Arash Pourkia- Hopf Cyclic Cohomology In Braided Monoidal Categories

8/3/2019 Arash Pourkia- Hopf Cyclic Cohomology In Braided Monoidal Categories

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Hopf Cyclic Cohomology In Braided

Monoidal Categories

(Spine title: Hopf Cyclic Cohomology In Braided Monoidal Categories)

(Thesis format: Monograph)

by

Arash Pourkia

Graduate Programin

Mathematics

A thesis submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy

School of Graduate and Postdoctoral StudiesThe University of Western Ontario

London, Ontario, Canada

c Arash Pourkia 2009

i


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Certificate of Examination

THE UNIVERSITY OF WESTERN ONTARIO

SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES

Chief Adviser: Examining Board:

Professor Masoud Khalkhali Professor Tatyana Foth

Advisory Committee: Professor Jan Minac

Professor John Bell

Professor Piotr M. Hajac

The thesis by

Arash Pourkia

entitled:

Hopf Cyclic Cohomology In Braided Monoidal Categories

is accepted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Date:Chair of Examining BoardFirstname Lastname

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Abstract

In the first three chapters of this thesis we recall basics of Hopf algebras, cyclic

cohomology and braided monoidal categories.

Chapters four and five form the heart of this thesis. In Chapter four, we ex-

tend the whole theory of Hopf cyclic cohomology with coefficients [18, 19, 25, 26], to

symmetric braided monoidal abelian categories. We also obtain a braided version ofConnes-Moscovicis Hopf cyclic cohomology [9, 10, 11] in any (not necessarily sym-

metric) braided monoidal abelian category. We use our theory to define a Hopf cyclic

cohomology for super Hopf algebras and for quasitriangular quasi-Hopf algebras.

In Chapter five, we define a superversion of Connes-Moscovici Hopf algebra H1

[9]. For that we define a super-bicrossproduct Hopf algebra k[Gs2]U(gs1), analogous

to the non super case [9, 17]. We call this super-bicrossproduct Hopf algebra the

super version of H1 and denote it by Hs1.

Keywords: Noncommutative geometry, Hopf algebra, braided monoidal categories,

Hopf cyclic cohomology, super mathematics.

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Acknowledgments

First and foremost, my great gratitude goes to my supervisor, Professor MasoudKhalkhali, for his invaluable guidance and his constant supports and encouragement.In the absence of his direction and assistance, I could not imagine myself even startingthis work.

I want to thank my external examiner Professor Piotr M. Hajac and examinersProfessors Jan Minac, Tatyana Foth and John Bell for carefully reading my thesis

and for their valuable suggestions.I am also grateful for all supports and helps I have constantly received from the

chair of graduate studies of the Department of Mathematics, Professor Andre Boivin.In addition, I want to thank Professor Stuart A. Rankin for helping me creating

my thesis file and for his valuable technical supports.I would also like to thank the depaerments administrative staff, Janet Williams,

Debbie Mayea, and Terry Slivinski.Last, but not least, there are those who have supported me through the hardest

times, with their love and spirit, and I cannot express what I owe them: my parents,my sisters, my brother and his family.

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To my familySpecially to my beautiful sisters Sanaz,

Haleh, Sima and to my lovely niece Zahra

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Table of Contents

Certificate of Examination . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Hopf Algebra Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hopf module algebras and coalgebras . . . . . . . . . . . . . . . . . . 51.3 Quasitriangular quasi-Hopf algebras . . . . . . . . . . . . . . . . . . . 6

2 Cyclic Cohomology Basics . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Cyclic cohomology of algebras . . . . . . . . . . . . . . . . . . . . . . 132.2 Cyclic and cocyclic modules . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Para-cyclic and cocyclic modules . . . . . . . . . . . . . . . . . . . . 222.4 Cyclic (co)homology of (co)cyclic objects . . . . . . . . . . . . . . . 252.5 Periodic cyclic cohomology of cocyclic objects . . . . . . . . . . . . . 292.6 Hopf cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7 Dual Hopf cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . 372.8 Yetter-Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . 402.9 Stable anti Yetter-Drinfeld modules . . . . . . . . . . . . . . . . . . . 422.10 Hopf cyclic cohomology with coefficients . . . . . . . . . . . . . . . . 45

3 Braided Monoidal Categories . . . . . . . . . . . . . . . . . . . . . 493.1 Braided monoidal categories . . . . . . . . . . . . . . . . . . . . . . . 493.2 Examples of braided monoidal categories . . . . . . . . . . . . . . . . 553.3 Braided algebras, coalgebras and Hopf algebras . . . . . . . . . . . . 59

4 Braided Hopf Cyclic Cohomology . . . . . . . . . . . . . . . . . . . 654.1 The cocyclic module of a braided triple (H, C, M) . . . . . . . . . . . 664.2 The braided version of Connes-Moscovicis Hopf cyclic cohomology . . 804.3 Hopf cyclic cohomology for super Hopf algebras . . . . . . . . . . . . 854.4 Hopf cyclic cohomology of the enveloping algebra of a super Lie algebra 864.5 Hopf cyclic cohomology in non-symmetric monoidal categories . . . . 944.6 A Hopf cyclic theory for quasitriangular quasi-Hopf algebras . . . . . 106

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5 A super version of the Connes-Moscovici Hopf algebra . . . . . . . 1105.1 The Connes-Moscovici Hopf algebra H

1. . . . . . . . . . . . . . . . . 111

5.2 The super group Gs = Diff+(R1,1) and its factorisation . . . . . . . 1165.3 Two super Hopf algebras U(gs1) and F(G

s2) . . . . . . . . . . . . . . . 120

5.3.1 The super Hopf algebra U(gs1) . . . . . . . . . . . . . . . . . . 1205.3.2 The super Hopf algebra F(Gs2) . . . . . . . . . . . . . . . . . 122

5.4 Actions and coactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.4.1 Actions ofX . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.4.2 Actions ofY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.4.3 Actions ofZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4.4 Actions ofU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.4.5 Actions ofV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4.6 Actions ofW . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.4.7 Coactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.5 Compatibilities and the super Hopf algebra Hs1 . . . . . . . . . . . . 152

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

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Chapter 1

Hopf Algebra Basics

In this chapter we review the basics about Hopf algebras and quasitriangular quasi-

Hopf algebra. We also recall the notions of Hopf module algebra and coalgebra.

1.1 Hopf algebras

We assume basic notions of algebra and coalgebra in the category of vector spaces

over a field k, as explained in [22, 31, 37]. In this section we recall the notions of

bialgebra and Hopf algebra over a field k, and give some examples.

A bialgebra is simultaneously an algebra and a coalgebra satisfying some com-

patibility conditions. More precisely:

Definition 1.1.1. A bialgebra is a quadruple (H,m,, , ) where H is a vector space

over a fieldk, m : H H H, : k H, : H H H and : H k are linear

maps called multiplication, unit, comultiplication and counit, respectively, such that

(H,m,) is a unital associative algebra, (H, , ) is a counital coassociative coalgebra,

and the two structures are compatible in the sense that and are morphisms of

algebras or, equivalently, m and are morphisms of coalgebras.

Compatibility conditions can be expressed in terms of commutative diagrams

as below:

1


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2

The map is a morphism of algebras:

H Hm //

H H

H H H Hidid

// H H H H

mm

OO

where : H H H H is the flip map defined by (g h) = h g, for all g and

h in H, and

H // H H

k k = k

OO

55kkkkkkkkkkkkkkk

The map is a morphism of algebras:

H Hm

//

H

uulll

llllllllllll

k k = k

kid //

k

H

77oooooooooooooo

Throughout this thesis, (except for slight changes in the last chapter), we use

the Sweedlers notation, with summation understood, i.e., we write h = h(1) h(2)

to denote the comultiplication of bialgebras. Similarly for higher comultiplications

we write:

nh = h(1) h(2) h(3) . . . h(n+1).

Definition 1.1.2. A Hopf algebra (H,m,, , , S ) consists of a bialgebra


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(H,m,, , ) and a linear map S : H H, called antipode, satisfying:

S(h(1))h(2) = h(1)S(h(2)) = (h)1, h H,

or equivalently,

m(S id) = m(id S) = . (1.1.1)

The antipode axioms (1.1.1) can be expressed by commutative diagrams as

below:

H HSid // H H

m

((QQQQ

QQQQ

QQQQ

QQQ

H //

OO

k // H

H HidS // H H

m66mmmmmmmmmmmmmmm

In this thesis Halways denotes a Hopf algebra. A Hopf algebra (H,m,, , , S )is called commutative if it is commutative as an algebra, i.e.,

ab = ba a, b H,

or equivalently

m = m.

H is called cocommutative if it is a cocommutative coalgebra, i.e.,

h(1) h(2) = h(2) h(1) h H,

or equivalently

= .

Proposition 1.1.1. [37] The antipode S of a Hopfalgebra H is unique. It is an


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antialgebra map and an anticoalgebra map. More precisely, for all g and h in H:

S(gh) = S(h)S(g), S(1) = 1,

S(h)(1) S(h)(2) = S(h(2)) S(h(1)),

S(h) = (h).

Proposition 1.1.2. [37] If H is commutative or cocommutative then S2 = id.

Example 1.1.1. LetG be a discrete group and H = kG the group algebra of G over

the field k. Let

(g) = g g, S(g) = g1

, and (g) = 1,

for allg G and extend them by linearity to kG. Then(H, , , S ) is a cocommutative

Hopf algebra. It is commutative if and only if G is commutative.

Example 1.1.2. Letg be a Lie algebra over the fieldk andH = U(g) be the universal

enveloping algebra of g. Using the universal property of U(g) one checks that there

are uniquely defined algebra homomorphisms : U(g) U(g) U(g), : U(g) k

and an antialgebra map S : U(g) U(g), determined by

(X) = X 1 + 1 X, (X) = 0, and S (X) = X,

for all X g. Then(U(g), , , S ) is a cocommutative Hopf algebra. It is commu-

tative if and only if g is an abelian Lie algebra, in which case U(g) = S(g) is the

symmetric algebra of g.


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1.2 Hopf module algebras and coalgebras

The idea of symmetry in noncommutative geometry is encoded by action or coaction

of a Hopf algebra on an algebra or a coalgebra. There are four possibilities of which

we recall only two in this section.

Definition 1.2.1. A left H-module algebra is an algebra (A, mA, A) equipped with

a left H-module structure via the left action

A : H A A, A(h a) = ha

such that , mA and A are H-module maps, i.e.,

h(ab) = h(1)ah(2)b,

or equivalently,

A(idH mA) = mA(A, A)(idH idA)(H idA idA),

and

h1A = (h)1A,

or equivalently,

A(idH A) = AH.

Definition 1.2.2. A leftH-module coalgebra is a coalgebra (C, C, C) equipped with

a left H-module structure via the left action

C : H C C, C(h c) = hc


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such that C and C are H-module maps i.e. :

(hc)(1) (hc)(2) = h(1)c(1) h(2)c(2),

or equivalently,

CC = (C C)(idH idC)(H C),

and

C(hc) = H(h)C(c),

or equivalently,

CC = H C.

Example 1.2.1. The conjugation action H H H, g h g(1)hS(g(2)) gives

H the structure of a left H-module algebra.

Example 1.2.2. The multiplication mH : H H H turns H into a leftH-module

coalgebra.

1.3 Quasitriangular quasi-Hopf algebras

In this section we recall the definitions of a quasitriangular Hopf algebra, quasi-Hopf

algebra, quasitriangular quasi-Hopf algebra, and provide some examples.

Definition 1.3.1. [22, 31] A quasitriangular Hopf algebra consists of a Hopf algebra

H and an invertible element R = R1 R2 of H H, with the inverse R1 =

(R1)1 (R1)2, satisfying the following relations: (note that we have used the

Sweedlers notation, therefore R andR1 are not necessarily simple tensors inHH)

( id)(R) = R13R23,


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(id )(R) = R13R12,

(h) = R((h))R1, h H, (1.3.2)

where

R13 = R1 1 R2, R23 = 1 R1 R2, R12 = R1 R2 1.

Definition 1.3.2. In Definition (1.3.1), if R1 = (R), i.e., if (R1)1 (R1)2 =

R2 R1, then (H, R) is called a triangular Hopf algebra.

Remark 1.3.1. One can define a coquasitriangular Hopf algebra in a dual fashion

[22, 31].

Proposition 1.3.1. If (H, R) is a quasitrangular Hopf algebra, then:

( id)(R) = (id )(R) = 1,

(S id)(R) = R1, (id S)(R) = R,

R12R13R23 = R23R13R12.

Example 1.3.1. Every cocommutative Hopf algebra is quasitriangular with the trivial

quasitriangular structure R = 1 1.

We note that every commutative quasitrangular Hopf algebra is cocommutative.

Example 1.3.2. [31] LetCZn be the group algebra of the finite cyclic group of order

n. By Example (1.1.1) it is a commutative and cocommutative Hopf algebra. One can

define a nontrivial quasitriangular structure onCZnby:

R = (1/n)n1

a,b=0

e(2iab)/nga gb,


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where g is the generator ofZn.

Example 1.3.3. If in Example (1.3.2) we let n = 2 then we have H = CZ2 with the

non-trivial quasitriangular structure R = R1 R2 defined by

R := (1

2)(1 1 + 1 g + g 1 g g),

where g is the generator of the cyclic group Z2.

If in the definition of a bialgebra we weaken the coassociativity of the comulti-

plication in a suitable way, we obtain the notion of a quasi-bialgebra.

Definition 1.3.3. A quasi-bialgebra consists of a quadruple (H,m,, , ) where

(H,m,) is an unital associative algebra and (H, , ) is a counital coalgebra in which

the coassociativity property is replaced by a weaker version:

(id ) = [( id)]1. (1.3.3)

Here = 1 2 3, called an associator, is an invertible element of H H H

with inverse 1 = (1)1 (1)2 (

1)3, and satisfying :

[(id id )()] [( id id)()] = (1 ) [(id id)()] ( 1), (1.3.4)

(id id)() = 1. (1.3.5)

As we shall see in Chapter (3), the above properties (1.3.3)-(1.3.5) are equivalent

to saying that the category of representations of the algebra (H,m,) is a monoidal

category [22].

Definition 1.3.4. A quasi-Hopf agebra consists of a quasi-bialgebra(H,m,, , , ),

a bijective linear map S : H H and elements and of H such that:

S(h(1))h(2) = (h), h H,


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h(1)S(h(2)) = (h), h H,

1S(2)3 = 1,

S((1)1)(1)2S((

1)3) = 1,

where 1 = (1)1 (1)2 (

1)3 is the inverse of in H H H.

Remark 1.3.2. By dualizing the above two definitions one can define a coquasi-

bialgebra or a coquasi-Hopf algebra [22, 31].

Definition 1.3.5. A quasitriangular quasi-Hopf algebra is a quasi-Hopf algebra

(H,m,, ,,,S,,) equipped with an invertible elementR = R(1)R(2) ofHH

satisfying :

( id)(R) = 312R131132R23,

(id )(R) = 1231R13213R121,

(h) = R((h))R1, h H.

As we shall see in Chapter (3), the above properties are equivalent to saying

that that the monoidal category of H-modules is a braided monoidal category [22].

Recall that a pair (X, Y) of Hopf algebras (bialgebras) is called a matched pair

if X is a left Y-module coalgebra via : Y X X and Y is a right X-module

coalgebra via : Y X Y such that:

y (xx) = (y(1) x(1))((y(2) x(2)) x)), y 1 = (y)1,

(yy ) x = (y (y(1) x(1)))(y(2) x(2)), 1 x = (x)1,

y(1) x(1) y(2) x(2) = y(2) x(2) y(1) x(1),

for all x, x in X and y, y in Y. For example for any finite-dimensional Hopf algebra

(H,m,, , , S ) with S, the pair ((Hop), H) is a matched pair. Here (Hop) =


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(H, (mop), , , , S ). For any matched pair (X, Y) of Hopf algebras there exist

a unique Hopf algebra structure on the vector space X Y defined by:

(x y)(x y) = x(y(1) x(1)) (y(2) x(2))y,

(x y) = (x(1) y(1)) (x(2) y(2)),

1XY = 1 1, (x y) = (x)(y),

S(x y) = S(y(2)) S(x(2)) S(y(1)) S(x(1)).

This Hopf algebra is called the bicrossed product of X and Y and denoted by

XY [22]. Using this structure for the matched pair ((Hop), H), when H is finite-

dimensional, one obtains the Drinfelds quantum double of H, D(H) = (Hop)H.

Its structure is explicitly given in the next example.

Example 1.3.4. [22] Let H be a finite-dimensional Hopf algebra with an invertible

anipode, and(Hop) be its dual Hopf algebra as mentioned above. The quantum double

of D(H) = (Hop)H is the vector space (Hop) H with the structure:

(f a)(g b) = f g(S1(a(3))a(1))

a(2)b, 1D(H) = 1 1,

(f a) = (f(1) a(1)) (f(2) a(2)), (f a) = (a)f(1).

In the first formula the map g(S1(a(3))a(1))

is defined by:

g(S1(a(3))a(1))

(x) = g(S1(a(3))xa(1)).

The quantum double D(H) is a quasitriangular Hopf algebra via R D(H)

D(H) defined by:

R = iI(1 ei) (ei 1),


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where {ei}iI is a basis for H, and {ei }iI is the dual basis for H

.

Example 1.3.5. [22] Let G be a finite group, the quantum double of G is defined to

be D(kG) = ((kG)op) kG. It is usually denoted by D(G), and has the following

structure:

(g a)(h b) = g(aha1) ab = g,aha1(g ab),

(g

a) = st=g s

a t

a,

(g a) = g,1,

S(g a) = (a1g1a) a1,

R =gG

(1 g) (g 1).

Example 1.3.6. [22] Let G be a finite group and let : G G G k be a

normalized 3-cocycl, i.e, for all a,b,c and d in G :

(a,b,c)(d,ab,c)(d,a,b)

(da,b,c)(d,a,bc)= 1,

and

(a,b,c) = 1 if a, b, or c = 1.

For any a,b,g G, we define

g(a, b) =(g,a,b)(a,b, (ab)1gab)

(a, a1ga,b),

g(a, b) =(a,b,g)(g, g1ag,g1bg)

(a,g,g1bg).

The twisted quantum double D(G) of G with respect to is (kG) kG as a vector


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space with the further structures given by:

(g a)(h b) = g(a, b)g(aha1) ab,

(g a) =st=g

a(s, t)s a t a,

(g a) = g,1,

S(g a) = g1(a, a1)1a(g, g

1)1(a1g1a) a1,

R =gG

(g 1) ((

h

h) g),

=

g,s,tG

(g,s,t)1g 1 s 1 t 1,

= 1D(G) = 1 1 , =

gG

(g, g1, g)g 1,

where {g

|g G} is the dual basis of the canonical basis of kG and g,1 is theKronecker delta.

The above structure defines D(G) = (kG) kG as a quasitriangular quasi-

Hopf algebra, and is isomorphic to the quantum double D(G) if is trivial, i.e, if

(a,b,c) = 1, for all a,b,c in G.


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and the Hochschild coboundary b : Cn(A) Cn+1(A) is defined by

b(a0,...,an+1) =n

i=0

(1)i(a0,...,aiai+1, ...an+1) + (1)n+1(an+1a0,...,an).

(2.1.2)

The Hochschild cohomology ofA denoted by HH(A) is defined to be the cohomology

of the cochain complex (2.1.1) [28].

We now turn to the definition of cyclic cohomology of algebras.

Definition 2.1.1. LetA be a unital algebra. The cyclic cohomology of A denoted by

HC(A) is the cohomology of the the following cochain complex known as Connes

complex

C0(A)b

C1(A)b

C2(A)b

C3(A) . . .

where Cn (A) is the subcomplex of Cn(A) containing those Cn(A) that satisfy

(a0,...,an) = (1)n(an, a0,...,an1), n = 0, 1, 2,...

for all a0,...,an in A, and b : Cn (A) C

n+1 (A) is the Hochschild coboundary as

defined in formula (2.1.2).

There are alternative ways to define cyclic cohomology. Consider the Hochschild

complex Cn(A) = HomC(A(n+1), C). Define the so called faces i : C

n1(A)

Cn

(A) and cyclic maps n : Cn

(A) Cn

(A) by:

i(a0,...,an) =

(a0,...,aiai+1,...,an) 0 i < n

(ana0, a1,...,an1) i = n

and

n(a0,...,an) = (an, a0,...,an1).


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Let

b =

n0

(1)ii,

b =n10

(1)ii,

n = (1)nn,

and denoting n by ,

N = 1 + + 2

+ ... + n

.

The relations

b2 = b2 = 0, (1 )b = b(1 ), and (1 )N = N(1 ) = 0,

can be verified [6, 28]. Therefore one can construct the so called (b, b)-bicomplex of

A, denoted by C(A), as:

......

...

C2(A)1

C2(A)N

C2(A)1

b b bC1(A)

1 C1(A)

N C1(A)

1

b

b

b

C0

(A)1

C0

(A)N

C0

(A)1

It can be shown that the cyclic cohomology of A is isomorphic to the cohomology of

the total complex TotC(A) [28].

Let us define degeneracies i : Cn+1(A) Cn(A) by,

i(a0,...,an) = (a0, ...ai, 1, ai+1,...,an), 0 i n.


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Now let

B = Ns(1 ) : Cn+1(A) Cn(A),

where the operator s : Cn+1(A) Cn(A) is defined by

s(a0,...,an) = (1, a0,...,an).

Then we have the relations: [7]

b2 = B2 = 0, and bB + Bb = 0.

We construct the (b, B)-bicomplex of A, denoted by B(A), as:

......

...

C2(A)B

C1(A)B

C0(A)

b bC1(A)

B C0(A)

b

C0(A)

Again it can be shown that the cyclic cohomology of A is isomorphic to the cohomol-

ogy of the total complex TotB(A) [7, 28].

Remark 2.1.1. One can use the (b, b)-bicomplex to drive the long exact sequence of

Connes as in [28]. Alternatively, consider the short exact sequence:

0 C(A) C(A)

C(A)/C(A) 0

One can show that the cohomology of the quotient complex C(A)/C(A) is isomorphic

to the cyclic cohomology of A with a shift in dimension by one. The resulting long


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exact sequence is Connes long exact sequence:

HCn(A)I

HHn(A)B

HCn1(A)S

HCn+1(A)

The operator S is known as Connes periodicity operator.

2.2 Cyclic and cocyclic modules

Yet another approach to cyclic (co)homology is based on the notion of (co)cyclic

modules [6].

Definition 2.2.1. The simplicial category is a small category whose objects are

the totally ordered sets (cf. e.g. [28])

[n] = {0 < 1 < < n}, n = 0, 1, 2, . . . .

A morphismf : [n] [m] of is an order preserving, i.e. monotone non-decreasing,

map f : {0, 1, . . . , n} {0, 1, . . . , m}.

Of particular interest among the morphisms of are faces i and degeneracies

j ,

i : [n 1] [n], i = 0, 1, . . . , n

j : [n + 1] [n], j = 0, 1, . . . , n .

By definition, i is the unique injective morphism missing i, and j is the unique

surjective morphism identifying j with j + 1. It can be checked that they satisfy the


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following simplicial identities:

j i = ij1 if i < j,

ji = ij+1 if i j,

j i =

ij1 if i < j

id if i = j or i = j + 1

i1j if i > j + 1.

(2.2.3)

Every morphism of can be uniquely decomposed as a product of faces followed by

a product of degeneracies [28].

Definition 2.2.2. The cyclic category has the same set of objects as and in fact

contains as a subcategory. Morphisms of are generated by simplicial morphisms

i and j as above and new morphisms n : [n] [n] for n 0 defined by

n(i) =

n i = 0

i 1 i = 0

In other words n is the following cyclic permutation

n =

0, 1, 2, ..., n

n, 0, 1, ..., n 1

.

The following extra relations hold between i, i and n in :

ni = i1n1 1 i n

n0 = n

ni = i1n+1 1 i n

n0 = n

2

n+1 (2.2.4)

n+1n = id.


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Let op denote the opposite category of the category .

Definition 2.2.3. Let C be a category. A cyclic object in C is a functor op C.

Equivalently, a cyclic object in C is given by the data

X = (Xn, i, i, n), n 0,

where Xn, n 0, is a family of objects of C, i : Xn Xn1, 0 i n,

i : Xn Xn+1, 0 i n, and n : Xn Xn called faces, degeneracies andcyclic maps, respectively, are morphisms of C satisfying the relations, dual to the

relations (2.2.3) and (2.2.4):

ij = j1i if i < j,

ij = j+1i if i j,

ij =

j1i if i < j

id if i = j or i = j + 1

j i1 if i > j + 1

(2.2.5)

and

in = n1i1 1 i n,

0n = n,

in = n+1i1 1 i n,

0n = 2n+1n,

n+1n = id. (2.2.6)

Definition 2.2.4. Let k be a commutative ring. A cyclic object in the category ofk-modules is called a cyclic k-module or just a cyclic module.


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Definition 2.2.5. A cocyclic object in C is a functor C. Equivalently, a cocyclic

object in C is given by the data

X = (Xn, i, i, n), n 0,

where Xn, n 0, are objects of C, and faces i : Xn1 Xn, 0 i n,

degeneracies i : Xn+1 Xn, 0 i n, and cyclic maps n : X

n Xn are

morphisms of C satisfying all the relations (2.2.3) and (2.2.4).

Definition 2.2.6. Letk be a commutative ring. A cocyclic object in the category of

k-modules is called a cocyclic k-module or just a cocyclic module.

For any commutative ring k, we denote the category of cyclic k-modules by

k. A morphism of cyclic k-modules is a natural transformation between the cor-

responding functors. Equivalently, a morphism f : X Y consists of a sequence

of k-linear maps fn : Xn Yn compatible with faces, degeneracies, and cyclic op-erators. One can of course talk about k the category of cocyclic modules in the

same manner. It is clear that k is an abelian category. The kernel and coker-

nel of a morphism f are defined pointwise: (Ker f)n = Ker fn : Xn Yn and

(Coker f)n = Coker fn : Xn Yn. More generally, if A is any abelian category then

the category A of cyclic objects in A is itself an additive category.

Let Algk denote the category of unital k-algebras and unital algebra homomor-

phisms. Note that Algk is not an additive category.

Example 2.2.1. To an algebraA in Algk, we associate the cyclic k-module A defined

by

An = A

(n+1), n 0,


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with face, degeneracy and cyclic operators given by

i(a0,...,an) =

(a0,...,aiai+1,...,an) 0 i < n

(ana0,...,an1) i = n

i(a0,...,an) = (a0, ...ai, 1, ai+1,...,an), 0 i n,

n(a0,...,an) = (an, a0,...,an1).

In above formulas by (a0,...,an) we mean (a0 a1 ... an) A(n+1). We

will use this convention in sequel when there is no confusion. From the above cyclic

module one defines the cyclic homology of the algebra A (cf. Section 2.4).

A unital algebra map f : A B induces a morphism of cyclic modules f :

A B by f(a0,...,an) = (f(a0), , f(an)), and this defines a functor .

: Algk k,

Theorem 2.2.1. [6] For any unital k-algebra A, there is a canonical isomorphism:

HCn(A) Extnk(A, k), for all n 0. (2.2.7)

In fact the isomorphism (2.2.7) can be made explicit for n = 0 [6]. Given a

trace t : A k, one defines a map of cyclic modules t : A k by:

t(a0, a1,...,an) := t(a0a1...an).

Example 2.2.2. Let(C, , ) be a coalgebra. To C, we associate the cocyclic module

C defined by

Cn = C(n+1), n 0,


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with face, degeneracy and cyclic operators defined by

i(c0,...,cn1) =

(c0,...,c(1)i , c

(2)i ,...,cn1) 0 i < n

(c(2)0 , c1,...,cn1, c

(1)0 ) i = n

i(c0,...,cn+1) = (c0, ...ci, (ci+1),...,cn+1), 0 i n,

n(c0,...,cn) = (c1,...,cn, c0).

In the next sections we shall see more examples of cyclic and cocyclic modules.

2.3 Para-cyclic and cocyclic modules

One can define the notion of para-(co)cyclic object by eliminating the last relation,

n+1n = id in the definition of a (co)cyclic object.

Definition 2.3.1. A para-cocyclic object in a category C is given by the data

X = (Xn, i, i, n), n 0,

where Xn, n 0, are objects of C, faces i : Xn1 Xn, 0 i n, degeneracies

i : Xn+1 Xn, 0 i n, and cyclic maps n : X

n Xn are morphisms of C

satisfying relations (2.2.3) and:

ni = i1n1 1 i n

n0 = n

ni = i1n+1 1 i n

n0 = n2n+1 (2.3.8)


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One also defines the notion of para-cyclic object in the same manner, i.e., by

eliminating the condition n+1n = id from the relations (2.2.5) and (2.2.6).

Let k be a commutative ring. A para-(co)cyclic object in the category of k-

modules is called a para-(co)cyclic k-module.

Example 2.3.1. LetA be a unital algebra and : A A be an automorphism of A.

One can associate to (A, ) the para-cyclic module A defined by

(A)n = A

(n+1), n 0,


i(a0,...,an) =

(a0,...,aiai+1,...,an) 0 i < n

((an)a0,...,an1) i = n

i(a0,...,an) = (a0, ...ai, 1, ai+1,...,an), 0 i n,

n(a0,...,an) = ((an), a0,...,an1).

Note that in this example

n+1n (a0,...,an) = ((a0), (a1),...,(an)).

Therefore n+1n = id except when = id. In fact for = id we get the cyclic module

of Example (2.2.1).

Example (2.3.1) is an special case of the following more general one.

Example 2.3.2. Let A be a unital algebra and G be a group acting on A by auto-

morphisms. We can associate to (A, G) the para-cyclic module AG defined by [16]

(AG)n = CG A

(n+1), n 0,


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i(g, a0,...,an) =

(g, a0,...,aiai+1,...,an) 0 i < n

(g, (g1an)a0,...,an1) i = n

i(a0,...,an) = (a0, ...ai, 1, ai+1,...,an), 0 i n,

n(g, a0,...,an) = (g, g1an, a0,...,an1).

Example 2.3.3. The above example can be further generalized as follows [1]. Let H

be a Hopf algebra and A be an H-module algebra. One can introduce the para-cyclic

module

(AH)n = H A

(n+1),

by defining face, degeneracy and cyclic operators as follows:

i(g, a0,...,an) =

(g, a0,...,aiai+1,...,an) 0 i < n

(g(1), (S(g(2))an)a0,...,an1) i = n

i(a0,...,an) = (a0, ...ai, 1, ai+1,...,an), 0 i n,

n(g, a0,...,an) = (g(1), (S(g(2)an), a0,...,an1).

Given a para-cocyclic object (Xn, i, i, n), n 0, in an abelian category C,

we can always define a cocyclic object by considering

Xn

:= ker(id n+1n ), (2.3.9)

and restricting the faces, degeneracies, and cyclic operators to these subspaces.


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In a dual fashion given a para-cyclic object (Xn, i, i, n), n 0, in an abelian

category C, we define a cyclic object by

Xn :=Xn

Im(id n+1n ), (2.3.10)

and the faces, degeneracies, and cyclic operators are naturally induced by the given

i, i and n to these quotients.

2.4 Cyclic (co)homology of (co)cyclic objects

To any (co)cyclic object in an abelian category, one can assign a cyclic (co)homology

in different, but equivalent, ways, analogous to what we defined for a unital algebra

in section (2.1).

Definition 2.4.1. Let C be an abelian category and X = (Xn, i, i, n), n 0

be a cocyclic object in C. The cyclic cohomology of X denoted by HC(X) is the

cohomology of the cochain complex

X0b

X1b

X2b

X3 . . . ,

Here

Xn = Ker(1 n),

where

n = (1)nn,

and b : Xn1 Xn is defined by

b =n

i=0

(1)ii.


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Definition 2.4.2. Let X = (Xn, i, i, n), n 0 be a cyclic object in an abelian

category C. The cyclic homology of X denoted by HC(X) is the homology of the

chain complex

X0b

X1b

X2b

X3 ,

where

Xn =Xn

Im(1 n),

and b : Xn Xn1 is defined by

b =n

i=0

(1)ii.

When the ambient categry is the category of vector spaces over a field k of

characteristic zero, to obtain the cyclic cohomology of the cocyclic object X =

(Xn, i, i, n) of C = V ectk one can alternatively construct the (b, b)-bicomplex,

denoted by C(X), as:

......

...

X21

X2N

X21

b b bX1

1 X1

N X1

1

b

b

b

X0 1 X0 N X0 1

where

b =n1i=0

(1)ii

n = (1)nn,

N = 1 + + 2 + ... + n.


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The relations

b2 = b2 = 0, (1 )b = b(1 ), and (1 )N = N(1 ) = 0,

are verified. It can be shown that the cyclic cohomology of X is isomorphic to the

cohomology of the total complex TotC(X).

When C = V ectk where k is a field of characteristic zero, one can also construct

the (b, B)-bicomplex of X, denoted by B(X), as

......

...

X2B

X1B

X0

b

bX1

B X0

b

X0

with B defined by

B = Ns(1 ),

where the operator s called the extra degeneracy is given by

s = nn+1 : Xn+1 Xn.

Note that the relations:

b2 = B2 = 0, and bB + Bb = 0.

are satisfied. Again it can be shown that the cyclic cohomology of X is isomorphic

to the cohomology of the total complex TotB(X).

Remark 2.4.1. The same statements are true for the cyclic homology of a cyclic


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object X = (Xn, i, i, n) in C = V ectk with dualized diagrams as follows.

......

...

X21

X2N

X21

b b bX1

1 X1

N X1

1 b b b

X01

X0N

X01

......

...

X2B

X1B

X0

b

bX1

B X0

b

X0

and

B = (1 )sN,

where

s = n+1n : Xn Xn+1.

Example 2.4.1. Following Sections (2.1) and (2.2), the cyclic cohomology of a unital

k-algebraA is an example of the cyclic cohomology of a cocyclic module. Also Example

(2.2.1) gives a cyclic object in the category of k-modules, and from that one defines

the cyclic homology of a unital k-algebra A.

Example 2.4.2. Considering the cocyclic object given in Example (2.2.2) and using

methods of this section one can define the cyclic cohomology of a coalgebra C.


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2.5 Periodic cyclic cohomology of cocyclic objects

Definition 2.5.1. LetC be an abelian category and X = (Xn, i, i, n), n 0, be a

cocyclic object in C. The periodic cyclic cohomology of X denoted by HP(X), where

= 0, 1, is defined as the following direct limit:

HPi(X) = limn

HCi+2n(X), i = 0, 1,

where the direct limit, limn

, is taken by using the Connes periodicity operator S :

HCn(X) HCn+2(X).

When C = V ectk where k is a field of characteristic zero, alternatively, HP(X)

can be defined as the cohomology of the total complex of either of the following two

bicomplexes:

The bicomplex

C(X)

......

......

......

X21

X2N

X21

X21

X2N

b b b b b X1

1 X1

N X1

1 X1

1 X1

N b b b b b

X01

X0N

X01

X01

X0N


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Or the bicomplex

B(X)

......

......

...

X4B

X3B

X2B

X1B

X0

b

b b b X3

B X2

B X1

B X0

b

b b X2

B X1

B X0

b b X1

B X0

b

X0

Note that the the total complexes Tot C(X) and Tot B(X) are as follows, respec-tively:

i0

Xi i0

Xi i0

Xi

i0

X2i(b+B)

i0

X2i+1(b+B)

i0

X2i(b+B)

i0

X2i+1

We shall provide some examples in the next section.

2.6 Hopf cyclic cohomology

Hopf cyclic cohomology is a cohomology theory of cyclic type for any Hopf algebra

equipped with a modular pair in involution. This theory was introduced by Connes

and Moscovici in [9, 10, 11]. Their first motivation came from the computation of

the index of transversally elliptic operators on foliations. For that they constructed,


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for any n 1, a Hopf algebra H(n), and computed the Hopf cyclic cohomology of

H(n) [9]. They also developed a theory of characteristic classes for actions of Hopf

algebras on algebras. Hopf cyclic cohomology can in fact be regarded as the right

noncommutative analogue of both group and Lie algebra homology. Here is a quick

recall:

Let (H, , , S) be a Hopf algebra. A character of H is a unital algebra map

: H k. A grouplike element of H is a nonzero element in H such that

() = . Let be a character of H and a group like element in H. We say

(, ) is a modular pair if () = 1. We can define a -twisted antipode by S = S,i.e.

S(h) = (h(1))S(h(2)), h H.A modular pair (, ) is called a modular pair in involution (MPI), [9, 10, 11] if

1 S2 = id, that is:1 S2(h) = h, h H.

Example 2.6.1. If H is commutative or cocommutative then S2 = id which implies

that(, 1) is an MPI for H. Conversely if(, 1) is an MPI for H, thenS2 = id. Note

that the condition S2 = id does not imply that H is commutative or cocommutative.

Example 2.6.2. The quantum universal enveloping algebra Uq(sl(2, k)) is a k-Hopf

algebra which is generated as a k- algebra by symbols , 1, x, y subject to the

following relations

1 = 1 = 1, x = q2x, y = q2y, xy yx = 1

q q1.

The coproduct, counit and antipode of Uq(sl(2, k)) are defined by:

(x) = x + 1 x, (y) = y 1 + 1 y, () = ,

S() = 1, S(x) = x1, S(y) = y,


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() = 1, (x) = (y) = 0.

It is easy to check thatS2(h) = h1 for al lh inH. Therefore (, 1) is a modular

pair in involution for Uq(sl(2, k)).

Example 2.6.3. Letk be a field of characteristic zero and q k, q = 0 and q not a

root of unity. The Hopf algebra H = A(SLq(2, k)) is defined as follows [24]. As an

algebra it is generated by symbols a,b,c,d, with the following relations:

ab = qba, ac = qca, ad da = (q q1)bc,

bc = cb, bd = qdb, cd = qdc,

ad qbc = da q1bc = 1.

Comultlipication, counit, and antipode on H are defined by:

(a) = a a + b c, (b) = a b + b d,

(c) = c a + d c, (d) = c b + d d,

(a) = (d) = 1, (b) = (c) = 0,

S(a) = d, S(b) = q1b,

S(c) = qc, S (d) = a.

Note that S2 = id. A modular pair (, ) for H is defined by:

(a) = q, (b) = (c) = 0, (d) = q1,

and = 1. Then one can check that

S2 = id, and hence (, 1) is modular pair in

involution for H.


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Theorem 2.6.1. [9, 10, 11] Let(H, , , S) be a Hopf algebra endowed with a modu-

lar pair in involution(, ). SetCn(H) = Hn, n 0, and define faces, degeneracies,

and cyclic maps by:

i(h1,...,hn1) =

(1, h1,...,hn1) i = 0

(h1,..., hi,...,hn1) 1 i n 1

(h1,...,hn1, ) i = n

i(h1,...,hn+1) = (hi+1)(h1,...,hi, hi+2,...,hn+1), 0 i n

n(h1,...,hn) = n1 S(h1) (h2,...,hn, ). (2.6.11)

Then (C(H), i, i, ) is a cocyclic module.

The action in Formula (2.6.11) is the diagonal action of Hn on itself defined

by:

(g1, g2,...,gn).(h1, h2,...,hn) := (g1h1, g2h2,...,gnhn).

Therefore

n1 S(h1) (h2,...,hn, ) = S(h(n)1 )h2,...,S(h(2)1 )hn, S(h(1)1 )Definition 2.6.1. The cohomology of the cocyclic module defined in Theorem (2.6.1)

is called the Hopf cyclic cohomology of H and is denoted byHC(,)(H), or sometimes

just HC(H), if there is no confusion.

Example 2.6.4. [9, 10, 11, 12] LetG be a discrete group andCG be its group algebra

(cf. Example 1.1.1). We recall from Example (2.6.1) that (, 1) is an MPI forCG.


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It can be shown that:

HCn(CG) =

C if n = even

0 if n = odd

This implies that:

HP0(CG) = C, HP1(CG) = 0.

Example 2.6.5. [9, 10, 11] Letg be a Lie algebra andU(g) be its universal enveloping

algebra (cf. Example 1.1.2). For any character : U(g) C, one can show that(, 1)

is an MPI for U(g), and we have:

HP0(U(g)) =i0

HLie2i (g,C), HP1(U(g)) =

i0

HLie2i+1(g,C).

Proof. [9, 10, 11, 12] One shows that the antisymmetrization map

A :ng U(g)n,

A(x1 ... xn) = (

Sn

sign()(x(1),...,x(n)))/n!,


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induces an quasi-isomorphism between the following two bicomplexes;

......

......

...

4 g dLie 3 g dLie 2 g dLie 1 g dLie C

0

0 b 0

3 g dLie 2 g dLie 1 g dLie C0

0 0

2 gdLie

1 gdLie C

0 0

1 g dLie C

0

C

......

......

...

U(g)4B

U(g)3B

U(g)2B

U(g)B

C

b

b b b U(g)3

B U(g)2

B U(g)

B C

b

b b U(g)2

B U(g)

B C

b

b

U(g) B Cb

C

The rows in the first bicomplex are the Chevalley-Eilenberg complex

C goo 2 gdLieoo 3 gdLieoo ...dLieoo


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dLie(x1 ... xn) = (n

i=1(1)i+1(xi)x1 ... xi ... xn)

+ (i


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We also have:

HC1(H) = Ker(B) Ker(b)Im(b) .

In the special case when = 1 we will have:

HC1(H) = Ker(B) Ker(b) = Ker( + ) P(H),

where P(H) is the space of primitive elemnts of H, i.e. those elements h in H such

that (h) = 1 h + h 1. In the very special case when = and = 1, we have:

HC1(H) = Ker() P(H) = P(H).

2.7 Dual Hopf cyclic cohomology

In [25] a dual Hopf cyclic theory for Hopf algebras is introduced. Their motivations

include the study of the coactions of Hopf algebras, and the fact that for group

algebras and in general for Hopf algebras with a normalized Haar integral the Hopf

cyclic cohomology is trivial [12]. They define a cyclic module for Hopf algebras wich

is a dual of the cocyclic module introduced in [9, 10, 11] by Connes and Moscovici

(cf. Section 2.6). In [25] they compute their dual theory for group algebras and some

quantum groups. In this subsection we, shortly, review this dual theory.

Let (H,m,, , , S ) be a Hopf algebra over a commutative ring k, be a

nonzero grouplike element ofH and : H k be a character for H. The pair (, )

is called a modular pair if () = 1, and a modular pair in involution, in the dual

sense, if

S2 = id, (2.7.12)where

S(h) = (h) (h(2))S(h(1)). (2.7.13)


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n(h1, h2,...,hn) = (h(2)n )

S(h

(1)1 h

(1)2 ...h

(1)n ), h

(2)1 ,...,h

(2)n1

.

The cyclic homology obtained from this cyclic module is denoted by HC(,) (H), orin the perodic case by HP(,) (H).Example 2.7.1. (Compare with Example 2.6.4) Let G be a discrete group and kG

be its group algebra over k. . It is obvious that kGn can be identified with kGn, the

free k module generated by Gn. One can check that

i(g1,...,gn) =

(g2,...,gn) if i = 0

(g1,...,gigi+1, ...gn) 1 i < n

(g1,...,gn1) if i = n

i(g1,...,gn) =

(1, g1,...,gn) if i = 0

(g1,...,gi, 1, gi+1, ...gn) 1 i n 1

(g1,...,gn1, gn, 1) if i = n

(g1, g2,...,gn) = ((g1g2...gn)1, g1,...,gn1),

and

HP(,1)

n

(kG) =

i0

H2i(G; k) n = 0

i0

H2i+1(G; k) n = 1

Example 2.7.2. (Compare with Example 2.6.5) Let g be a Lie algebra over k and

U(g) be its enveloping algebra. For any group like element the pair (, ) is an MPI

for U(g). One can prove that:

HC(,)n (U(g)) = k0 H

Lien2k(g; k).


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and a left H-comodule structure on M:

M H M, m m(1) m(0).

The left action and coaction are supposed to satisfy the Yetter-Drinfeld compatibility

condition:

(hm)(1) (hm)(0) = h(1)m(1)S(h

(3)) h(2)m(0),

for allh H and m M.

Definition 2.8.2. A right-left Yetter-Drinfeld H-module is a rightH-module and left

H-comodule M such that

(mh)(1) (mh)(0) = S1(h(3))m(1)h

(1) m(0)h(2),

for allh H and m M.

There are of course analogous definitions for left-right, and right-right Yetter-

Drinfeld modules.

The category HHYD of all left-left Yetter-drinfeld H-modules is the center of

the monoidal category H Mod. Recall that the (left) center ZC of a monoidal

category is a category whose objects are pairs (X, X,), where X is an object of C

and X, : X X is a natural isomorphism satisfying certain compatibility

conditions with the associativity and unit constraints of C. It can be shown that the

center of a monoidal category is a braided monoidal category, and: [22]

Z(H Mod) =HH YD.

Example 2.8.1. Let H = kG be the group algebra of a discrete group G. A left


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kG-comodule is simply a G-graded vector space

M =gG

Mg

where the coaction M kG M is defined by m g m, for all m in Mg. An

action kG M M, (g, m) gm, defines a Yetter-Drinfeld module structure on

M iff for all g, h G and m Mg, hm Mhgh1 .

2.9 Stable anti Yetter-Drinfeld modules

Stable anti Yetter-Drinfeld modules were introduced in [18]. Motivation was to find

a right definition for Hopf cyclic cohomology with coefficients [19] (cf. Section 2.10).

This class of H-modules is the right choice for the coefficients in Hopf cyclic coho-

mology. It turns out that one dimensional stable anti Yetter-Drinfeld modules are

exactly modular pairs in involution.

Definition 2.9.1. A left-left anti-Yetter-Drinfeld H-module is a left H-module and

left H-comodule M such that

(hm)(1) (hm)(0) = h(1)m(1)S

1(h(3)) h(2)m(0), (2.9.14)

for allh H and m M. We say that M is stable if in addition we have

m(1)m(0) = m,

for allm M.

Definition 2.9.2. A right-left anti-Yetter-Drinfeld H-module is a right H-module

and left H-comodule M such that

(mh)(1) (mh)(0) = S(h(3))m(1)h

(1) m(0)h(2), (2.9.15)


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43

for allh H and m M. We say that M is stable if in addition we have

m(0)m(1) = m,

for allm M.

Remark 2.9.1. Notice that if one changes S1 to S in the formula (2.9.14), or, S

to S1 in (2.9.15), one gets the relations for left-left and right-left Yetter-Drinfeld

modules as in Definitions (2.8.1) amd (2.8.2). Therefore if S

2

= id the the notionsof Yetter-Drinfeld and anti Yetter-Drinfeld modules coincide.

There are of course analogous definitions for left-right and right-right stable

anti-Yetter-Drinfeld (SAYD) modules [18].

The following lemma shows that 1-dimensional right-left SAYD modules corre-

spond to Connes-Moscovicis modular pairs in involution:

Lemma 2.9.1. [18] There is a one-one correspondence between modular pairs in

involution (MPI) (, ) onH and right-left SAYD module structure onM = k, defined

by

rh = (h)r, r r,

for allh H and r k. We denote this module by M =k.

Technically, we want to show that the two conditions

S2(h) = h1, h H, (2.9.16)and

(rh)(1) (rh)(0) = S(h(3))r(1)h

(1) r(0)h(2), h H, r k, (2.9.17)

are equivalent.


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Proof. (2.9.16) = (2.9.17): we have

(h)1 = S((h)) = S(S(h(1))h(2))= S(h(2))S2(h(1)) = (h(2))S(h(3))S2(h(1))(2.9.16)

= (h(2))S(h(3))h(1)1

r(h) = r(h(2))S(h(3))h(1)

(rh)(1) (rh)(0) = S(h(3))r(1)h

(1) r(0)h(2).

(2.9.17) = (2.9.16):

S2(h) = S(S(h))) = S((h

(1))S(h(2)))

= (h(1))S(S(h(2))) = (h(1))((S(h(2)))(1))S((S(h(2)))(2))= (h(1))(S(h(2)(2)))S(S(h(2)(1)))

= (h(1))(S(h(3)))S2(h(2)) = (S(h(3)))S2(h(2))(h(1))

(2.9.17)= (S(h(3)))S2(h(2))S(h(1)(3))h(1)(1)1(h(1)(2))

= (S(h(5)))S2(h(4))S(h(3))h(1)1(h(2))

= (S(h(5)))S(h(3)S(h(4)))h(1)1(h(2))

= (S(h(4)))(h(3))h(1)1(h(2))

= (S(h(3)))h(1)1(h(2))

= (h(2)S(h(3)))h(1)1

= (h(2))h(1)1

= h1.


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45

Example 2.9.1. The above lemma gives all one-dimensional right-left SAYD H-

modules as modular pairs in involution.

Example 2.9.2. In the special case where S2 = id (e.g., whenH is (co)commutative),

(, 1) is an MPI.

2.10 Hopf cyclic cohomology with coefficients

Hopf cyclic cohomology was generalized to Hopf cyclic cohomology with coefficients in

[18, 19] by introducing the notion ofstable anti Yetter-Drinfeld modules (SAYD). They

assigned a cocyclic module to a triple (H, C, M) where C is a H-module coalgebra

and M is a SAYD H-module:

Theorem 2.10.1. [18, 19] Let H be a Hopf algebra, C an H-module coalgebra and

M a right-left SAYDH-module. Then the complex CnH(C, M) := MHC(n+1) with

the following faces, degeneracies and cyclic maps, defines a cocyclic module attached

to the triple (H, C, M):

i(m H c0 cn1) =

m H c0 c(1)i c

(2)i cn1 0 i < n

m(0) H c(2)0 c1 cn1 m(1)c(1)0 i = n

i(m H c0 cn+1) = m H c0 (ci+1) cn+1, 0 i n

n(m H c0 cn) = m(0) H c1 cn m(1)c0.


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Example 2.10.1. Consider C = H as a H-module coalgebra via mH, and M =k

as in Lemma (2.9.1). Then the above cocyclic module reduces to the cocyclic module

of Connes and Moscovici mentioned in Theorem (2.6.1).

Proof. First we notice that the map f : H(n+1) H(n+1), defined by

f(h0

hn

) := h(1)

0 S(h

(2)

0)(h

1 h

n)

= h(1)0 S(h

(2)0 )

(1)h1 S(h(2)0 )

(n)hn,

= h(1)0 S(h

(n+1)0 )h1 S(h

(2)0 )hn,

defines an H-module isomorphism, where H(n+1), on the left, is considered as an H-

module via diagonal action and H(n+1), on the right, is considered as an H-module

via multiplication by the first term. The inverse of f is defined by

f1(h0 hn) := h(1)0 h

(2)0 (h1 hn)

= h(1)0 h

(2)(1)0 h1 h

(2)(n)0 hn,

= h(1)0 h

(2)0 h1 h

(n+1)0 hn.

For any M, the map f induces the isomorphism

f : M H H(n+1) = M H H(n+1) = (M H H) H(n).If we combine this map with the natural isomorphism M H H = M, we have the

isomorphism

: M H H(n+1) = M H(n),

(m H h0 hn) = mh(1)0 S(h

(n+1)0 )h1 S(h

(2)0 )hn. (2.10.18)


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It is also easy to see that 1 : M H(n) = M H H(n+1), composed of the

natural isomorphism M = M H H, and f1, is defined by:

1(m h1 hn) = m H 1 h1 hn.

Now if M =k then : k H H(n+1) = k H(n) = H(n) is defined by

(r H h0 hn) = r(h(1)0 )S(h

(n+1)0 )h1 S(h

(2)0 )hn

= rS(h(n)0 )h1 S(h(1)0 )hn= rS(h0)(h1 hn), (2.10.19)

and

1 : H(n) = k H H(n+1),

1(h1 hn) = 1k H 1 h1 hn.

Next it is easy to check that the map in (2.10.19), is an isomorphism of cocyclic

modules between the cocyclic module CnH(H, k) := k H H(n+1) defined in Theo-

rem (2.10.1), for M = k and C = H, and the Connes-Moscovicis cocyclic module

Cn(H) = Hn, n 0, defined in Theorem (2.6.1).

Example 2.10.2. More generally, for any SAYD H-module M and for C = H, the

isomorphism in (2.10.18), reduces the cocyclic module defined in Theorem (2.10.1)

to the following simpler one. The complex CnH(H, M) := M Hn with the faces,

degeneracies and cyclic maps as follows:

i(m h1 hn1) =

m 1 h1 hn1 i = 0

m h1 h(1)i h

(2)i hn1 1 i < n

m(0) h1 hn1 m(1) i = n


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i(m h1 hn+1) = m h1 (hi+1) hn+1, 0 i n

n(m h1 hn) = m(0)h(1)1 S(h

(2)1 )(h2 hn m(1)).

Definition 2.10.1. The cyclic cohomology of this cocyclic module is by definition the

Hopf cyclic cohomology of H with coefficients in M.


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Chapter 3

Braided Monoidal Categories

In this chapter we recall basic notions of braided monoidal categories and review the

notions of algebras, coalgebras, bialgebras, and Hopf algebras in those categories.

We also recall modules and comodules over a Hopf algebra in a braided monoidal

category. We provide some examples in each case.

3.1 Braided monoidal categories

Braided monoidal categories are the proper underlying context to define notions like

braided algebra, coalgebra and Hopf algebra and many more (cf. Section 3.3). When

it comes to practicing the homological algebra, one also needs the braided monoidal

category to be abelian (cf. Chapter 4). In this section we give a quick review of the

concept of braided monoidal category. In next section we provide some examples.

Basic references are [20, 21, 22, 29, 30, 31].

Definition 3.1.1. A monoidal, or tensor, category (C, , I , a , l , r) consists of a cat-

egory C, a functor : C C C (called tensor product), an object I C (called unit

object), and three natural isomorphisms, defined for all objects A, B, C, of C,

a = aA,B,C : A (B C) (A B) C,

l = lA : I A A, r = rA : A I A,

49


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50

called the associativity isomorphism, the left unit isomorphism, and the right unit

isomorphism, respectively, such that the following pentagon and triangle diagrams

commute [29, 31]:

((A B) C) D

ttiiiiiiiiiiiiiiii

**UUUUU

UUUU

UUUU

UUU

(A (B C)) D

(A B) (C D)

A ((B C) D) // A (B (C D))

(A I) B //

''NNNN

NNNN

NNN

A (I B)

wwppppppppppp

A B

The coherence theorem of Mac Lane [29] asserts that all diagrams formed by

a,l,r by tensoring and composing, commute. More precisely, it asserts that any two

natural transformations defined by a,l,r between any two functors defined by and

I are equal.

Definition 3.1.2. A braided monoidal category is a monoidal category C endowed

with a natural family of isomorphisms

A,B : A B B A,

called braiding such that for all objects A,B,C of C the following hexagon diagrams

commute :

A (B C) // (B C) A

a1

((QQQQ

QQQQ

QQQQ

(A B) C

a166mmmmmmmmmmmm

id

((QQQQ

QQQQ

QQQQ

B (C A)

(B A) C a1

// B (A C)

id66mmmmmmmmmmmm


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51

(A B) C // C (A B)

a((QQ

QQQQ

QQQQQQ

A (B C)

a

66mmmmmmmmmmmm

id

((QQQQ

QQQQ

QQQQ

(C A) B

A (C B) a // (A C) B

id66mmmmmmmmmmmm

Definition 3.1.3. A braiding is called a symmetry if we have

A,B B,A = idBA,

for all objects A andB ofC. Sometimes we just write 2 = id to signify the symmetry

condition. A symmetric monoidal category is a monoidal category endowed with a

symmetry.

Definition 3.1.4. A monoidal category C is called strict if its associativity and unit

isomorphisms, a,l,r, are equalities, i.e., for all objects A,B,C of C,

(A B) C = A (B C),

I A = A I = A.

By a theorem of Mac Lane [29] (cf. also [31]), any braided monoidal abelian

category is monoidal equivalentto a braided strict monoidal abelian category in which

the hexagon commuting diagrams, in Definition (3.1.2), are reduced to the following

equalities:

A,BC = (idB A,C)(A,B idC),

AB,C = (A,C idB)(idA B,C),

for all objects A,B,C of C.

Here we, roughly, review the procedure of strictificationby recalling the notion

of monoidal equivalence. We show how one can go back and forth between a given


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52

non-strict category, C, and its strictified version, Cst. This guarantees that one always

can work in a strict braided monoidal category without losing any information [22,

29, 30, 31].

Let us first recall that a monoidal functor between two monoidal category

(C, , I , a , l , r) and (C, , I, a, l, r) consists of following data [29] :

(i) A functor F : C C,

(ii)A morphism Ftensor(x, y) : F(x) F(y) F(x y) in C, for any two objects x

and y in C,

(iii) A morphism Funit : I F(I) in C,

all subject to certain commutative diagrams, which guarantee that the functor F pre-

serves the monoidal structure. A monoidal functor (F, Ftensor, Funit), from C to C,

is called a monoidal equivalence between C and C, if there exist a monoidal functor

(G, Gtensor, Gunit) : C C,

such that F G and GF are naturally isomorphic to the identity. In such a case, we

say, C and C are monoidal equivalent or gauge equivalent.

Starting from a braided monoidal (abelian) category (C, , I , a , l , r , ) one can

produce a strict braided monoidal (abelian) category (Cst, st, Ist, st) monoidal

equivalent to C [29]. Cst is called the strictification of C.

Roughly speaking, objects of Cst are all finite strings of objects of C shown by

[A1,...,An], where n = 1, 2,..., and if n > 1 then Ai = I for all 1 i n. We shall

denote [A1,...,An] by Ast. Let us define a map

LR : Cst C

LR[A1,...,An] := (A1 (A2 (A3...(An1 An)))...).

The object (A1(A2(A3...(An1An)))...))) is called the left to rightrepresentation


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of the string [A1,...,An] in C.

The morphisms between two strings Ast = [A1,...,An] and Bst = [B1,...,Bm]

ofCst are defined to be the morphisms between LR[A1,...,An] and LR[B1,...,Bm] in

C, with compositions just as in C.

The tensor product st in Cst is defined as follows.

[I] st Ast = Ast st [I] = Ast, Ast Cst,

and for any other two strings [A1,...,An] and [B1,...,Bm] is just their concatenation

:

[A1,...,An] st [B1,...,Bm] := [A1,...,An, B1,...,Bm].

By above definition it is clear that [I] is the unit object of Cst, i.e., Ist := [I].

Let f : [A1,...,An] [B1,...,Bm] and g : [C1,...,Cs] [D1,...,Dt] be two

morphisms in Cst. As we mentioned above they are actually the morphisms f :

LR[A1,...,An] LR[B1,...,Bm] and g : LR[C1,...,Cs] LR[D1,...,Dt] in C. The

tensor product

f st g : [A1,...,An, C1,...,Cs] [B1,...,Bm, D1,...,Dt],

is defined by:

LR[A1,...,An, C1,...,Cs] = LR[A1,...,An] LR[C1,...,Cs]

fg LR[B1,...,Bm] LR[D1,...,Dt] = LR[B1,...,Bm, D1,...,Dt].

The first and last isomorphisms in the above formula are the natural isomorphisms

in C generated by the associativity map a.

It is easy to see that Cst is a strict braided monoidal category where ast, lst and

rst

are equalities. Following the definition of morphisms in Cst

, for any two strings


61/176


62/176

55

for all objects A and morphisms f in C. The morphism Gtensor(A, B) : G(A) st

G(B) G(A B) is just the identity map idAB : A B A B, and Gunit :

G(I) = [I] [I] is also the identity, id[I] = idI.

Remark 3.1.1. We remark that if the original category is symmetric then its stric-

tification is symmetric as well. This plays an important role in our approach. In

fact, using this result, we can safely assume that our symmetric monoidal categories

are strict and symmetric. Working with strict categories drastically simplifies the

formalism and that is what we shall do in this thesis.

3.2 Examples of braided monoidal categories

Example 3.2.1. The category V ectk of vector spaces over a field k with the usual

tensor product k and with the braiding = usual flip, is a non-strict symmetric

braided monoidal abelian category. More generaly for any unital ring R, the category

of (left) R-modules is a symmetric monoidal abelian category.

Example 3.2.2. The category (Set, , I) with the Cartesian product as its tensor,

the one point setI = {1} as the unit, and with the usual flip map defined by, A,B(a

b) := ba, for any two sets A andB, as the braiding map, is a non-abelian, non-strict,

symmetric braided monoidal category.

Most of the categories that we mention or work with in this thesis are abelian.

Here we have to recall the notion of braid groups which we need to define the braid

category in the next example. The braid group Bn, for n 3, is defined to be the

group generated by n 1 generators bi, 1 i n 1, called braids, with relations:

bibj = bjbi, |i j| = 1,

bibi+1bi = bi+1bibi+1, 1 i n 2.


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56

The braid group B2 is defined to be the infinite cyclic group generated by one gener-

ator b1, and B1 is defined to be the identity group.

It is helpful and instructive to have the following intuition about the braids,

for a fixed n. Suppose there are n identical strings hanging down from the ceiling

with their top ends all tightened up to an straight line (e.g., a bar) attached to the

ceiling. Let assign to those end points numbers 1 to n, in order. The braid bi can be

visualized by crossing the ith string over the (i + 1)th string, leaving all other strings

untouched, and keeping all the bottom end points fixed. It is important to notice

that, the braid bi actually changes the position of the ith string with the position of

the (i + 1)th string. The braid b1i then would be produce in a similar way but by

crossing the (i + 1)th string over the ith string. It is again important to notice that,

the braid b1i , like bi, changes the position of the ith string with the position of the

(i + 1)th string, but in a reverse fashion. The identity braid idn is to simply leave all

the strings hanging down, untouched. The multiplication bibj of two braids means to

do the procedure bi followed by bj , with the change of positions understood. [23, 29]

Example 3.2.3. (Braid category [23, 29])

The braid categoryB is defined as follows. Objects ofB are natural numbers, 0, 1, 2,....

For any n 1 in B, HomB(n, n) := Bn. The only morphism from 0 to 0 is defined to

be the identity, and is called the empty braid. There is no morphism between n and

m if n = m. Tensor product of two objects n and m is defined to be n + m.

Having the above intuition in mind, n + m means to simply put the set of m

strings beside the set of n strings. Now it is easy to see that, any pairs of braids

f : n n and g : m m, uniquely define a braid called f g : n + m n + m, by

just putting them side by side. One can also define the braiding map

n,m : n + m m + n,

by crossing the whole set of n strings over the set of m strings.


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The braid category B defined as above, is a braided monoidal category, with the

empty braid as its unit. It is clearly strict, but is not symmetric, i.e.,

m,nn,m = idn+m

.

Example 3.2.4. Let (H, R = R1 R2) be a quasitriangular Hopf algebra and H-

Mod be the category of all left H-modules. ThenH-Mod is a braided monoidal abeliancategory. It is symmetric if and only if R1 = R2R1 [31]. In this latter case, (H, R)

is actually called a triangular Hopf algebra. The monoidal structure on H-Mod is

defined by

h (v w) = h(1) v h(2) w,

and the braiding map VW acts by

VW(v w) := (R2 w R1 v),

for anyV and W in C, where denotes the action of H.

Example 3.2.5. In a dual manner if we consider a co-quasitriangular Hopf algebra

(H, R), then the category of left H-comodules is a braided monoidal abelian category.

Example 3.2.6. As a very special case of Example (3.2.4), let H = CZ2 with thenon-trivial quasitriangular structure R = R1 R2 defined by

R := (1

2)(1 1 + 1 g + g 1 g g),

where g is the generator of the cyclic group Z2. Then the category C = Z2- Mod is the

category of super vector spaces with even morphisms [31]. The braiding map VW


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for anyV = V0 V1 and W = W0 W1 in C acts as below:

VW(v w) = (1)|v||w| (w v). (3.2.1)

This is a symmetric, braided, monoidal abelian category, with the unitC = C 0.

There is also a category of differential graded (DG) super vector spaces whose

objects are super complexes V0d

dV1, and its morphisms are even chain maps. It is

a braided monoidal category with the same braiding map as (3.2.1).

Example 3.2.7. One can extend Example (3.2.6) to CZn for any n > 2 with non-

trivial quasitriangular structure onCZn given by:

R = (1/n)n1

a,b=0

e(2iab)/nga gb,

where g is the generator of Zn. Notice that the braided monoidal abelian category

CZn-Mod is not symmetric for any n > 2. Therefore, this provides a good source of

non-symmetric braided monoidal categories[31].

Example 3.2.8. LetH be a Hopf algebra over a field k with comultiplication h =

h(1) h(2) and a bijective antipode S. We recall from Definition (2.8.1), that a left-

left Yetter-Drinfeld (YD) H-module consists of a vector space V, a left H-module

structure on V [33, 40]:

H V V, h v hv,

and a left H-comodule structure on V:

V H V, v v(1) v(0).

The left action and coaction are supposed to satisfy the Yetter-Drinfeld (YD) com-


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59

patibility condition:

(hv)(1) (hv)(0) = h(1)v(1)S(h

(3)) h(2)v(0),

for all h H and v V. The category of all YD H-modules is called the Yetter-

Drinfeld category of H, and is usually denoted by HHYD. It is a braided monoidal

abelian category with the braiding map:

VW(v w) = v(1)w v(0).

This category is in general not symmetric. In fact the inverse of the braiding is given

by :

1VW(w v) = v(0) S1(v(1))w.

3.3 Braided algebras, coalgebras and Hopfalgebras

In this section we recall the notions of algebra, coalgebra, bialgebra and Hopf algebra

in a braided monoidal category. By a braided algebra we mean an algebra in a braided

monoidal category. We use the similar convention for braided coalgebras, bialgebras

and Hopf algebras. We recall some basic properties of braided Hopf algebras and

provide some important examples of them. We also give the definitions ofH-modules

and comodules for a Hopf algebra H in a braided monoidal category C. We fix a

strict braided monoidal category (C, , I , ) throughout this section.

Definition 3.3.1. An algebra (H,m,) in C consists of an object H objC and

morphisms m : H H H and : I H called multiplication and unit maps

satisfying the commutation relations:


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H H m // H

H H H

idm

OO

mid // H H

m

OO

associativity

H Hm // H H H

moo

I H = H

id

OO

id

55llllllllllllllllH = H I

id

OOid

iiSSSSSSSSSSSSSSSS

unit

Definition 3.3.2. A coalgebra in C is a triple (H, , ) consisting of an object H

objC and morphisms : H H H and : H I called comultiplication and

counit maps satisfying the commutation relations:

H //

H H

id

H H id // H H Hcoassociativity

H H

id

H //oo

id ))SSSS

SSSS

SSSS

SSSS

id

uullllllllllllllll H H

id

I H = H H = H Icounit

Definition 3.3.3. A bialgebra (H,m,, , ) in C is an algebra and a coalgebra si-

multaneously, satisfying the compatibility conditions:

H Hm //

H H

H H H Hidid

// H H H H

mm

OO


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61

H // H H

I I = I

OO

55kkkkkkkkkkkkkkk

H Hm //

H

uulll

llllllllllll

I I = I

Iid //

I

H

77oooooooooooooo

Definition 3.3.4. A Hopf algebra (H,m,, , , S ) in C consists of a bialgebra plus

a morphism S : H H called the antipode map satisfying the relations:

m(S id) = m(id S) = . (3.3.2)

The relations (3.3.2), in terms of commutative diagrams, are as follows:

H HSid // H H

m

((QQQQ

QQQQ

QQQQ

QQQ

H //

OO

I // H

H HidS // H H

m

66mmmmmmmmmmmmmmm

The following proposition shows that the standard properties of Hopf algebras

hold for braided Hopf algebras.

Proposition 3.3.1. If (H,m,, , , S ) is a braided Hopf algebra (in C), then:

Sm = m(S S) = m(S S), (3.3.3)


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62

S = ,

S = (S S) = (S S), (3.3.4)

S = .

Example 3.3.1. A Hopf algebra in the category V ectk is a Hopf algebra in the stan-

dard sense.

Example 3.3.2. For any V in HHYD of Example (3.2.8), the tensor algebra T(V) is

a braided Hopf algebra in HHYD. Its comultiplication, counit, and antipode are defined

by (v) = 1 v + v 1, (v) = 0, and S(v) = v, for all v in V.

Hopf algebras in HHYD are called Yetter-Drinfeld Hopf algebras. They are stud-

ied in [36].

We recall from Example (3.2.6) that a super vector space V = V0 V1 is an

object ofZ2-Mod. The degree of a homogeneous element a in V will be denoted by

|a|.

Example 3.3.3. A super Hopf algebra is a Hopf algebra H inZ2-Mod. Thus a super

Hopf algebra is a super vector space H = H0 H1 which is simultaneously a super

algebra and a super coalgebra, the two structures are compatible, and H also has a

degree preserving antipode map. More precisely: H is aZ2-graded algebra (or a super algebra), i.e.,

|ab| = |a| + |b|,

for all homogeneous elements a and b of H.

H is a super coalgebra, i.e.,

|a| = |a(1)| + |a(2)|,


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for all homogeneous elements a of H.

H is a super bialgebra, i.e., we have the following compatibility condition between

the above two structures:

(ab) = (ab)(1) (ab)(2) = (1)|a(2)||b(1)| (a(1)b(1) a(2)b(2)) (3.3.5)

Furthermore, there is an even map S : H H, the antipode, such that:

S(h(1))h(2) = h(1)S(h(2)) = (h)1, for all h in H.

Remark 3.3.1. We emphasize that the compatibility condition (3.3.5), shows that a

super Hopf algebra is, in general, not a Hopf algebra in the category of vector spaces.

Proposition 3.3.2. One easily checks that formulas (3.3.3) and (3.3.4)of Proposition

(3.3.1) reduce to the following ones for a super Hopf algebra H:

S(ab) = (1)|a||b| S(b)S(a),

S(a) = S(a)(1) S(a)(2) = (1)|a(1)||a(2)| S(a(2)) S(a(1)).

Example 3.3.4. [31] Recall from Example (3.2.4) that for any quasitriangular Hopf

algebra (H, R = R1 R2), the category, H-Mod, of all (left) H-modules is a braided

monoidal abelian category. One can prove that, every (H, R) can be turned into a

braided Hopf algebra H in H-Mod as follows. As a vector space H = H, with H-

module structure given by conjugation

a h = a(1)hS(a(2)).

It has the same multiplication, unit, and counit as H, but the comultiplication and


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antipode are replaced by:

(h) = h(1) h(2) = h(1)S(R2) R1 h(2),

S(h) = R2S(R1 h).

Definition 3.3.5. Let H be a braided Hopf algebra in C. A right H-module is an

object M in C equipped with a morphism M : M H M, called H action, such

that:()(idM mH) = ()( idH),

()(idM H) = idM.

Definition 3.3.6. A left H-comodule is an object M in C equipped with a morphism

M : M H M, called H coaction, such that:

(H idM)() = (idM )(),

(H idM)() = (idH H)(H,M)() = idM.

Left H-modules and right H-comodules can be defined in a similar manner.


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As a special case of the theory that we provide, we will define a Hopf cyclic

cohomology for a (differential graded) super Hopf algebra and relate it to the co-

homology of super Lie algebras by considering the enveloping algebra of a super Lie

algebra. We also provide a Hopf cyclic theory for quasitriangular quasi-Hopf algebras.

We should mention that the cyclic cohomology of (ribbon)-algebras in braided

monoidal abelian categories has been introduced and studied in [2], motivated by

non-associative geometry.

Throughout this chapter we shall use the following conventions to denote ob-

jects and morphisms of a braided monoidal category (C, , I , ):

An for An,

1 for id, e.g. we write 1A or just 1 for idA,

(f, g) for (f g),

idn or just 1n for idAn,

1A,B for 1A 1B ,

for A,A.

For example instead of writing H mH = (mHmH)(idHH,HidH)(HH),

which expresses the fact that the comultiplication of a Hopf algebra is an anti-

algebra map, we just write m = (m, m)(1, , 1)(, ); or instead of U,UU =

(idU U,U)(U,U idU), we simply write U,U2 = (1, )(, 1) when there is no

chance of confusion, and so on.

4.1 The cocyclic module of a braided triple

(H, C, M)

In this section we extend the notion of a stable anti-Yetter-Drinfeld (SAYD) module

[18, 19] (cf. Section 2.9) to braided monoidal categories and assign a cocyclic object

to a braided triple (H, C, M) in a symmetric braided monoidal abelian category C.


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In the last section of this chapter we treat the general non-symmetric case which is

much more subtle. Recall from section (2.3) that, by definition, in a para-cocyclic

object all axioms of a cocyclic object are satisfied except the relations n+1n = id.

Given a para-cocyclic object Xn, n 0, in an abelian category, we can always define

a cocyclic object by considering

Xn

:= ker(id n+1n ), (4.1.1)

and restricting the faces, degeneracies, and cyclic operators to these subspaces. For

general notion of (co)cyclic objects and (co)cyclic modules we also refer to sections

(2.2)-(2.4).

We fix a strict, braided monoidal category (C, , I , ), and a Hopf algebra

(H,m,, , , S ) in C. For the following definitions C need not be symmetric or

additive.

Definition 4.1.1. (compare with Definition 2.9.2) A right-left braided stable anti-

Yetter-Drinfeld H-module in C is an object M in C such that:

(i) M is a right H-module via an action M : M H M,

(ii) M is a left H-comodule via a coaction M : M H M,

(iii) M satisfies the braided anti-Yetter-Drinfeld condition, i.e. :

()() = [(m)(S m) ][(H2,H

idM

idH

)(idH2

M,H

idH

)

(idH2 idM H,H)(idH M,H idH2)][ 2]. (4.1.2)

(iv) M is stable, i.e. :

()(H,M)() = idM.

Remark 4.1.1. To deal with large expressions like (4.1.2) we break them into two

lines.


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Definition 4.1.2. A quadruple (C, C, C, C) is called a left (braided) H-module-

coalgebra in C if (C, C, C) is a coalgebra in C, and C is a left H-module via an

action C : H C C such that C is a coalgebra map in C i.e. we have:

CC = (C C)(idH H,C idC)(H C),

CC = H C.

Definition 4.1.3. Let (C, C) be a left H-module. The diagonal action of H onCn+1 := C(n+1) is defined by:

Cn+1 : H Cn+1 Cn+1

Cn+1 := (C, C,...,C) n+1 times

(F(H,C))(nH 1Cn+1),

where,F(H,C) :=

ni=1

(idHi , H,C, H,C,...,H,C n+1i times

, idCi).

Now we are going to associate a para-cocyclic object to any triple (H,C,M),

where H is a Hopf algebra, C is an H-module coalgebra and M is a SAYD H-

module, all in a symmetric monoidal category C. Notice that C need not be additive.

For n 0, let

Cn = Cn(C, M) := M Cn+1.

We define faces i : Cn1 Cn, degeneracies i : C

n+1 Cn and cyclic maps

n : Cn Cn by:


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i =

(1M, 1Ci , C, 1Cni1) 0 i < n

(1M, C,Cn)(1M, C, 1Cn)(H,M, 1Cn+1)(M, C, 1Cn1) i = n

i = (1M, 1Ci+1 , C, 1Cni), 0 i n

n = (1M, C,Cn)(1M, C, 1Cn)(H,M, 1Cn+1)(M, 1Cn+1)

Proposition 4.1.1. If C is a symmetric monoidal category, then (C, i, i, ) is a

para-cocyclic object in C.

Remark 4.1.2. Note that by just using the faces we can construct the Hochschild

cohomology of the coalgebra C with coefficients in the C-bi-comodule M C with the

left and right coactions defined as:

lM,C = (C, 1M, 1C)(1M, M,C, 1C)(M, C) : (M C) C (M C),

rM,C = (1M, C) : (M C) (M C) C.

For instance in a special case, if we put M = I with M = (H, 1I) the trivial

coaction, since I C = C, we get the Hochschild cohomology of the coalgebra C withthe coefficients in the C-bi-comodule C with the left and right coaction defined by

C.

Now let us assume in addition that C is an abelian category. Then we can form

the balanced tensor products

CnH = CnH(C, M) := M H C

n+1, n 0,


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with induced faces, degeneracies and cyclic maps denoted by the same letters i, i

and n.

Remark 4.1.3. Note that by just using the faces we can construct the Hochschild

cohomology of the coalgebra C with coefficients in the C-bi-comodule M H C with

the left and right coactions induced by l and r defined in Remark (4.1.2).

For example, as a special case, if we put (C, C) = (H, H), since MH H =

M, we get the Hochschild cohomology of the coalgebra H with coefficients in the

H-bi-comodule M with the left coaction defined by M and the right trivial coaction

(1M, H) : M M H.

Now we proceed to the main theorem of this thesis.

Theorem 4.1.1. If C is a symmetric monoidal abelian category, then (CH, i, i ,

) is a cocyclic object in C.

Proof. The most difficult part is to show that the cyclic map n is well defined

on CnH(C, M) for all n. For this, we use the following diagram:

M H Cn+1fM,Cn+1

M Cn+1coker(f

M,Cn+1)

M H Cn+1

M H Cn+1fM,Cn+1

M Cn+1coker(f

M,Cn+1)

M H

Cn+1

where fM,Cn+1 = ((M 1Cn+1) (1M Cn+1)). So we have to show that

coker(fM,Cn+1) () (fM,Cn+1) = 0,

i.e.

coker(fM,Cn+1) [(1M, C,Cn)(1M, C, 1Cn)(H,M, 1Cn+1)

(M, 1Cn+1)] (M 1Cn+1 1M Cn+1) = 0. (4.1.3)


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It is not hard to see that the equality (4.1.3) is equivalent to:

coker(fM,H2) [ [(1M,H, mH)(1M, H2,H)(H,M, 1H2)(M, H)]

[(1M, H, 1H)(H,M)(M)(M)] ] = 0,

i.e. if we put:

= [(1M,H, mH)(1M, H2,H)(H,M, 1H2)(M, H)],

and

= [(1M, H, 1H)(H,M)(M)(M)],

then:

coker(fM,H2) ( ) = 0.

To prove this equality, we will define an isomorphism

: M H H2 M H H2 = M Hand will show that,

() coker(fM,H2) ( ) = 0.More explicitly:

Step 1:

We claim that defined as below is a H-linear isomorphism, where the domain

H2 is considered as a H-module via diagonal action (H2), and the codomain H2 is

considered as a H-module via multiplication in the first factor (H2

).

:= (1H, mH)(1H, SH, 1H))(H, 1H) : H2 H2.


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It is easy to see that is an isomorphism and in fact its inverse map is

1 := (1, m)(, 1).

To see that is H-linear, we have to show that the following diagram commutes:

H2 // B

H H2 1//

H2

OO

H H2

H2

OO

i.e

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Arash Pourkia- Hopf Cyclic Cohomology In Braided Monoidal Categories