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Malaysian Journal of Mathematical Sciences 3(2): 147-159 (2009)

On Invariants of Complex Filiform Leibniz Algebras

Isamiddin S.Rakhimov

Department of Mathematics, Faculty of Science and

Institute for MathematicalResearch, Universiti Putra

Malaysia,

43400 UPM Serdang, Selangor, Malaysia

E-mail: is a m i d d i n @ s c ie n ce . up m . e du . my

ABSTRACT

The paper intends to survey the subject of the title for an audience of

mathematicians not necessarily expert in the areas of commutative algebra and

algebraic geometry. It is devoted to the isomorphism invariants of low

dimensional Complex filiform Leibniz Algebras under the action of the

general linear group (transport of structure). The description of the field

of invariant rational functions in low dimensional cases ispresented.

Keywords: field of invariant function, group action, Leibniz algebra,

orbit.2000 MSC: 17A32, 17B30,13A50.

1. Group Action. Orbits.

INTRODUCTION

Let Gbe a group and X be a nonempty set.

Definition 1.1. An action of the group G on X is a function : G X Xwith:

(i) (e,x) =x , where e is the unit element ofGand

x X

(ii) (g,(h,x)) =(gh,x) , forany

g, h Gand

x X .

We shortly writegx for(g,x) , and call X a G-set.

LetKbe a field K[X] ={f : X K} be the set of all functions on X. It is an

algebra over K with respect to point wise addition, multiplicationand multiplication by scalar operations.
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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Definition 1.2. A function f : X K is said to be invariant if

f(gx) =f(x) for any gG and xX. The set of invariant functions on X,denoted byK[X]

G, is a subalgebra ofK[X].

O(x) ={yX| gG such thaty =gx} is called the orbit of the elementx

under the action ofG.

Theorem 1.3. Let X be a G-set. Define a relation on X by for allx,y X , x y if and only ifgx =y for some g

G .

Then is an equivalent relation onX.

It is evident that the equivalence classes with respect to equivalent relation

are the same with the orbits of the action of the groupG.

If we consider the factor set X/ then the invariant functions arefunctions on X/ .

2. Affine Algebraic Variety [Hartshorne,

1977].

Definition and examples Algebra of regular function on A.A.V. Irreducibility

Field of rational function Morphism of A.A.V.

LetKbe a fixed algebraically closed field. We define affine space over

K denoted An, to be the set of all n-tuples of elements ofK. An element

PAn will be called a point, and if P = (a1,...,an) with aiK, then the aiwill be called the coordinates of P.

Let A = K[x1, ..., xn] be the polynomial ring in n variables overK. We willinterpret the elements ofA as functions from the affine n-space to K, by

defining f(P) = f(al,..,a

n), where fA and PAn. Thus if fA is a

polynomial, we can talk about the set of zeros of f, namely Z(f) = {

PAnf(P) = 0}. More generally, if T is any subset ofA, we define thezero set of T to be the common zeros of all the elements of T:

Z(T) = {P An | f(P) = 0 for all

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f T} .

148 Malaysian Journal of Mathematical Sciences

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Clearly if is the ideal of A generated by T, then Z(T) = Z()

. Furthermore, since A is a noetherian ring, any ideal has a finite setofgenerators f1,..., fr. Thus Z(T) can be expressed as the common zeros ofthe

finite set of polynomials

f1,...,fr.

Definition 2.1. A subset Y of An

is an algebraic set if there exists a

subset

T A such that Y = Z(T).

Proposition 2.2. The union of two algebraic sets is an algebraic set. The

intersection of any family of algebraic sets is an algebraic set. The empty

set and the whole space are algebraic sets.

Definition 2.3. We define the Zariski topology on An

by taking the open

subsets to be the complements of the algebraic sets. This is a topology,

because according to the proposition, the intersection of two open sets is

open, and the union of any family of open sets is open. Furthermore, the

empty set and the whole space are both open.

Definition 2.4. A nonempty subset Y of a topological space X is

irreducible if it cannot be expressed as the union Y = Y1 Y2 of twoproper subsets, each one of which is closed in Y. The empty set is notconsidered to be irreducible.

Example 2.5. A1

is irreducible, because its only proper closed subsets

are finite, yet it is infinite (becauseKis algebraically closed, hence infinite).

Example 2.6. Any nonempty open subset of an irreducible space isirreducible and dense.

Example 2.7. If Y is an irreducible subset of X, then its closure in X is alsoirreducible.

Definition 2.8. An affine algebraic variety or simply affine variety is annirreducible closed subset ofA (with the induced topology).

Now we need to explore the relationship between subsets ofAn

and ideals

inA more deeply. So for any subset Y An

, let us define the ideal of Y inAby I(Y) ={fA | f(P) = 0 for all P Y} .

Now we have a function Z that maps subsets A to algebraic sets, and a

function I which maps subsets of An

to ideals. Their properties are

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summarized in the following proposition.

Malaysian Journal of Mathematical Sciences 149

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n

n


Proposition 2.9.

(a) If T1T2 are subsets ofA, then Z(T1)Z(T2).(b) If Y1Y2 are subsets ofA , thenI(Y1)I(Y2).(c) For any two subsets Y1, Y2 ofA , we have I(Y1 Y2) = I(Y1)I(Y2).(d) For any ideal A,I(Z())=

the radical of.

(e) For any subset Y An, Z(I(Y)) = the closure of Y.

Theorem 2.10 (Hilbert's Nullstellensatz). Let K be an algebraically

closed field, let be an ideal in A = K[x1, ..., xn], and let fA be apolynomial which vanishes at all points of Z(). Then f r for someintegerr>0.

Proof. Atiyah-Macdonald [Atiyah, Macdonald, 1969] or Zariski-

Samuel

[Zariski, Samuel, (1958, 1960)].

Corollary 2.11. There is a one-to-one inclusion-reversing correspondence

between algebraic sets in An

and radical ideals (i.e., ideals which are equal

to their own radical) in A, given by YI(Y) and Z (). Furthermore,an algebraic set is irreducible if and only if its ideal is a prime ideal.

Example 2.12. Let fbe an irreducible polynomial in A = K[x,y]. Then fgenerates a prime ideal in A, since A is a unique factorization domain, sothe zero set Y = Z(f) is irreducible. It is called the affine curve definedby the equationf(x,y) = 0. Iffhas degree d, than Y is said to be a curve ofdegree d.

Example 2.13. More generally, if f is an irreducible polynomial in

A =K[x1

,,xn

] , we obtain an affine variety Y = Z(f), which is called a

surface ifn = 3, or a hypersurface ifn > 3.

Example 2.14. A maximal ideal ofA=K[x1, ..., xn] corresponds to aminimal irreducible closed subset of A

n, which must be a point, say

P = (a1

,, an

) . This shows that every maximal ideal of A is of the

form

= ( x1

a1

,, xn

an

) , for some a1,

,anK.

Example 2.15. IfK is not algebraically closed, these results do not hold.For example, ifK= R, the curvex

2+y

2+ 1 = 0 in A

2has no points.

Definition 2.16. If YAn is an affine algebraic set, we define the affinecoordinate ringK[Y] (sometimeA(Y)) of Y to beA/I(Y).

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Remark 2.17. If Y is an affine variety, then A(Y) is an integral domain.

Furthermore, A(Y) is a finitely generated K-algebra. Conversely, any

finitely


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+y

=

y

x

+yMalaysian Journal of Mathematical Sciences 151

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y


Example 1. Let

t 0 G = :tK*.

0 t

Then the orbits under this action are drawn on the picture below:

All the orbits, except for zero, are one dimensional and the closure of all

contains the zero.

The invariant regular functions are constants only:K[X]G=K.

The field of invariant rational function is generated by x :y

K(X)G =K

x .

Example 2..

t 0 G =

0 t1:tK*.

Orbits:

All generic orbits are one dimensional and closed. There are two onedimensional orbits the closure of them contains the zero.

Invariant Regular functions:K[X]G=K[xy].

Invariant Rational functions:K(X)G=K(xy

).


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Example 3. Let

1 uG = :uK

Orbits:

0 1

Invariant Regular functions:K[X]G=K[y].

Invariant Rational function:K(X)G=K(y

).

Let a group G acts on an algebraic variety X regularly. For simplicity,we assume that the base field K is algebraically closed and ofcharacteristic zero. We can define an action of G on the K-algebraK[X] of regular functions on X: (gf)(x)=f(gx). Of special interest is

the subalgebra of invariant functions which will be denoted by K[X]G.

It carries a lot of information about the orbit structure and its geometry.

The algebra of invariants was a major object of research in the last two

centuries. There are a number of natural questions in this context:

Is the invariant algebraK[X]G finitely generated as aK-algebra? If so, can one determine an explicit upper bound for the degrees of

a system of generators ofK[X]G?

Are there algorithms to calculate a system of generators and whatis their complexity?

If X is an irreducible how to describeK(X)G? WhenK(X)G=QK[X]G? Describe the orbits closures of the action ofG?

The first question is essentially HILBERT'S 14th problem, although his

formulation was more general. The answer is positive for reductivegroups by results of HILBERT, WEYL, MUMFORD, NAGATA andothers, but


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negative in general due to the famous counter example of NAGATA (in

1959).

The fourteenth Hilbert problem: The problem of description of all

linear algebraic groups for which the algebra of invariant regular functions

is finite generated as aK-algebra.

Actually, we deal with the fourth and sixth questions of the above list forvariety of non-Lie complex filiform Leibniz algebras in low dimensionalcase. In this case the field of rational invariant functions is described.

4. Complex filiform Leibniz algebrasvariety

Leibniz algebras. Nilpotent and Filiform Leibniz Algebras Variety of algebras and Subvarieties The isomorphism action GLn(C) on the variety of algebras

(transport of structure)

Main results

Let V be a vector space of dimension n over an algebraically closed field

K (charK=0). The bilinear maps VVVform a vector space

Hom(VV,V) ofdimension

n3 , which can be considered together with its

natural structure of an affine algebraic variety over K and denoted by3

Algn(K) Kn . An n -dimensional algebra L overK may be considered

as an element (L) ofAlgn

(K) via the bilinear mapping :L L Ldefining an binary algebraic operation on L : let {e1,e2 ,,en } be a basis ofthe algebra L. Then the table of multiplication ofL is represented by point

n

ij i j

ij k ij

(k) of this affine space as follow: (e ,e )

=

k=1ke ,

k

are called

structural constants of L. The linear reductive group GLn

(K) acts on

Algn

(K) by (g)(x,y) =g((g

1(x),

g

1(y))) (transport of structure).

Two algebras 1

and 2

are isomorphic if and only if they belong to the

same orbit under this action. It is clear that elements of the given orbitare isomorphic to each other algebras. The classification means to specify

the representatives of the orbits.

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ij

jk il ij lk ik lj


Definition 4.1. An algebra L over a field K is called aLeibniz algebra if it

satisfies the following Leibniz identity:

[x,[y,z]] = [[x,y],z] [[x,z],y],

where [,] denotes the multiplication inL. Let Leibn

(K) be a subvariety of

Algn

(K) consisting of all n -dimensional Leibniz algebras overK. It is

invariant under the above mentioned action of GLn(K) . As a subset of

Algn(K

)

the set Leibn(K

)

is specified by system of equations with

respect to structural constants k :

n

(l m lm +l m ) = 0l=1

It is easy to see that if the bracket in Leibniz algebra happens to be

anticommutative then it is Lie algebra. So Leibniz algebras are

noncommutative generalization of Lie algebras.

Further all algebras assumed to be over the field of complex

numbers. Let L be a Leibniz algebra. We put:

L1 =L, Lk+1 = [Lk ,

L],kN.

Definition 4.2. A Leibniz algebra L is said to be nilpotentif there exists an

integersN

,

such that L1 L2 ...Ls =

{0}.

The smallest integers for

that Ls =

0is called the nilindex ofL .

Definition 4.3. An n -dimensional Leibniz algebra L is said to befiliform if

dimLi =n

i,where 2 i n.

There are two sources to get classification of non-Lie complex filiform

Leibniz algebras. The first of them is the naturally graded non-Liefiliform Leibniz algebras and the another one is the naturally gradedfiliform Lie algebras [Gomez, Omirov, 2006]. Here we considerLeibniz algebras appearing from the naturally graded non-Lie filiformLeibniz algebras. According to the theorem presented in [Ayupov,Omirov, 2001] this class can be divided into two disjoint subclasses.

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Theorem 4.4. Any (n + 1) -dimensional complex non-Lie filiform

Leibniz algebra obtained from the naturally graded filiform Leibnizalgebras can be included in one of the following two classes of Leibniz

algebras:

a) (The first class):

[e0 ,e0 ] =e2,

[ei ,e0 ] =ei +1,

1i n 1

e ,e =e +e +...+ e +e ,[ 0 1 ] 3 3 4 4 n 1 n1 n[e

j,e

1] =

3e

j + 2 +4 ej + 3 +...+

n+1jen ,

1j n 2

(omitted products are supposed to be zero)

b) (The second class):

[e0 ,e0 ] =e2 ,

[ei ,e0 ] =ei +1 ,

[e0 ,e1 ] =3e3 +4e4 +...+nen ,

[e1,e

1] =e

n,

2 i n 1

[ej,e

1] =

3e

j + 2 +4ej +3 +...+n +1je

n,

2 j n 2

omitted products are supposed to be zero, where e0, e1,, en is

a basis.

In other words, the above theorem means that the above mentioned type

of

(n + 1) -dimensional non-Lie complex filiform Leibniz algebras canbe

represented as a disjoint union of two subsets and the algebras from the

difference classes never are isomorphic to each other. The basis in thetheorem is called adapted.

Let denote by FLeibn+1 the variety of all non-Lie complex filiform Leibnizalgebras from the first class and by SLeibn+1 the variety of all non-Liecomplex filiform Leibniz algebras from the second class. Each of them is

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invariant under isomorphism action (transport of structure) of GLn+1. Itis known [Gomez, Omirov, 2006] that the transport of structure actionof GLn+1 can be reduced to the action of subgroup of GLn+1 calledadapted transformations. The next series of theorems are devoted to thedescription


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2 2 2


of the field of invariant rational function. The generators are written as a

function of structural constantsrespect to the adapted basis.

3 ,4 ,...,n , and 3 ,4 ,...,n , with

MAIN RESULTS

For the simplification and computational purpose we establish

the following notations:

2 3 4 53

=3,

4=

4+ 2

3,

5=

5

53 ,

i=

i

i,

6

=6

+143,

i = 4, 5, 6, 7.

7

=7

423

,

Theorem 4.5. The transcendental degree of the field of invariant rational

functions C(FLeib5)G

of FLeib5 under the action of the adapted subgroup

G of the group GL5 is one and it is generated by the following function:

2

3

F=

4.

4

Theorem 4.6. The transcendental degree of the field of invariant rational

functions C(FLeib6)G

of FLeib6 under the action of the adapted subgroupG of the group GL6 is two and it is generated by the following functions:

( + 5 )F= 3 5 3 4 ,

3

F = 3 .1 2 24

5

4

Theorem 4.7. The transcendental degree of the field of invariant rationalfunctions C(FLeib7)

Gof FLeib7 under the action of the adapted subgroup

G of the group GL7 is three and it is generated by the following function:

( + 5 )F = 3 5 3 4 ,F =(

(6 +635 +93 4 +34 )

,

4

3

3 F = .1 2 2 3

4 4

3 6 4

Theorem 4.8. The transcendental degree of the field of invariant rationalfunctions C(FLeib8)

Gof FLeib8 under the action of the adapted subgroup

G of the group GL8 is four and it is generated by the following function:

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2 2 2

2

G =2

3

4+

1

1

1 1 3 4 2 4 1 3 4

,

G = .

( + 5 )

F =3 5 3 4 ,

Isamiddin S. Rakhimov

3 (6 +635 +93 4 +34 )

1 2F

2 = 34 4

+ 7 + 282 + 142 + 7 + 73 + 425F =3 7 3 6 3 4 3 5 4 5 3 4 3 ,3 3 4

4

5

F

4

=

3

7.

4

Later on = 4 52

, =42

73

, =83

214

.1 3 5 4 2 3 6 4 3 3 7 4

Theorem 4.9. The transcendental degree of the field of invariant rationalfunctions C(SLeib5)

Gof SLeib5 under the action of the adapted subgroup

G of the group GL5 is one and it is generated by the following function:

G =

.3

Theorem 4.10. The transcendental degree of the field of invariantrational functions C(SLeib6)

Gof SLeib6 under the action of the adapted

subgroup G of the group GL6 is one and it is generated by the followingfunction:

23

2



subgroup G of the group GL7 is two and it is generated by the followingfunctions:

31

(2 34 1 +24) , G =2

(2 34 1 +24)2 2



subgroup G of the group GL8 is three and it is generated by the followingfunctions:

G =3

( 3 ), G =

4 (7 72+4),

1 2 2 ( 3 )2 4 1

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1 .G3 =

3

3

(23

4

1)3

2 4 1


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REFERENCES

Atiyah, M.F., Macdonald, I.G. 1969. Introduction to CommutativeAlgebra.Addison-Wesley series in Mathematics.

Ayupov, Sh.A., Omirov, B.A. 2001. On some classes of nilpotent Leibniz

algebras. Sib. Math. J. 42(1):18-29.(in Russian).

Gomez, J.R., Omirov, B.A. 2006. On classification of complex filiform

Leibniz Algebras. arXiv:math/0612735 v1 [math.R.A.].

Hartshorne, R. 1977. Algebraic geometry. Springer, Graduate Texts in

Mathematics, 52.

Zariski, O., Samuel, P. 1958,1960. Commutative Algebra (Vol. I,II), Van

Nostrand, Princeton


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