎ

119869 (18)

Remark 15 Note that the centre of the odd Jacobi algebraconsists entirely of 119876-closed functions This is evident aswe have [[119891 1]]

119869= (minus1)

119891119876(119891) Thus the restriction of the

Loday-Poisson bracket to the centre of (119862infin(119872) [[sdot sdot]]119869) is

identically zero or in other words the trivial Poisson bracket

In local coordinates the Loday-Poisson bracket is givenby

119891 119892119869= (minus1)

119861(119891+1)+1((minus1)

119860119878119861119860119876119862 1205972119891

120597119909119862120597119909119860

+ 119878119861119860 120597119876

119862

120597119909119860120597119891

120597119909119862

minus 119876119861119876119860 120597119891

120597119909119860)

120597119892

120597119909119861

(19)

Directly from this local expression we see that the bracketsatisfies the right Leibniz rule and is not skew symmetric ingeneral Moreover the Loday-Poisson bracket is a second-order differential operator in the first argument From thedefinitions it is clear that

119891 119892119869minus (minus1)

119891119892119892 119891

119869= (minus1)

119891+1119876([119891 119892]

119869) (20)

4 Some Examples

In this section we briefly present four examples of odd Jacobimanifolds and the Loday-Poisson brackets associated withthem These examples are taken straight from [1]

Schouten Manifolds can be considered as odd Jacobimanifolds with the homological vector field being the zerovector In this case the corresponding Loday-Poisson bracketis also trivial

119891 119892119869= 0 (21)

for all 119891 119892 isin 119862infin(119872) Examples of Schouten manifoldsinclude odd symplectic manifolds which have found appli-cations in physics via the Batalin-Vilkovisky formalism

Q-manifolds are understood as odd Jacobi manifoldswith the almost Schouten structure being zero Q-manifoldshave found important applications in the Batalin-Vilkoviskyformalism along-side Schouten structures [17 18] The oddJacobi bracket on a Q-manifold is then given by

[119891 119892]119876= (minus1)

119891119876 (119891119892) = (minus1)

119891119876119860 120597119891

120597119909119860119892 + 119891119876

119860 120597119892

120597119909119860

(22)

The associated Loday-Poisson bracket is thus

119891 119892119876= (minus1)

119891+1119876 (119891)119876 (119892) = (minus1)

119861(119891+1)119876119861119876119860 120597119891

120597119909119860120597119892

120597119909119861

(23)

Journal of Mathematics 5

which is in fact skew symmetric and thus a genuine Poissonbracket One can see this directly or from the fact that119876([[119891 119892]]

119876) = 0

Remark 16 There is nothing really new here In essenceall we have is a Poisson structure (bivector) given by 119875 =

plusmn(12)(120589119876)2 where 120589 Vect(119872) rarr 119862infin(Π119879lowast119872) is the odd

isomorphism between vector fields and ldquoone-vectorsrdquo Notethat 120589119876 is now even and the homological property becomes[[120589119876 120589119876]] = 0 where the bracket here is the canonicalSchouten-Nijenhuis bracket Thus due to the Leibniz rule itis clear that [[119875 119875]] = 0 and we have a Poisson structure Onecould also build higher Poisson structures from Q-manifoldsin this way

Recall that a Lie Algebroid 119864 rarr 119872 can be describedas a weight one homological vector field on the total spaceof Π119864 This understanding in terms of graded Q-manifoldsis attributed to Vaıntrob [19] The weight is assigned as zeroto the base coordinates and one to the fibre coordinatesIn natural local coordinates (119909119860 120585120572) the homological vectorfield is of the form

119876 = 120585120572119876119860

120572(119909)

120597

120597119909119860+1

2120585120572120585120573119876120574

120573120572(119909)

120597

120597120585120574isin Vect (Π119864)

(24)

The associated weight one odd Jacobi bracket is given by

[120601 120595]119864= (minus1)

120601(120585

120572119876119860

120572(119909)

120597120601

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120601

120597120585120574)120595

+ 120601(120585120572119876119860

120572(119909)

120597120595

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120595

120597120585120574)

(25)

where 120601 120595 isin 119862infin(Π119864) are ldquoLie algebroid differential formsrdquoThe weight of two-Poisson bracket is given in local coordi-nates by

120601 120595119864= (minus1)

119861120601+(119861+1)120574120585120574120585120572119876119861

120572119876119860

120574

120597120601

120597119909119860120597120595

120597119909119861

+1

2120585120574120585120575120585120572((minus1)

119861(120601+1)+(120574+120575)(119861+1)

times 119876119861

120572119876120598

120575120574

120597120601

120597120585120598120597120595

120597119909119861

+ (minus1)(120598+1)(120601+1)+120598(120574+1)

times 119876120598

120572120575119876119861

120574

120597120601

120597119909119861120597120595

120597120585120598)

+ (minus1)(120573+1)(120601+1)+120573(120598+120588)

times1

4120585120598120585120588120585120574120585120575119876120573

120575120574119876120572

120588120598

120597120601

120597120585120572120597120595

120597120585120573

(26)

Remark 17 The constructions here directly generalize to 119871infin-

algebroids understood as the pair (Π119864119876) where 119876 isin

Vect(Π119864) is a homological vector field now inhomogeneousin weight

Consider the manifold 119872 = Π119879lowast119873 times R0|1 where 119873 is

a pure even classical manifold Let us equip this manifoldwith natural local coordinates (119909119886 119909lowast

119886 120591) where (119909lowast

119886) are fibre

coordinates on Π119879lowast119873 which are Grassmann odd and 120591 isthe coordinate on the factor R0|1 The manifold 119872 is an oddcontact manifold That is it comes with an odd contact formgiven by

120572 = 119889120591 minus 119909lowast

119886119889119909

119886 (27)

Associated with this odd contact form is an odd Jacobistructure given by

119878 = 119901119886

lowast(119901119886+ 119909

lowast

119886120587) isin 119862

infin(119879lowast119872)

119876 = minus120597

120597120591isin Vect (119872)

(28)

where we have employed fibre coordinates (119901119886 119901119886lowast 120587) on

119879lowast119872 The corresponding odd Jacobi bracket is given by

[119891 119892]119869= (minus1)

119891+1 120597119891

120597119909lowast119886

120597119892

120597119909119886minus

120597119891

120597119909119886120597119892

120597119909lowast119886

+ 119909lowast

119886

120597119891

120597119909lowast119886

120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119909lowast

119886

120597119892

120597119909lowast119886

+ 119891120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119892

(29)

Then the Loday-Poisson bracket is given by

119891 119892119869=

1205972119891

120597119909lowast119886120597120591

120597119892

120597119909119886minus (minus1)

119891 1205972119891

120597119909119886120597120591

120597119892

120597119909lowast119886

+ (minus1)119891119909lowast

119886

1205972119891

120597119909lowast119886120597120591

120597119892

120597120591+ (minus1)

119891 120597119891

120597120591

120597119892

120597120591

(30)

5 Hamiltonian Vector Fields

We now continue this paper with a study of the algebraicrelations between the Hamiltonian vector fields with respectto the odd Jacobi bracket and the (even) Loday-Poissonbracket

Definition 18 Let 119891 isin 119862infin(119872) be an arbitrary function The

associatedHamiltonian vector fieldwith respect to the Loday-Poisson bracket 119884

119891isin Vect(119872) is defined namely

119884119891(119892) = 119891 119892

119869 (31)

Throughout this section we will denote Hamiltonianvector fields with respect to the odd Jacobi structure as 119883

119891

and those with respect to the Loday-Poisson structure as 119884119891

in order to distinguish the two Note that119883119891= 119891+1 and that

119884119891= 119891

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Mathematics

This paper is arranged as follows In Section 2 werecall the definition of an odd Jacobi manifold and presentsome results needed throughout the rest of this paper InSection 3 we construct the Loday-Poisson bracket on an oddJacobi manifold and in Section 4 we present some standardexamples The Hamiltonian vector fields with respect to theodd Jacobi brackets and the Loday-Poisson brackets are thesubject of Section 5 The derived product generated by ahomological vector field is defined and studied in Section 6Here it is shown that the Loday-Poisson bracket satisfies anappropriate Leibniz rule over the noncommutative derivedproduct In Section 7 the previous constructions are appliedto the specific case of Jacobi algebroids We conclude thispaper with some final remarks in Section 8 The remainderof the introduction is devoted to some preliminary notionsand nomenclature

11 Preliminaries All vector spaces and algebras will beZ2-graded We will generally omit the prefix super By

manifold we will mean a smooth supermanifold We denotethe Grassmann parity of an object by tilde 119860 isin Z

2 By even

or oddwewill be referring explicitly to theGrassmann parityA Poisson (120576 = 0) or Schouten (120576 = 1) algebra is

understood as a vector space 119860 with a bilinear associativemultiplication and a bilinear operation sdot sdot

120576 119860 otimes 119860 rarr 119860

such that

Grading 119886 119887120576= 119886 + + 120576

Skew symmetry 119886 119887120576= minus(minus1)

(119886+120576)(+120576)119887 119886

120576

Jacobi identity sumcyclic(119886119887119888) (minus1)(119886+120576)(119888+120576)

119886 119887 119888120576120576

=

0

Leibniz rule 119886 119887119888120576= 119886 119887

120576119888 + (minus1)

(119886+120576)119887119886 119888

120576

For all homogenous elements 119886 119887 and 119888 isin 119860 Extension toinhomogeneous elements is via linearity If the Leibniz ruledoes not hold identically but is modified as

119886 119887119888120576= 119886 119887

120576119888 + (minus1)

(119886+120576)119887119886 119888

120576minus 119886 1

120576119887119888 (2)

then we have even (120598 = 0) or odd (120598 = 1) Jacobi algebrasA manifold 119872 such that 119862infin(119872) is a PoissonSchouten

algebra is known as a PoissonSchouten manifold In partic-ular the cotangent of a manifold comes equipped with acanonical Poisson structure

Let us employ natural local coordinates (119909119860 119901119860) on119879lowast119872

with 119909119860 = 119860 and 119901119860

= 119860 Local diffeomorphisms on 119872

induce vector bundle automorphism on 119879lowast119872 of the form

1199091198601015840

= 1199091198601015840

(119909) 1199011198601015840 = (

120597119909119861

1205971199091198601015840)119901

119861 (3)

We will in effect use the local description as a natural vec-tor bundle to define the cotangent bundle of a supermanifoldThe canonical Poisson bracket on the cotangent is given by

119865 119866 = (minus1)119860119865+119860 120597119865

120597119901119860

120597119866

120597119909119860minus (minus1)

119860119865 120597119865

120597119909119860120597119866

120597119901119860

(4)

A manifold equipped with an odd vector field 119876 suchthat the nontrivial condition 119876

2 = (12)[119876 119876] = 0 holdsis known as a Q-manifold The vector field 119876 is known as ahomological vector field for obvious reasons

The notion of a Loday-Poisson bracket is important inthis paper We define such a bracket as ldquoa Poisson bracketbut without the skewsymmetryrdquo Note that the Jacobi identityas presented above for an even bracket (with or without aLeibniz rule) implies that the adjoint endomorphism 119891 999492999484

ad119891= 119891 sdot is a derivation over the bracket itself That is

ad119891119892 ℎ = ad

119891(119892) ℎ + (minus1)

119891119892119892 ad

119891 (ℎ) (5)

Note that the above has direct meaning even if the skewsymmetry of the bracket is weakened Thus the most usefuldefinition of the Jacobi identity for an even bracket when thesymmetry is lost is

119891 119892 ℎ = 119891 119892 ℎ + (minus1)119891119892

119892 119891 ℎ (6)

We refer to this form of the Jacobi identity as the (right)Jacobi-Loday identity [4] If we have an even bracket thatsatisfies the Jacobi-Loday identity and the right Leibniz rulebut not necessarily the skew symmetry then we call such abracket as a right Loday-Poisson bracket Left Loday-Poissonbrackets can be similarly defined as can Loday-Poissonbrackets that satisfy the Leibniz rule in both directions Forexample classical Poisson brackets are specific examples ofLoday-Poisson bracket albeit skew symmetricWewill exam-ine only right Loday-Poisson brackets but we will typicallydrop explicit reference to the word right To the authorrsquosknowledge (noncommutative) Loday-Poisson algebra wasfirst discussed by Casas and Datuashvili [8]

Remark 2 Much of this paper will generalize directly to QS-manifolds in the sense of Voronov [9] A primary example ofa QS-manifold is the antitangent bundle Π119879119872 of a Poissonmanifold 119872 The Schouten structure is supplied by theKoszul-Schouten bracket and the homological vector field bythe de Rham differential [10] The associated Loday-Poissonbracket is a natural extension of the Poisson bracket on119862infin(119872) to differential forms over119872 but note this extension

is as a Loday bracket only [2] The constructions here willnot directly generalize to even Jacobi supermanifolds dueto incompatibility of the Grassmann parities There is nocanonical choice of a homological Jacobi vector field on aneven Jacobi supermanifold

2 Odd Jacobi Manifolds

Lichnerowicz [11 12] introduced the notion of a Poissonman-ifold as well as a Jacobi manifold Such manifolds have foundapplications in classical mechanics and play an importantrole in quantisation In this section we recall some of thebasic notions as pertaining to odd Jacobi (super) manifoldsNo proofs are given here and can be found in [1] or followdirectly from the definitions For a modern discussion ofJacobi structures including Z-graded versions see [13ndash15]

Journal of Mathematics 3

Definition 3 An odd Jacobi structure (119878 119876) on a manifold119872

consists of

(i) an odd function 119878 isin 119862infin(119879lowast119872) of degree two in fibrecoordinates

(ii) an odd vector field 119876 isin Vect(119872)

such that the following conditions hold

(1) the homological condition 1198762 = (12)[119876 119876] = 0(2) the invariance condition 119871

119876119878 = 0

(3) the compatibility condition 119878 119878 = minus2Q119878

Here Q isin 119862infin(119879lowast119872) is the principal symbol or ldquoHamil-tonianrdquo of the vector field119876The brackets sdot sdot are the canoni-cal Poisson brackets on the cotangent bundle of themanifold

A manifold equipped with an odd Jacobi structure (119878 119876)

is known as an odd Jacobi manifold

Remark 4 At first glance the above definition seems dif-ferent from the classical notion of a Jacobi manifold Thecompatibility condition involves the principal symbol of thehomological vector field rather than the vector field itselfHowever all we are really doing is thinking of the vector fieldas a linear function on the cotangent bundle

Definition 5 The odd Jacobi bracket on 119862infin(119872) is defined as

[119891 119892]119869= (minus1)

119891+1119878 119891 119892 minus (minus1)

119891+1Q 119891119892

= (minus1)(119861+1)119891+1

119878119861119860 120597119891

120597119909119860120597119892

120597119909119861

+ (minus1)119891(119876

119860 120597119891

120597119909119860)119892 + 119891(119876

119860 120597119892

120597119909119860)

(7)

with 119891 119892 isin 119862infin(119872)

The odd Jacobi bracket makes the algebra of smoothfunctions on119872 into an odd Jacobi algebra

Remark 6 The definition of an odd Jacobi manifold givenhere is not quite the most general and one can include anodd function in the construction of an odd Jacobi structuresee Grabowski and Marmo [14] for details Note that thedefinition given by Grabowski andMarmo coincide with thatgiven here (up to conventions) up on setting the odd functionto zero Thus there are examples of odd Jacobi brackets onsupermanifolds not covered by the constructions here

Definition 7 Given a function 119891 isin 119862infin(119872) the associated

Hamiltonian vector field is given by

119891 999492999484 119883119891isin Vect (119872)

119883119891(119892) = (minus1)

119891[[119891 119892]]

119869minus 119876 (119891) 119892

(8)

Note that the homological vector field 119876 is itself Hamil-tonian with respect to the unit constant function 119876 = 1198831 =

[[1 sdot]]119869

Definition 8 Avector field119883 isin Vect(119872) is said to be a Jacobivector field if and only if

119871119883119878 = 120594 119878 = 0 119871

119883119876 = 120594Q = 0 (9)

where 120594 isin 119862infin(119879lowast119872) is the symbol or ldquoHamiltonianrdquo of thevector field 119883 Note that the homological vector field 119876 is aJacobi vector field

Proposition 9 Let 119885 isin Vect(119872) be a vector field on an oddJacobi manifold Then the following are equivalent

(1) 119885 is a Jacobi vector field(2) 119885 is a derivation over the odd Jacobi bracket

119885([119891 119892]119869) = [119885 (119891) 119892]

119869+ (minus1)

119885(119891+1)[119891 119885 (119892)]

119869(10)

(3) [119885119883119891] = (minus1)

119885119885119883(119891)

for all Hamiltonian vectorfields119883

119891

Proposition 10 AHamiltonian vector field119883119891isin Vect(119872) is

a Jacobi vector field if and only if 119891 isin 119862infin(119872) is Q-closed

Proposition 11 The assignment 119891 999492999484 119883119891is a morphism

between the odd Lie algebra on 119862infin(119872) provided by the oddJacobi brackets and the Lie algebra of vector fields Specificallythe following holds

[119883119891 119883

119892] = minus119883

[119891119892]119869

(11)

for all 119891 119892 isin 119862infin(119872)

Proposition 12 On an odd Jacobi manifold the followingidentity holds

119883119891119892

= (minus1)119891119891119883

119892+ (minus1)

119892(119891+1)119892119883

119891+ (minus1)

119891+119892+1119891119892119876 (12)

Remark 13 An even or odd Lie bracket defined on sectionsof an even line bundle over a manifold which is a first-orderdifferential operator with respect to each argument is knownas a Kirillov bracket [16] Jacobi brackets are Kirillov bracketson trivial line bundles In this respect Jacobi brackets serveas a local description of Kirillov brackets but should be seenas a secondary notion We will not pursue the more generalsituation of Kirillov brackets here and focus only on Jacobistructures

3 Loday-Poisson Brackets

We are now in a position to state the main theorem of thispaper We draw heavily on the notion of derived brackets inthe sense of Kosmann-Schwarzbach see [2 3] for details

Theorem 14 Let (119872 119878 119876) be an odd Jacobi manifold then119862infin(119872) comes equipped with a canonical Loday-Poisson

bracket

Proof We prove the theorem by direct construction of theLoday-Poisson bracket The bracket is canonical up to minor

4 Journal of Mathematics

issues of conventions Following Kosmann-Schwarzbach [2]we define the Loday-Poisson bracket as

119891 119892119869= (minus1)

119891+1[119876(119891) 119892]

119869 (13)

The vector field 119876 is homological and Jacobi This facttogether with the Jacobi identity for the odd Jacobi bracketimplies that the Loday-Poisson is even and satisfies theJacobi-Loday identity For completeness we outline the stepshere and urge the reader to consult [2]

From the definitions and the Jacobi identity for the oddJacobi bracket we have

119891 119892 ℎ119869119869= (minus1)

119891+119892[119876 (119891) [119876 (119892) ℎ]

119869]119869

= (minus1)119891+119892

([[[[119876(119891) 119876(119892)]]119869 ℎ]]

119869

+ (minus1)119891119892[[119876(119892) [[119876(119891) ℎ]]

119869]]119869)

(14)

Using the fact that the homological vector field is a Jacobivector field we have

119891 119892 ℎ119869119869

= (minus1)119892+119891+1

[(minus1)119891+1

119876([119876 (119891) 119892]119869) ℎ]

119869

+ (minus1)119891119892+119891+119892

[119876 (119892) [119876 (119891) ℎ]119869]119869

minus (minus1)119892[[119876

2(119891) 119892]

119869 ℎ]

119869

(15)

As 1198762 = 0 the above gives

119891 119892 ℎ119869119869= 119891 119892

119869 ℎ

119869+ (minus1)

119891119892119892 119891 ℎ

119869119869 (16)

Note however that the bracket is not automatically skewsymmetric The Leibniz rule follows from the definitionsdirectly

119891 119892ℎ119869= (minus1)

119891+1[119876(119891) 119892ℎ]

119869

= (minus1)119891+1

[119876(119891) 119892]119869ℎ

+ (minus1)119891+1+119891119892

119892[119876(119891) ℎ]119869

minus [119876 (119891) 1]119869119892ℎ

= 119891 119892119869ℎ + (minus1)

119891119892119892119891 ℎ

119869

minus [119876(119891) 1]119869119892ℎ

(17)

which follows from the modified Leibniz rule for the oddJacobi bracket and the definition of the Loday-Poissonbracket Then as [119891 1]

119869= (minus1)

119891119876(119891) we have [119876(119891) 1]

119869=

plusmn119876(119876(119891)) = 0 as 119876 is homological This could also be shownvia direct application of the Jacobi identity for the odd Jacobibracket Then

119891 119892ℎ119869= 119891 119892

119869ℎ + (minus1)

119891119892119892119891 ℎ

119869 (18)

Remark 15 Note that the centre of the odd Jacobi algebraconsists entirely of 119876-closed functions This is evident aswe have [[119891 1]]

119869= (minus1)

119891119876(119891) Thus the restriction of the

Loday-Poisson bracket to the centre of (119862infin(119872) [[sdot sdot]]119869) is

identically zero or in other words the trivial Poisson bracket

In local coordinates the Loday-Poisson bracket is givenby

119891 119892119869= (minus1)

119861(119891+1)+1((minus1)

119860119878119861119860119876119862 1205972119891

120597119909119862120597119909119860

+ 119878119861119860 120597119876

119862

120597119909119860120597119891

120597119909119862

minus 119876119861119876119860 120597119891

120597119909119860)

120597119892

120597119909119861

(19)

Directly from this local expression we see that the bracketsatisfies the right Leibniz rule and is not skew symmetric ingeneral Moreover the Loday-Poisson bracket is a second-order differential operator in the first argument From thedefinitions it is clear that

119891 119892119869minus (minus1)

119891119892119892 119891

119869= (minus1)

119891+1119876([119891 119892]

119869) (20)

4 Some Examples

In this section we briefly present four examples of odd Jacobimanifolds and the Loday-Poisson brackets associated withthem These examples are taken straight from [1]

Schouten Manifolds can be considered as odd Jacobimanifolds with the homological vector field being the zerovector In this case the corresponding Loday-Poisson bracketis also trivial

119891 119892119869= 0 (21)

for all 119891 119892 isin 119862infin(119872) Examples of Schouten manifoldsinclude odd symplectic manifolds which have found appli-cations in physics via the Batalin-Vilkovisky formalism

Q-manifolds are understood as odd Jacobi manifoldswith the almost Schouten structure being zero Q-manifoldshave found important applications in the Batalin-Vilkoviskyformalism along-side Schouten structures [17 18] The oddJacobi bracket on a Q-manifold is then given by

[119891 119892]119876= (minus1)

119891119876 (119891119892) = (minus1)

119891119876119860 120597119891

120597119909119860119892 + 119891119876

119860 120597119892

120597119909119860

(22)

The associated Loday-Poisson bracket is thus

119891 119892119876= (minus1)

119891+1119876 (119891)119876 (119892) = (minus1)

119861(119891+1)119876119861119876119860 120597119891

120597119909119860120597119892

120597119909119861

(23)

Journal of Mathematics 5

which is in fact skew symmetric and thus a genuine Poissonbracket One can see this directly or from the fact that119876([[119891 119892]]

119876) = 0

Remark 16 There is nothing really new here In essenceall we have is a Poisson structure (bivector) given by 119875 =

plusmn(12)(120589119876)2 where 120589 Vect(119872) rarr 119862infin(Π119879lowast119872) is the odd

isomorphism between vector fields and ldquoone-vectorsrdquo Notethat 120589119876 is now even and the homological property becomes[[120589119876 120589119876]] = 0 where the bracket here is the canonicalSchouten-Nijenhuis bracket Thus due to the Leibniz rule itis clear that [[119875 119875]] = 0 and we have a Poisson structure Onecould also build higher Poisson structures from Q-manifoldsin this way

Recall that a Lie Algebroid 119864 rarr 119872 can be describedas a weight one homological vector field on the total spaceof Π119864 This understanding in terms of graded Q-manifoldsis attributed to Vaıntrob [19] The weight is assigned as zeroto the base coordinates and one to the fibre coordinatesIn natural local coordinates (119909119860 120585120572) the homological vectorfield is of the form

119876 = 120585120572119876119860

120572(119909)

120597

120597119909119860+1

2120585120572120585120573119876120574

120573120572(119909)

120597

120597120585120574isin Vect (Π119864)

(24)

The associated weight one odd Jacobi bracket is given by

[120601 120595]119864= (minus1)

120601(120585

120572119876119860

120572(119909)

120597120601

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120601

120597120585120574)120595

+ 120601(120585120572119876119860

120572(119909)

120597120595

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120595

120597120585120574)

(25)

where 120601 120595 isin 119862infin(Π119864) are ldquoLie algebroid differential formsrdquoThe weight of two-Poisson bracket is given in local coordi-nates by

120601 120595119864= (minus1)

119861120601+(119861+1)120574120585120574120585120572119876119861

120572119876119860

120574

120597120601

120597119909119860120597120595

120597119909119861

+1

2120585120574120585120575120585120572((minus1)

119861(120601+1)+(120574+120575)(119861+1)

times 119876119861

120572119876120598

120575120574

120597120601

120597120585120598120597120595

120597119909119861

+ (minus1)(120598+1)(120601+1)+120598(120574+1)

times 119876120598

120572120575119876119861

120574

120597120601

120597119909119861120597120595

120597120585120598)

+ (minus1)(120573+1)(120601+1)+120573(120598+120588)

times1

4120585120598120585120588120585120574120585120575119876120573

120575120574119876120572

120588120598

120597120601

120597120585120572120597120595

120597120585120573

(26)

Remark 17 The constructions here directly generalize to 119871infin-

algebroids understood as the pair (Π119864119876) where 119876 isin

Vect(Π119864) is a homological vector field now inhomogeneousin weight

Consider the manifold 119872 = Π119879lowast119873 times R0|1 where 119873 is

a pure even classical manifold Let us equip this manifoldwith natural local coordinates (119909119886 119909lowast

119886 120591) where (119909lowast

119886) are fibre

coordinates on Π119879lowast119873 which are Grassmann odd and 120591 isthe coordinate on the factor R0|1 The manifold 119872 is an oddcontact manifold That is it comes with an odd contact formgiven by

120572 = 119889120591 minus 119909lowast

119886119889119909

119886 (27)

Associated with this odd contact form is an odd Jacobistructure given by

119878 = 119901119886

lowast(119901119886+ 119909

lowast

119886120587) isin 119862

infin(119879lowast119872)

119876 = minus120597

120597120591isin Vect (119872)

(28)

where we have employed fibre coordinates (119901119886 119901119886lowast 120587) on

119879lowast119872 The corresponding odd Jacobi bracket is given by

[119891 119892]119869= (minus1)

119891+1 120597119891

120597119909lowast119886

120597119892

120597119909119886minus

120597119891

120597119909119886120597119892

120597119909lowast119886

+ 119909lowast

119886

120597119891

120597119909lowast119886

120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119909lowast

119886

120597119892

120597119909lowast119886

+ 119891120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119892

(29)

Then the Loday-Poisson bracket is given by

119891 119892119869=

1205972119891

120597119909lowast119886120597120591

120597119892

120597119909119886minus (minus1)

119891 1205972119891

120597119909119886120597120591

120597119892

120597119909lowast119886

+ (minus1)119891119909lowast

119886

1205972119891

120597119909lowast119886120597120591

120597119892

120597120591+ (minus1)

119891 120597119891

120597120591

120597119892

120597120591

(30)

5 Hamiltonian Vector Fields

We now continue this paper with a study of the algebraicrelations between the Hamiltonian vector fields with respectto the odd Jacobi bracket and the (even) Loday-Poissonbracket

Definition 18 Let 119891 isin 119862infin(119872) be an arbitrary function The

associatedHamiltonian vector fieldwith respect to the Loday-Poisson bracket 119884

119891isin Vect(119872) is defined namely

119884119891(119892) = 119891 119892

119869 (31)

Throughout this section we will denote Hamiltonianvector fields with respect to the odd Jacobi structure as 119883

119891

and those with respect to the Loday-Poisson structure as 119884119891

in order to distinguish the two Note that119883119891= 119891+1 and that

119884119891= 119891

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

Definition 3 An odd Jacobi structure (119878 119876) on a manifold119872

consists of

(i) an odd function 119878 isin 119862infin(119879lowast119872) of degree two in fibrecoordinates

(ii) an odd vector field 119876 isin Vect(119872)

such that the following conditions hold

(1) the homological condition 1198762 = (12)[119876 119876] = 0(2) the invariance condition 119871

119876119878 = 0

(3) the compatibility condition 119878 119878 = minus2Q119878

Here Q isin 119862infin(119879lowast119872) is the principal symbol or ldquoHamil-tonianrdquo of the vector field119876The brackets sdot sdot are the canoni-cal Poisson brackets on the cotangent bundle of themanifold

A manifold equipped with an odd Jacobi structure (119878 119876)

is known as an odd Jacobi manifold

Remark 4 At first glance the above definition seems dif-ferent from the classical notion of a Jacobi manifold Thecompatibility condition involves the principal symbol of thehomological vector field rather than the vector field itselfHowever all we are really doing is thinking of the vector fieldas a linear function on the cotangent bundle

Definition 5 The odd Jacobi bracket on 119862infin(119872) is defined as

[119891 119892]119869= (minus1)

119891+1119878 119891 119892 minus (minus1)

119891+1Q 119891119892

= (minus1)(119861+1)119891+1

119878119861119860 120597119891

120597119909119860120597119892

120597119909119861

+ (minus1)119891(119876

119860 120597119891

120597119909119860)119892 + 119891(119876

119860 120597119892

120597119909119860)

(7)

with 119891 119892 isin 119862infin(119872)

The odd Jacobi bracket makes the algebra of smoothfunctions on119872 into an odd Jacobi algebra

Remark 6 The definition of an odd Jacobi manifold givenhere is not quite the most general and one can include anodd function in the construction of an odd Jacobi structuresee Grabowski and Marmo [14] for details Note that thedefinition given by Grabowski andMarmo coincide with thatgiven here (up to conventions) up on setting the odd functionto zero Thus there are examples of odd Jacobi brackets onsupermanifolds not covered by the constructions here

Definition 7 Given a function 119891 isin 119862infin(119872) the associated

Hamiltonian vector field is given by

119891 999492999484 119883119891isin Vect (119872)

119883119891(119892) = (minus1)

119891[[119891 119892]]

119869minus 119876 (119891) 119892

(8)

Note that the homological vector field 119876 is itself Hamil-tonian with respect to the unit constant function 119876 = 1198831 =

[[1 sdot]]119869

Definition 8 Avector field119883 isin Vect(119872) is said to be a Jacobivector field if and only if

119871119883119878 = 120594 119878 = 0 119871

119883119876 = 120594Q = 0 (9)

where 120594 isin 119862infin(119879lowast119872) is the symbol or ldquoHamiltonianrdquo of thevector field 119883 Note that the homological vector field 119876 is aJacobi vector field

Proposition 9 Let 119885 isin Vect(119872) be a vector field on an oddJacobi manifold Then the following are equivalent

(1) 119885 is a Jacobi vector field(2) 119885 is a derivation over the odd Jacobi bracket

119885([119891 119892]119869) = [119885 (119891) 119892]

119869+ (minus1)

119885(119891+1)[119891 119885 (119892)]

119869(10)

(3) [119885119883119891] = (minus1)

119885119885119883(119891)

for all Hamiltonian vectorfields119883

119891

Proposition 10 AHamiltonian vector field119883119891isin Vect(119872) is

a Jacobi vector field if and only if 119891 isin 119862infin(119872) is Q-closed

Proposition 11 The assignment 119891 999492999484 119883119891is a morphism

between the odd Lie algebra on 119862infin(119872) provided by the oddJacobi brackets and the Lie algebra of vector fields Specificallythe following holds

[119883119891 119883

119892] = minus119883

[119891119892]119869

(11)

for all 119891 119892 isin 119862infin(119872)

Proposition 12 On an odd Jacobi manifold the followingidentity holds

119883119891119892

= (minus1)119891119891119883

119892+ (minus1)

119892(119891+1)119892119883

119891+ (minus1)

119891+119892+1119891119892119876 (12)

Remark 13 An even or odd Lie bracket defined on sectionsof an even line bundle over a manifold which is a first-orderdifferential operator with respect to each argument is knownas a Kirillov bracket [16] Jacobi brackets are Kirillov bracketson trivial line bundles In this respect Jacobi brackets serveas a local description of Kirillov brackets but should be seenas a secondary notion We will not pursue the more generalsituation of Kirillov brackets here and focus only on Jacobistructures

3 Loday-Poisson Brackets

We are now in a position to state the main theorem of thispaper We draw heavily on the notion of derived brackets inthe sense of Kosmann-Schwarzbach see [2 3] for details

Theorem 14 Let (119872 119878 119876) be an odd Jacobi manifold then119862infin(119872) comes equipped with a canonical Loday-Poisson

bracket

Proof We prove the theorem by direct construction of theLoday-Poisson bracket The bracket is canonical up to minor

4 Journal of Mathematics

issues of conventions Following Kosmann-Schwarzbach [2]we define the Loday-Poisson bracket as

119891 119892119869= (minus1)

119891+1[119876(119891) 119892]

119869 (13)

The vector field 119876 is homological and Jacobi This facttogether with the Jacobi identity for the odd Jacobi bracketimplies that the Loday-Poisson is even and satisfies theJacobi-Loday identity For completeness we outline the stepshere and urge the reader to consult [2]

From the definitions and the Jacobi identity for the oddJacobi bracket we have

119891 119892 ℎ119869119869= (minus1)

119891+119892[119876 (119891) [119876 (119892) ℎ]

119869]119869

= (minus1)119891+119892

([[[[119876(119891) 119876(119892)]]119869 ℎ]]

119869

+ (minus1)119891119892[[119876(119892) [[119876(119891) ℎ]]

119869]]119869)

(14)

Using the fact that the homological vector field is a Jacobivector field we have

119891 119892 ℎ119869119869

= (minus1)119892+119891+1

[(minus1)119891+1

119876([119876 (119891) 119892]119869) ℎ]

119869

+ (minus1)119891119892+119891+119892

[119876 (119892) [119876 (119891) ℎ]119869]119869

minus (minus1)119892[[119876

2(119891) 119892]

119869 ℎ]

119869

(15)

As 1198762 = 0 the above gives

119891 119892 ℎ119869119869= 119891 119892

119869 ℎ

119869+ (minus1)

119891119892119892 119891 ℎ

119869119869 (16)

Note however that the bracket is not automatically skewsymmetric The Leibniz rule follows from the definitionsdirectly

119891 119892ℎ119869= (minus1)

119891+1[119876(119891) 119892ℎ]

119869

= (minus1)119891+1

[119876(119891) 119892]119869ℎ

+ (minus1)119891+1+119891119892

119892[119876(119891) ℎ]119869

minus [119876 (119891) 1]119869119892ℎ

= 119891 119892119869ℎ + (minus1)

119891119892119892119891 ℎ

119869

minus [119876(119891) 1]119869119892ℎ

(17)

which follows from the modified Leibniz rule for the oddJacobi bracket and the definition of the Loday-Poissonbracket Then as [119891 1]

119869= (minus1)

119891119876(119891) we have [119876(119891) 1]

119869=

plusmn119876(119876(119891)) = 0 as 119876 is homological This could also be shownvia direct application of the Jacobi identity for the odd Jacobibracket Then

119891 119892ℎ119869= 119891 119892

119869ℎ + (minus1)

119891119892119892119891 ℎ

119869 (18)

Remark 15 Note that the centre of the odd Jacobi algebraconsists entirely of 119876-closed functions This is evident aswe have [[119891 1]]

119869= (minus1)

119891119876(119891) Thus the restriction of the

Loday-Poisson bracket to the centre of (119862infin(119872) [[sdot sdot]]119869) is

identically zero or in other words the trivial Poisson bracket

In local coordinates the Loday-Poisson bracket is givenby

119891 119892119869= (minus1)

119861(119891+1)+1((minus1)

119860119878119861119860119876119862 1205972119891

120597119909119862120597119909119860

+ 119878119861119860 120597119876

119862

120597119909119860120597119891

120597119909119862

minus 119876119861119876119860 120597119891

120597119909119860)

120597119892

120597119909119861

(19)

Directly from this local expression we see that the bracketsatisfies the right Leibniz rule and is not skew symmetric ingeneral Moreover the Loday-Poisson bracket is a second-order differential operator in the first argument From thedefinitions it is clear that

119891 119892119869minus (minus1)

119891119892119892 119891

119869= (minus1)

119891+1119876([119891 119892]

119869) (20)

4 Some Examples

In this section we briefly present four examples of odd Jacobimanifolds and the Loday-Poisson brackets associated withthem These examples are taken straight from [1]

Schouten Manifolds can be considered as odd Jacobimanifolds with the homological vector field being the zerovector In this case the corresponding Loday-Poisson bracketis also trivial

119891 119892119869= 0 (21)

for all 119891 119892 isin 119862infin(119872) Examples of Schouten manifoldsinclude odd symplectic manifolds which have found appli-cations in physics via the Batalin-Vilkovisky formalism

Q-manifolds are understood as odd Jacobi manifoldswith the almost Schouten structure being zero Q-manifoldshave found important applications in the Batalin-Vilkoviskyformalism along-side Schouten structures [17 18] The oddJacobi bracket on a Q-manifold is then given by

[119891 119892]119876= (minus1)

119891119876 (119891119892) = (minus1)

119891119876119860 120597119891

120597119909119860119892 + 119891119876

119860 120597119892

120597119909119860

(22)

The associated Loday-Poisson bracket is thus

119891 119892119876= (minus1)

119891+1119876 (119891)119876 (119892) = (minus1)

119861(119891+1)119876119861119876119860 120597119891

120597119909119860120597119892

120597119909119861

(23)

Journal of Mathematics 5

which is in fact skew symmetric and thus a genuine Poissonbracket One can see this directly or from the fact that119876([[119891 119892]]

119876) = 0

Remark 16 There is nothing really new here In essenceall we have is a Poisson structure (bivector) given by 119875 =

plusmn(12)(120589119876)2 where 120589 Vect(119872) rarr 119862infin(Π119879lowast119872) is the odd

isomorphism between vector fields and ldquoone-vectorsrdquo Notethat 120589119876 is now even and the homological property becomes[[120589119876 120589119876]] = 0 where the bracket here is the canonicalSchouten-Nijenhuis bracket Thus due to the Leibniz rule itis clear that [[119875 119875]] = 0 and we have a Poisson structure Onecould also build higher Poisson structures from Q-manifoldsin this way

Recall that a Lie Algebroid 119864 rarr 119872 can be describedas a weight one homological vector field on the total spaceof Π119864 This understanding in terms of graded Q-manifoldsis attributed to Vaıntrob [19] The weight is assigned as zeroto the base coordinates and one to the fibre coordinatesIn natural local coordinates (119909119860 120585120572) the homological vectorfield is of the form

119876 = 120585120572119876119860

120572(119909)

120597

120597119909119860+1

2120585120572120585120573119876120574

120573120572(119909)

120597

120597120585120574isin Vect (Π119864)

(24)

The associated weight one odd Jacobi bracket is given by

[120601 120595]119864= (minus1)

120601(120585

120572119876119860

120572(119909)

120597120601

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120601

120597120585120574)120595

+ 120601(120585120572119876119860

120572(119909)

120597120595

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120595

120597120585120574)

(25)

where 120601 120595 isin 119862infin(Π119864) are ldquoLie algebroid differential formsrdquoThe weight of two-Poisson bracket is given in local coordi-nates by

120601 120595119864= (minus1)

119861120601+(119861+1)120574120585120574120585120572119876119861

120572119876119860

120574

120597120601

120597119909119860120597120595

120597119909119861

+1

2120585120574120585120575120585120572((minus1)

119861(120601+1)+(120574+120575)(119861+1)

times 119876119861

120572119876120598

120575120574

120597120601

120597120585120598120597120595

120597119909119861

+ (minus1)(120598+1)(120601+1)+120598(120574+1)

times 119876120598

120572120575119876119861

120574

120597120601

120597119909119861120597120595

120597120585120598)

+ (minus1)(120573+1)(120601+1)+120573(120598+120588)

times1

4120585120598120585120588120585120574120585120575119876120573

120575120574119876120572

120588120598

120597120601

120597120585120572120597120595

120597120585120573

(26)

Remark 17 The constructions here directly generalize to 119871infin-

algebroids understood as the pair (Π119864119876) where 119876 isin

Vect(Π119864) is a homological vector field now inhomogeneousin weight

Consider the manifold 119872 = Π119879lowast119873 times R0|1 where 119873 is

a pure even classical manifold Let us equip this manifoldwith natural local coordinates (119909119886 119909lowast

119886 120591) where (119909lowast

119886) are fibre

coordinates on Π119879lowast119873 which are Grassmann odd and 120591 isthe coordinate on the factor R0|1 The manifold 119872 is an oddcontact manifold That is it comes with an odd contact formgiven by

120572 = 119889120591 minus 119909lowast

119886119889119909

119886 (27)

Associated with this odd contact form is an odd Jacobistructure given by

119878 = 119901119886

lowast(119901119886+ 119909

lowast

119886120587) isin 119862

infin(119879lowast119872)

119876 = minus120597

120597120591isin Vect (119872)

(28)

where we have employed fibre coordinates (119901119886 119901119886lowast 120587) on

119879lowast119872 The corresponding odd Jacobi bracket is given by

[119891 119892]119869= (minus1)

119891+1 120597119891

120597119909lowast119886

120597119892

120597119909119886minus

120597119891

120597119909119886120597119892

120597119909lowast119886

+ 119909lowast

119886

120597119891

120597119909lowast119886

120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119909lowast

119886

120597119892

120597119909lowast119886

+ 119891120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119892

(29)

Then the Loday-Poisson bracket is given by

119891 119892119869=

1205972119891

120597119909lowast119886120597120591

120597119892

120597119909119886minus (minus1)

119891 1205972119891

120597119909119886120597120591

120597119892

120597119909lowast119886

+ (minus1)119891119909lowast

119886

1205972119891

120597119909lowast119886120597120591

120597119892

120597120591+ (minus1)

119891 120597119891

120597120591

120597119892

120597120591

(30)

5 Hamiltonian Vector Fields

We now continue this paper with a study of the algebraicrelations between the Hamiltonian vector fields with respectto the odd Jacobi bracket and the (even) Loday-Poissonbracket

Definition 18 Let 119891 isin 119862infin(119872) be an arbitrary function The

associatedHamiltonian vector fieldwith respect to the Loday-Poisson bracket 119884

119891isin Vect(119872) is defined namely

119884119891(119892) = 119891 119892

119869 (31)

Throughout this section we will denote Hamiltonianvector fields with respect to the odd Jacobi structure as 119883

119891

and those with respect to the Loday-Poisson structure as 119884119891

in order to distinguish the two Note that119883119891= 119891+1 and that

119884119891= 119891

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

issues of conventions Following Kosmann-Schwarzbach [2]we define the Loday-Poisson bracket as

119891 119892119869= (minus1)

119891+1[119876(119891) 119892]

119869 (13)

The vector field 119876 is homological and Jacobi This facttogether with the Jacobi identity for the odd Jacobi bracketimplies that the Loday-Poisson is even and satisfies theJacobi-Loday identity For completeness we outline the stepshere and urge the reader to consult [2]

From the definitions and the Jacobi identity for the oddJacobi bracket we have

119891 119892 ℎ119869119869= (minus1)

119891+119892[119876 (119891) [119876 (119892) ℎ]

119869]119869

= (minus1)119891+119892

([[[[119876(119891) 119876(119892)]]119869 ℎ]]

119869

+ (minus1)119891119892[[119876(119892) [[119876(119891) ℎ]]

119869]]119869)

(14)

Using the fact that the homological vector field is a Jacobivector field we have

119891 119892 ℎ119869119869

= (minus1)119892+119891+1

[(minus1)119891+1

119876([119876 (119891) 119892]119869) ℎ]

119869

+ (minus1)119891119892+119891+119892

[119876 (119892) [119876 (119891) ℎ]119869]119869

minus (minus1)119892[[119876

2(119891) 119892]

119869 ℎ]

119869

(15)

As 1198762 = 0 the above gives

119891 119892 ℎ119869119869= 119891 119892

119869 ℎ

119869+ (minus1)

119891119892119892 119891 ℎ

119869119869 (16)

Note however that the bracket is not automatically skewsymmetric The Leibniz rule follows from the definitionsdirectly

119891 119892ℎ119869= (minus1)

119891+1[119876(119891) 119892ℎ]

119869

= (minus1)119891+1

[119876(119891) 119892]119869ℎ

+ (minus1)119891+1+119891119892

119892[119876(119891) ℎ]119869

minus [119876 (119891) 1]119869119892ℎ

= 119891 119892119869ℎ + (minus1)

119891119892119892119891 ℎ

119869

minus [119876(119891) 1]119869119892ℎ

(17)

which follows from the modified Leibniz rule for the oddJacobi bracket and the definition of the Loday-Poissonbracket Then as [119891 1]

119869= (minus1)

119891119876(119891) we have [119876(119891) 1]

119869=

plusmn119876(119876(119891)) = 0 as 119876 is homological This could also be shownvia direct application of the Jacobi identity for the odd Jacobibracket Then

119891 119892ℎ119869= 119891 119892

119869ℎ + (minus1)

119891119892119892119891 ℎ

119869 (18)

Remark 15 Note that the centre of the odd Jacobi algebraconsists entirely of 119876-closed functions This is evident aswe have [[119891 1]]

119869= (minus1)

119891119876(119891) Thus the restriction of the

Loday-Poisson bracket to the centre of (119862infin(119872) [[sdot sdot]]119869) is

identically zero or in other words the trivial Poisson bracket

In local coordinates the Loday-Poisson bracket is givenby

119891 119892119869= (minus1)

119861(119891+1)+1((minus1)

119860119878119861119860119876119862 1205972119891

120597119909119862120597119909119860

+ 119878119861119860 120597119876

119862

120597119909119860120597119891

120597119909119862

minus 119876119861119876119860 120597119891

120597119909119860)

120597119892

120597119909119861

(19)

Directly from this local expression we see that the bracketsatisfies the right Leibniz rule and is not skew symmetric ingeneral Moreover the Loday-Poisson bracket is a second-order differential operator in the first argument From thedefinitions it is clear that

119891 119892119869minus (minus1)

119891119892119892 119891

119869= (minus1)

119891+1119876([119891 119892]

119869) (20)

4 Some Examples

In this section we briefly present four examples of odd Jacobimanifolds and the Loday-Poisson brackets associated withthem These examples are taken straight from [1]

Schouten Manifolds can be considered as odd Jacobimanifolds with the homological vector field being the zerovector In this case the corresponding Loday-Poisson bracketis also trivial

119891 119892119869= 0 (21)

for all 119891 119892 isin 119862infin(119872) Examples of Schouten manifoldsinclude odd symplectic manifolds which have found appli-cations in physics via the Batalin-Vilkovisky formalism

Q-manifolds are understood as odd Jacobi manifoldswith the almost Schouten structure being zero Q-manifoldshave found important applications in the Batalin-Vilkoviskyformalism along-side Schouten structures [17 18] The oddJacobi bracket on a Q-manifold is then given by

[119891 119892]119876= (minus1)

119891119876 (119891119892) = (minus1)

119891119876119860 120597119891

120597119909119860119892 + 119891119876

119860 120597119892

120597119909119860

(22)

The associated Loday-Poisson bracket is thus

119891 119892119876= (minus1)

119891+1119876 (119891)119876 (119892) = (minus1)

119861(119891+1)119876119861119876119860 120597119891

120597119909119860120597119892

120597119909119861

(23)

Journal of Mathematics 5

which is in fact skew symmetric and thus a genuine Poissonbracket One can see this directly or from the fact that119876([[119891 119892]]

119876) = 0

Remark 16 There is nothing really new here In essenceall we have is a Poisson structure (bivector) given by 119875 =

plusmn(12)(120589119876)2 where 120589 Vect(119872) rarr 119862infin(Π119879lowast119872) is the odd

isomorphism between vector fields and ldquoone-vectorsrdquo Notethat 120589119876 is now even and the homological property becomes[[120589119876 120589119876]] = 0 where the bracket here is the canonicalSchouten-Nijenhuis bracket Thus due to the Leibniz rule itis clear that [[119875 119875]] = 0 and we have a Poisson structure Onecould also build higher Poisson structures from Q-manifoldsin this way

Recall that a Lie Algebroid 119864 rarr 119872 can be describedas a weight one homological vector field on the total spaceof Π119864 This understanding in terms of graded Q-manifoldsis attributed to Vaıntrob [19] The weight is assigned as zeroto the base coordinates and one to the fibre coordinatesIn natural local coordinates (119909119860 120585120572) the homological vectorfield is of the form

119876 = 120585120572119876119860

120572(119909)

120597

120597119909119860+1

2120585120572120585120573119876120574

120573120572(119909)

120597

120597120585120574isin Vect (Π119864)

(24)

The associated weight one odd Jacobi bracket is given by

[120601 120595]119864= (minus1)

120601(120585

120572119876119860

120572(119909)

120597120601

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120601

120597120585120574)120595

+ 120601(120585120572119876119860

120572(119909)

120597120595

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120595

120597120585120574)

(25)

where 120601 120595 isin 119862infin(Π119864) are ldquoLie algebroid differential formsrdquoThe weight of two-Poisson bracket is given in local coordi-nates by

120601 120595119864= (minus1)

119861120601+(119861+1)120574120585120574120585120572119876119861

120572119876119860

120574

120597120601

120597119909119860120597120595

120597119909119861

+1

2120585120574120585120575120585120572((minus1)

119861(120601+1)+(120574+120575)(119861+1)

times 119876119861

120572119876120598

120575120574

120597120601

120597120585120598120597120595

120597119909119861

+ (minus1)(120598+1)(120601+1)+120598(120574+1)

times 119876120598

120572120575119876119861

120574

120597120601

120597119909119861120597120595

120597120585120598)

+ (minus1)(120573+1)(120601+1)+120573(120598+120588)

times1

4120585120598120585120588120585120574120585120575119876120573

120575120574119876120572

120588120598

120597120601

120597120585120572120597120595

120597120585120573

(26)

Remark 17 The constructions here directly generalize to 119871infin-

algebroids understood as the pair (Π119864119876) where 119876 isin

Vect(Π119864) is a homological vector field now inhomogeneousin weight

Consider the manifold 119872 = Π119879lowast119873 times R0|1 where 119873 is

a pure even classical manifold Let us equip this manifoldwith natural local coordinates (119909119886 119909lowast

119886 120591) where (119909lowast

119886) are fibre

coordinates on Π119879lowast119873 which are Grassmann odd and 120591 isthe coordinate on the factor R0|1 The manifold 119872 is an oddcontact manifold That is it comes with an odd contact formgiven by

120572 = 119889120591 minus 119909lowast

119886119889119909

119886 (27)

Associated with this odd contact form is an odd Jacobistructure given by

119878 = 119901119886

lowast(119901119886+ 119909

lowast

119886120587) isin 119862

infin(119879lowast119872)

119876 = minus120597

120597120591isin Vect (119872)

(28)

where we have employed fibre coordinates (119901119886 119901119886lowast 120587) on

119879lowast119872 The corresponding odd Jacobi bracket is given by

[119891 119892]119869= (minus1)

119891+1 120597119891

120597119909lowast119886

120597119892

120597119909119886minus

120597119891

120597119909119886120597119892

120597119909lowast119886

+ 119909lowast

119886

120597119891

120597119909lowast119886

120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119909lowast

119886

120597119892

120597119909lowast119886

+ 119891120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119892

(29)

Then the Loday-Poisson bracket is given by

119891 119892119869=

1205972119891

120597119909lowast119886120597120591

120597119892

120597119909119886minus (minus1)

119891 1205972119891

120597119909119886120597120591

120597119892

120597119909lowast119886

+ (minus1)119891119909lowast

119886

1205972119891

120597119909lowast119886120597120591

120597119892

120597120591+ (minus1)

119891 120597119891

120597120591

120597119892

120597120591

(30)

5 Hamiltonian Vector Fields

We now continue this paper with a study of the algebraicrelations between the Hamiltonian vector fields with respectto the odd Jacobi bracket and the (even) Loday-Poissonbracket

Definition 18 Let 119891 isin 119862infin(119872) be an arbitrary function The

associatedHamiltonian vector fieldwith respect to the Loday-Poisson bracket 119884

119891isin Vect(119872) is defined namely

119884119891(119892) = 119891 119892

119869 (31)

Throughout this section we will denote Hamiltonianvector fields with respect to the odd Jacobi structure as 119883

119891

and those with respect to the Loday-Poisson structure as 119884119891

in order to distinguish the two Note that119883119891= 119891+1 and that

119884119891= 119891

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

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Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

which is in fact skew symmetric and thus a genuine Poissonbracket One can see this directly or from the fact that119876([[119891 119892]]

119876) = 0

Remark 16 There is nothing really new here In essenceall we have is a Poisson structure (bivector) given by 119875 =

plusmn(12)(120589119876)2 where 120589 Vect(119872) rarr 119862infin(Π119879lowast119872) is the odd

isomorphism between vector fields and ldquoone-vectorsrdquo Notethat 120589119876 is now even and the homological property becomes[[120589119876 120589119876]] = 0 where the bracket here is the canonicalSchouten-Nijenhuis bracket Thus due to the Leibniz rule itis clear that [[119875 119875]] = 0 and we have a Poisson structure Onecould also build higher Poisson structures from Q-manifoldsin this way

Recall that a Lie Algebroid 119864 rarr 119872 can be describedas a weight one homological vector field on the total spaceof Π119864 This understanding in terms of graded Q-manifoldsis attributed to Vaıntrob [19] The weight is assigned as zeroto the base coordinates and one to the fibre coordinatesIn natural local coordinates (119909119860 120585120572) the homological vectorfield is of the form

119876 = 120585120572119876119860

120572(119909)

120597

120597119909119860+1

2120585120572120585120573119876120574

120573120572(119909)

120597

120597120585120574isin Vect (Π119864)

(24)

The associated weight one odd Jacobi bracket is given by

[120601 120595]119864= (minus1)

120601(120585

120572119876119860

120572(119909)

120597120601

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120601

120597120585120574)120595

+ 120601(120585120572119876119860

120572(119909)

120597120595

120597119909119860

+1

2120585120572120585120573119876120574

120573120572(119909)

120597120595

120597120585120574)

(25)

where 120601 120595 isin 119862infin(Π119864) are ldquoLie algebroid differential formsrdquoThe weight of two-Poisson bracket is given in local coordi-nates by

120601 120595119864= (minus1)

119861120601+(119861+1)120574120585120574120585120572119876119861

120572119876119860

120574

120597120601

120597119909119860120597120595

120597119909119861

+1

2120585120574120585120575120585120572((minus1)

119861(120601+1)+(120574+120575)(119861+1)

times 119876119861

120572119876120598

120575120574

120597120601

120597120585120598120597120595

120597119909119861

+ (minus1)(120598+1)(120601+1)+120598(120574+1)

times 119876120598

120572120575119876119861

120574

120597120601

120597119909119861120597120595

120597120585120598)

+ (minus1)(120573+1)(120601+1)+120573(120598+120588)

times1

4120585120598120585120588120585120574120585120575119876120573

120575120574119876120572

120588120598

120597120601

120597120585120572120597120595

120597120585120573

(26)

Remark 17 The constructions here directly generalize to 119871infin-

algebroids understood as the pair (Π119864119876) where 119876 isin

Vect(Π119864) is a homological vector field now inhomogeneousin weight

Consider the manifold 119872 = Π119879lowast119873 times R0|1 where 119873 is

a pure even classical manifold Let us equip this manifoldwith natural local coordinates (119909119886 119909lowast

119886 120591) where (119909lowast

119886) are fibre

coordinates on Π119879lowast119873 which are Grassmann odd and 120591 isthe coordinate on the factor R0|1 The manifold 119872 is an oddcontact manifold That is it comes with an odd contact formgiven by

120572 = 119889120591 minus 119909lowast

119886119889119909

119886 (27)

Associated with this odd contact form is an odd Jacobistructure given by

119878 = 119901119886

lowast(119901119886+ 119909

lowast

119886120587) isin 119862

infin(119879lowast119872)

119876 = minus120597

120597120591isin Vect (119872)

(28)

where we have employed fibre coordinates (119901119886 119901119886lowast 120587) on

119879lowast119872 The corresponding odd Jacobi bracket is given by

[119891 119892]119869= (minus1)

119891+1 120597119891

120597119909lowast119886

120597119892

120597119909119886minus

120597119891

120597119909119886120597119892

120597119909lowast119886

+ 119909lowast

119886

120597119891

120597119909lowast119886

120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119909lowast

119886

120597119892

120597119909lowast119886

+ 119891120597119892

120597120591minus (minus1)

119891+1 120597119891

120597120591119892

(29)

Then the Loday-Poisson bracket is given by

119891 119892119869=

1205972119891

120597119909lowast119886120597120591

120597119892

120597119909119886minus (minus1)

119891 1205972119891

120597119909119886120597120591

120597119892

120597119909lowast119886

+ (minus1)119891119909lowast

119886

1205972119891

120597119909lowast119886120597120591

120597119892

120597120591+ (minus1)

119891 120597119891

120597120591

120597119892

120597120591

(30)

5 Hamiltonian Vector Fields

We now continue this paper with a study of the algebraicrelations between the Hamiltonian vector fields with respectto the odd Jacobi bracket and the (even) Loday-Poissonbracket

Definition 18 Let 119891 isin 119862infin(119872) be an arbitrary function The

associatedHamiltonian vector fieldwith respect to the Loday-Poisson bracket 119884

119891isin Vect(119872) is defined namely

119884119891(119892) = 119891 119892

119869 (31)

Throughout this section we will denote Hamiltonianvector fields with respect to the odd Jacobi structure as 119883

119891

and those with respect to the Loday-Poisson structure as 119884119891

in order to distinguish the two Note that119883119891= 119891+1 and that

119884119891= 119891

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

Lemma 19 Let 119884119891be the Hamiltonian vector fields associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

119884119891= 119883

119876(119891)= minus [119876119883

119891] (32)

where119883119891is theHamiltonian vector field associatedwith119891with

respect to the odd Jacobi structure

Proof From the definitions

119884119891(119892) = 119891 119892

119869= (minus1)

119891+1[[119876(119891) 119892]]

119869

= 119883119876(119891)

(119892) + 119876 (119876 (119891)) 119892

(33)

then given that 119876 is homological we get 119884119891

= 119883119876(119891)

Then using Proposition 9 we get 119883

119876(119891)= minus[119876119883

119891] which

establishes the lemma

The above lemma can be viewed as establishing a mildgeneralization of bi-Hamiltonian systems In particular anyvector field that is Hamiltonian with respect to the Loday-Poisson bracket is also Hamiltonian with respect to the oddJacobi structure and the Hamiltonians are related directly viathe homological field

Corollary 20 Let 119891 isin 119862infin(119872) be an even function that

satisfies the ldquoclassical master equationrdquo [119891 119891]119869= 0 Then this

implies that 119891 119891119869= 0 Furthermore we have [119876(119891) 119891]

119869= 0

and 119891 119876(119891)119869= 0

The above corollary naturally generalizes the statementthat for classical bi-Hamiltonian systems both Hamiltoni-ans are in involution with respect to both Poisson struc-tures Also note that the Hamiltonian vector fields withrespect to the Loday-Poisson bracket only depend on the Q-cohomology class of theHamiltonian function Specifically if119891 minus 119891

1015840 = 119876(119892) for for some 119892 isin 119862infin(119872) then 119884119891= 119884

1198911015840

Proposition 21 Let 119884119891be the Hamiltonian vector field associ-

ated with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then

[119876 119884119891] = 0 (34)

Proof From Lemma 19 and the Jacobi identity for the Liebracket we have

[119876 119884119891] = minus [119876 [119876119883

119891]] = minus

1

2[[119876 119876] 119883119891] = 0 (35)

as 119876 is a homological vector field

Lemma 22 Let 119884119891be the Hamiltonian vector field associated

with the function 119891 isin 119862infin(119872) with respect to the Loday-Poisson bracket Then 119884

119891is a Jacobi vector field

Proof It follows from Lemma 19 that Hamiltonian vectorfields with respect to the odd Jacobi bracket are Jacobi if andonly if the Hamiltonian function is 119876-closed As 119876

2 = 0

evidently 119884119891is Jacobi

Proposition 23 Let 119884119891and 119884

119892be the Hamiltonian vector

fields associated with the functions 119891 119892 isin 119862infin(119872) with respectto the Loday-Poisson bracket Then

[119884119891 119884119892] = 119884

119891119892119869

(36)

Proof Via direct computation

[119884119891 119884119892] = [119883

119876(119891) 119883

119876(119892)]

= minus 119883[[119876(119891)119876(119892)]]

119869

(37)

using the properties of Hamiltonian vector fields associatedwith the odd Jacobi bracket Then using

119876([119876(119891) 119892]119869) = (minus1)

119891[119876(119891) 119876(119892)]

119869 (38)

we arrive at

minus119883[119876(119891)119876(119892)]

119869

= (minus1)119891+1

119883119876([119876(119891)119892]

119869) (39)

which established the proposition

The above proposition is rather expected and moreinteresting are the ldquomixedrdquo commutators of the Hamiltonianvector fields In particular are there nice expressions for119883119891119892119869

and 119884[119891119892]119869

Proposition 24 Let 119883119892and 119884

119891be the Hamiltonian vector

fields associated with the functions 119892 119891 isin 119862infin(119872)with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

[119884119891 119883

119892] = (minus1)

119891119883119891119892119869

(40)

Proof The proposition follows from the fact that 119884119891is a

Jacobi vector field see Lemma 22 and Proposition 9

Proposition 25 Let 119883119891and 119884

119891be the Hamiltonian vector

fields associated with the function 119891 isin 119862infin(119872) with respectto the odd Jacobi bracket and the Loday-Poisson bracketrespectively Then

(minus1)119891+1

119884[119891119892]119869

= 119883119891119892119869

+ (minus1)119891119892119883119892119891119869

(41)

Proof From Lemma 19 we have

119884[119891119892]119869

= minus [119876119883[119891119892]119869

] = 119883119876([119891119892]

119869) (42)

Then using (20) and upon multiplication by overall signfactor we arrive at the above expression

Corollary 26 With the definitions previously given

(1) [119876119883119891119892119869

] = minus[119884119891 119884119892] = minus119884

119891119892119869

(2) [119876 119884[119891119892]119869

] = 0

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 7

Expressions for higher nested commutators of Hamilto-nian vector field can be worked out from the relations givenhere and the Jacobi identity for the commutator

The next natural thing to consider in this section ishow the Hamiltonian vector field with respect to the Loday-Poisson bracket behaves under the product of two functions

Proposition 27 On an odd Jacobi manifold the followingidentity holds

119884119891119892

= 119891119884119892+ (minus1)

119891119892119892119884

119891

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(43)

Proof First note from Lemma 19 and Leibniz rule for 119876 that

119884119891119892

= 119883119876(119891119892)

= 119883119876(119891)119892

+ (minus1)119891119883119891119876(119892)

(44)

Then application of Proposition 12 produces

119884119891119892

= 119891119883119876(119892)

+ (minus1)119891119892119892119883

119876(119891)

+ (minus1)119891+1

119876 (119891) (119883119892minus (minus1)

119892119892119876)

+ (minus1)119891119892+119892+1

119876 (119892) (119883119891minus (minus1)

119891119891119876)

(45)

which implies the proposition

It is easy to verify the ldquoconsistency conditionsrdquo 1198841119892 =

119884119892and 119884

1198911 = 119884119891 Furthermore Proposition 27 can be

interpreted as ldquomeasuringrdquo the violation of the left Leibnizrule of Loday-Poisson bracket The failure of the left Leibnizrule is of course a direct consequence of the Loday-Poissonbracket being a second-order differential operator in the firstargument Specifically we have

119891119892 ℎ119869= 119891119892 ℎ

119869+ (minus1)

119892ℎ119891 ℎ

119869119892

+ (minus1)119891+1

119876 (119891) ((minus1)119892[119892 ℎ]

119869minus 119876 (119892ℎ))

+ (minus1)119891119892+119892+1

119876 (119892) ((minus1)119891[119891 ℎ]

119869minus 119876 (119891ℎ))

(46)

Remark 28 TheLoday-Poisson bracket is then a biderivationif we restrict the left-hand entries of the bracket to be119876-closed However this condition implies that the Loday-Poisson bracket is trivial The other extreme is to insist that(minus1)

119891[[119891 ℎ]]

119869minus119876(119891ℎ) = 0 for all119891 ℎ isin 119862infin(119872)This implies

that [[119891 ℎ]]119869

= (minus1)119891119876(119891ℎ) and thus the underlying odd

Jacobi structure is (0 119876)That is we have ldquojustrdquo aQ-manifold

The similarity between the relations satisfied by the twoclasses of Hamiltonian vector field on an odd Jacobi manifoldand the Cartan identities is striking but not surprising asthe Cartan calculus can be understood in terms of derivedbrackets [3] In essence we have the associations

Hamiltonian vector fields wrt odd Jacobi structure larrrarr Interior derivative

Hamiltonian vector fields wrt Loday-Poisson structure larrrarr Lie derivative

Loday-Poisson bracket larrrarr Lie bracket

(47)

With these formal algebraic similarities in mind onecan interpret the constructions here as a (partially noncom-mutative) generalization of the Cartan calculus Howeveras the interior product cannot directly be understood as aHamiltonian vector field with respect to some odd Jacobistructure the Cartan calculus cannot be seen as a special caseof the constructions given in this work

6 The Derived Product

Definition 29 Let (119872119876) be a Q-manifold The derivedproduct is the binary operation lowast 119862infin(119872) times 119862infin(119872) rarr

119862infin(119872) defined as

119891 lowast 119892 = (minus1)119891+1

119876 (119891) 119892 (48)

where 119891 119892 isin 119862infin(119872)

It is easy to verify that this derived product is associativebut not (super) commutative The derived product is an odd

form of noncommutative multiplication on 119862infin(119872) 119891 lowast 119892 =

119891 + 119892 + 1 The notion of a derived product is also due toLoday and has its origin in his study of dialgebras [20] Thederived product on aQ-manifold can be viewed in the light ofdeformation quantisation That is the vector space structureof the smooth functions on theQ-manifold remains the sameit is only the product that is deformed Also note that thederived lowast-commutator is given by

[119891 119892]lowast= 119891 lowast 119892 minus (minus1)

(119891+1)(119892+1)119892 lowast 119891 = minus[119891 119892]

119876 (49)

We observe that up to a sign the odd Jacobi bracketgenerated by a homological vector field is the derived lowast-commutator This is in the same spirit as understandingPoisson brackets as the classical limit of commutators indeformation quantisation However note that 1 lowast 119891 = 0meaning that constant function 1 is not the identity (ldquobar-unitrdquo in Lodayrsquos language) for the derived product Also note119891 lowast 1 = plusmn119876(119891) Furthermore we do not have any parameterplaying the role of ℎ

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Mathematics

Remark 30 We will not have any course in this work toemploy ideas from the theory of dialgebras We only remarkthat ldquodialgebras are to Loday algebra what associative algebrais to Lie algebrardquoThe relation between thelowast-commutator andthe odd Jacobi bracket on aQ-manifold is an example of this

The derived lowast-commutator has the following easy toverify properties

(1) [119891 119892]lowast= minus(minus1)

(119891+1)(119892+1)[119892 119891]

lowast

(2) [119891 1]lowast= minus[1 119891]

lowast for all 119891

(3) [119891 119892]lowastlowast ℎ = 0

Lemma 31 Let (119872 119878 119876) be an odd Jacobi manifoldThen theodd Jacobi bracket satisfies a generalized Leibniz rule given by

[119891 119892 lowast ℎ]119869= [119891 119892]

119869lowast ℎ

+ (minus1)(119891+1)(119892+1)

119892 lowast [119891 ℎ]119869

+ 119891 lowast 119892 lowast ℎ + (minus1)119892119891 119892

119869ℎ

(50)

where 119891 119892 ℎ isin 119862infin(119872)

Proof Direct from the definitions and the modified Leibnizrule for the odd Jacobi bracket we have

[119891 119892 lowast ℎ]119869= (minus1)

119892+1[119891 119876(119892)ℎ]

119869

= (minus1)119892+1

[119891 119876(119892)]119869ℎ

+ (minus1)119892+1+(119892+1)(119891+1)

119876 (119892) [119891 ℎ]119869

minus (minus1)119892+1

[119891 1]119869119876 (119892) ℎ

(51)

Then using the fact that the homological vector field 119876 is aJacobi vector field the above can be cast in the form

[119891 119892 lowast ℎ]119869= (minus1)

119891+119892119876([119891 119892]

119869) ℎ

minus (minus1)119891+119892

[119876(119891) 119892]119869ℎ

+ (minus1)119891(119892+1)

119876 (119892) [119891 ℎ]119869

+ (minus1)119891+119892

[1 119891]119869119876 (119892) ℎ

(52)

Then using the definitions the lemma is established

Proposition 32 Let (119872119876) be a Q-manifold The lowast-commu-tator is a Schouten bracket with respect to the derived product

Proof The skew symmetry follows from Definition 29 Theappropriate Jacobi identity follows directly from the equiv-alence of the lowast-commutator with the odd Jacobi bracketderived from the homological vector field Thus the lowast-commutator gives an odd Lie bracket Only the Leib-niz rule is not immediate However this follows fromLemma 31 noting that for the case in hand (minus1)

119892119891 119892

119876ℎ =

(minus1)119891+119892+1

119876(119891)119876(119892)ℎ = minus119891 lowast 119892 lowast ℎ

The above proposition is the direct odd parallel of thewell-known result that a standard commutator on a (possibly)noncommutative algebra is in fact a Poisson bracket Indeedone could ignore the relation between odd noncommutativeproduct and odd Jacobi brackets and establish the Jacobiidentity directly from the definition

Theorem 33 Let (119872 119878 119876) be an odd Jacobi manifold Thenthe Loday-Poisson bracket obeys the Leibniz rule with thederived product

119891 119892 lowast ℎ119869= 119891 119892

119869lowast ℎ + (minus1)

119891(119892+1)119892 lowast 119891 ℎ

119869 (53)

where 119891 119892 ℎ isin 119862infin(119872)

Proof The above theorem follows directly from Lemma 31upon the replacement 119891 rarr 119876(119891) and the definition of theLoday-Poisson bracket

Statement Theorems 14 and 33 tell us that not only doesthe Loday-Poisson bracket on an odd Jacobi bracket obeythe right Leibniz rule over the standard product of smoothfunctions but also for the odd derived product

Corollary 34 Directly from the above theorem we get themixed Loday-Jacobi identity

119891 [119892 ℎ]lowast119869= [119891 119892

119869 ℎ]

lowast+ (minus1)

119891(119892+1)[119892 119891 ℎ

119869]lowast (54)

where 119891 119892 ℎ isin 119862infin(119872)

Proposition 35 With the definitions previously given

(1) 119883119891lowast119892

= (minus1)119891+1

(119891 lowast 1)119883119892

+ (minus1)119891119892119892119883

(119891lowast1) +

(minus1)119891+119892

(119891 lowast 119892)119876

(2) 119884119891lowast119892

= (119891lowast1)119884119892+(minus1)

(119891+1)(119892+1)(119892lowast1)119884

119891minus(119891lowast119892lowast1)119876

where 119891 119892 ℎ isin 119862infin(119872)

Proof The proof follows the definitions directly

(1) From Proposition 12 we have

119883119876(119891)119892

= (minus1)119891+1

119876 (119891)119883119892+ (minus1)

119891119892119892119883

119876(119891)

+ (minus1)119891+119892

119876 (119891) 119892119876

(55)

Then using the definition of the derived product and119891 lowast 1 = (minus1)

119891+1119876(119891) the first part of the proposition

is established

(2) From Proposition 27 we have

119884119876(119891)119892

= 119876 (119891)119884119892+ (minus1)

(119891+1)119892+119892+1

times 119876 (119892) (119883119876(119891)

+ (minus1)119891119876 (119891)119876)

(56)

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 9

taking into account that1198762 = 0 Then multiplying bythe correct sign factor and using the definition of 119884

119891

produce

119884119891lowast119892

= (119891 lowast 1) 119884119892+ (minus1)

(119891+1)(119892+1)(119892 lowast 1) 119884

119891

+ (minus1)119891119892119876 (119892)119876 (119891)119876

(57)

Then using 119876(119892)119876(119891) = (minus1)(119891+1)(119892+1)

119876(119891)119876(119892) andthe definition of the derived product the second partof the proposition is established

Remark 36 As far as the author is aware the case of Poisson-like brackets on algebra with an odd form of multiplicationhas not been studied in detail

7 Application to Jacobi Algebroids

An interesting class of odd Jacobimanifolds is the Jacobi alge-broids [13 21] We apply some of the previous constructionsto the setting of Jacobi algebroids

Definition 37 A vector bundle 119864 rarr 119872 is said to have thestructure of a Jacobi algebroid if and only if the total spaceof Π119864

lowast comes equipped with a weight minus one odd Jacobistructure

It is well known that Jacobi algebroids which are alsoknown as generalized Lie algebroids are equivalent to Liealgebroids in the presence of a 1-cocycle see [21] Let usemploy natural local coordinates (119909119860 120578

120572 119901119860 120587120572) on the total

space of 119879lowast(Π119864lowast) The weight is assigned as 119908(119909119860) = 0119908(119901

119860) = 0 119908(120578

120572) = +1 and 119908(120587120572) = minus1 This is the natural

weight associated with the vector bundle structure 119864lowast rarr 119872The parity of the coordinates is given by 119909119860 = 119860 120578

120572=

( + 1) 119901119860

= 119860 and 120572 = ( + 1) In these natural localcoordinates the odd Jacobi structure is given by

119878 = (minus1)120587120572119876119860

120572(119909) 119901119860 + (minus1)

+120573 1

2120587120572120587120573119876120574

120573120572120578120574

Q = 120587120572119876120572 (119909)

(58)

which are both functions on the total space of 119879lowast(Π119864lowast)The algebra of ldquomultivector fieldsrdquo119862infin(Π119864lowast) comes equippedwith an odd Jacobi bracket namely

[119883 119884]119864= (minus1)

+1119878 119883

119879lowast(Π119864lowast) 119884

119879lowast(Π119864lowast)

minus (minus1)+1

Q 119883119884119879lowast(Π119864lowast)

(59)

with119883119884 isin 119862infin(Π119864lowast)

In natural local coordinates this bracket is given by

[119883 119884]119864= 119876

119860

120572((minus1)

(++1)(119860+1) 120597119883

120597120578120572

120597119884

120597119909119860

minus (minus1)(+1) 120597119883

120597119909119860120597119884

120597120578120572

)

minus (minus1)(+1)+120573

119876120574

120572120573120578120574

120597119883

120597120578120573

120597119884

120597120578120572

+ (minus1)119876120572

120597119883

120597120578120572

119884 + 119883119876120572

120597119884

120597120578120572

(60)

where119883 = 119883(119909 120578) = 119883(119909)+119883120572(119909)120578120572+(12)119883120572120573(119909)120578

120573120578120572+sdot sdot sdot

and so forth Clearly this odd Jacobi bracket is of weightminus one If 119876

120572= 0 then the Jacobi algebroid reduces to

a genuine Lie algebroid and the above bracket is a weightminus one Schouten bracket The weight minus one oddJacobi bracket is a natural generalization of the weight minusone Schouten bracket associated with a Lie algebroid

Now let us proceed to the Loday-Poisson bracket derivedfrom the weight minus one odd Jacobi bracket and thehomological vector field 119876 = 119876

120572(120597120597120578

120572) In natural local

coordinates the Loday-Poisson bracket is given by

119883 119884119864= 119876

119860

120572((minus1)

119860(+)119876120575

1205972119883

120597120578120575120597120578120572

120597119884

120597119909119860

+ (minus1)(+1) 120597119876120575

120597119909119860120597119883

120597120578120575

120597119884

120597120578120572

+ (minus1)(+1)+119860

119876120575

1205972119883

120597120578120575120597119909119860

120597119884

120597120578120572

)

minus (minus1)(+1)

119876120574

120572120573120578120574119876120575

1205972119883

120597120578120575120597120578120573

120597119884

120597120578120572

+ (minus1)(+1)

119876120572119876120573

120597119883

120597120578120573

120597119884

120597120578120572

(61)

By construction which is easily verified in natural localcoordinates the associated Loday-Poisson bracket is ofweight minus two

Remark 38 The Loday-Poisson bracket on 119862infin(Π119864lowast) should

not be confused with the Poisson bracket on 119862infin(119864lowast) asso-ciated with the Lie algebroid structure ldquobehindrdquo the Jacobialgebroid Indeed if we have a Lie algebroid and the trivial1-cocycleQ = 0 then the associated Loday-Poisson bracket isobviously itself trivial

The algebra of ldquomultivector fieldsrdquo 119862infin(Π119864

lowast) comes

equipped with a derived product namely

119883 lowast 119884 = (minus1)+1

119876 (119883)119884 = (minus1)+1

119876120572

120597119883

120597120578120572

119884 (62)

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Mathematics

Note that 119891lowast119883 = 0 where 119891 isin 119862infin(119872) ViaTheorem 33the Loday-Poisson bracket satisfies the Leibniz rule from theleft over the derived product

119883 119884 lowast 119885119864= 119883 119884

119864lowast 119885 + (minus1)

(+1)119884 lowast 119883119885

119864

(63)

where119883119884 and 119885 isin 119862infin(Π119864lowast)

Statement ldquoMultivector fieldsrdquo on a Jacobi algebroid can beconsidered as elements of a noncommutative Loday-Poissonalgebra with an odd form of multiplication

8 Final Remarks

In this paper we used the derived bracket formalism toconstruct a Loday bracket on 119862infin(119872) from the initial datumof an odd Jacobi structure (119878 119876) on the supermanifold119872 The Loday bracket in question is the bracket derivedfrom the odd Jacobi bracket and the homological vectorfield 119876 Furthermore it was shown that this Loday bracketsatisfies the Leibniz rule acting to the right over the standardsupercommutative product of functions and the derivedproduct generated by the homological vector field Thus weemploy the nomenclature right Loday-Poisson bracket Someof the relations between the variousHamiltonian vector fieldswere also explored as were some specific examples of Loday-Poisson brackets such as those in the theory of Lie algebroidsand Jacobi algebroids

Is it important to remark that the construction of theLoday-Poisson bracket from the odd Jacobi bracket presentedhere makes use of only half the structure available namelyjust the homological vector field This is the obvious thing todo if one wants to pass from an odd Jacobi bracket to an evenLoday bracket A natural question here is as follows can onedo better and use the full odd Jacobi structure to pass from theodd Jacobi bracket to an even Loday bracket

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor J Grabowski andDr R A Mehta for their comments on earlier drafts of thiswork The author must also thank the anonymous refereesfor their invaluable comments and suggestions that havegreatly improved the presentation of this work The authorgraciously acknowledges the support of the Warsaw Centerof Mathematics and Computer Science

References

[1] A J Bruce ldquoOdd Jacobi manifolds general theory and applica-tions to generalised Lie algebroidsrdquo ExtractaMathematicae vol27 no 1 pp 91ndash123 2012

[2] Y Kosmann-Schwarzbach ldquoFrom Poisson algebras to Gersten-haber algebrasrdquo Annales de lrsquoInstitut Fourier vol 46 no 5 pp1243ndash1274 1996

[3] Y Kosmann-Schwarzbach ldquoDerived bracketsrdquo Letters in Math-ematical Physics vol 69 pp 61ndash87 2004

[4] J-L Loday Cyclic Homology Springer Berlin Germany 19922nd edition 1998

[5] J-L Loday ldquoUne version non commutative des algebres de Lieles algebres de Leibnizrdquo LrsquoEnseignement Mathematique vol 39no 3-4 pp 269ndash293 1993

[6] J Grabowski and G Marmo ldquoNon-antisymmetric versions ofNambu-Poisson and algebroid bracketsrdquo Journal of Physics Avol 34 no 18 pp 3803ndash3809 2001

[7] J Grabowski and GMarmo ldquoBinary operations in classical andquantum mechanicsrdquo in Classical and Quantum Integrability JGrabowski and P Urbanski Eds vol 59 pp 163ndash172 BanachCenter Warsaw Poland 2003

[8] J M Casas and T Datuashvili ldquoNoncommutative Leibniz-Poisson algebrasrdquo Communications in Algebra vol 34 no 7 pp2507ndash2530 2006

[9] T Voronov ldquoGraded manifolds and Drinfeld doubles for Liebialgebroidsrdquo in Quantization Poisson Brackets and Beyondvol 315 of Contemporary Mathematics pp 131ndash168 AmericanMathematical Society Providence RI USA 2002

[10] J-L Koszul ldquoCrochet de Schouten-Nijenhuis et cohomologierdquoinTheMathematical Heritage of Elie Cartan Numero Hors pp257ndash271 Asterisque Lyon France 1985

[11] A Lichnerowicz ldquoLes varietes de Poisson et leurs algebres deLie associeesrdquo Journal of Differential Geometry vol 12 no 2 pp253ndash300 1977

[12] A Lichnerowicz ldquoLes varietes de Jacobi et leurs algebres de Lieassocieesrdquo Journal de Mathematiques Pures et Appliquees vol57 no 4 pp 453ndash488 1978

[13] J Grabowski and G Marmo ldquoJacobi structures revisitedrdquoJournal of Physics A vol 34 no 49 pp 10975ndash10990 2001

[14] J Grabowski and G Marmo ldquoThe graded Jacobi algebras and(co)homologyrdquo Journal of Physics A vol 36 no 1 pp 161ndash1812003

[15] J Grabowski ldquoGraded contact manifolds and contact Courantalgebroidsrdquo Journal of Geometry and Physics vol 68 pp 27ndash582013

[16] A A Kirillov ldquoLocal Lie algebrasrdquo Uspekhi MatematicheskikhNauk vol 31 no 4 pp 57ndash76 1976 (Russian)

[17] M AlexandrovM Kontsevich A Schwarz andO ZaboronskyldquoThe geometry of themaster equation and topological quantumfield theoryrdquo International Journal of Modern Physics A vol 12no 7 pp 1405ndash1430 1997

[18] A Schwarz ldquoSemiclassical approximation in Batalin-Vilkoviskyformalismrdquo Communications in Mathematical Physics vol 158no 2 pp 373ndash396 1993

[19] A Y Vaıntrob ldquoLie algebroids and homological vector fieldsrdquoRussian Mathematical Surveys vol 52 no 2 pp 428ndash429 1997

[20] J-L Loday ldquoDialgebrasrdquo inDialgebras andRelatedOperads vol1763 of Lecture Notes inMathematics pp 7ndash66 Springer BerlinGermany 2001

[21] D Iglesias and J C Marrero ldquoGeneralized Lie bialgebroids andJacobi structuresrdquo Journal of Geometry and Physics vol 40 no2 pp 176ndash199 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of