New York Journal of Mathematics

New York J Math

Heisenberg Lie Bialgebras as Central Extensions

Miloud Benayed and El Mamoun Souidi

Abstract We determine and study all Lie bialgebra central extensions of

n

R by R admitting the Heisenb erg algebra H as the underlying Lie alge

n

bra structure The presentwork answers the question ab out the realizability

n

of Lie bialgebra structures on H as central extensions of R endowed

n

with the adapted Lie bialgebra structure by R

Contents

Intro duction

Central Structures on H

n

Exact Central Structures on H

n

Equivalence of Central Structures on H

n

Conclusion

References

 Intro duction

g equipp ed with a co cycle g g A Lie bialgebra is a Lie algebra

for the extended adjoint action of g on g whose transp ose g g

denes a Lie algebra structure on the dual vector space g of g A cob oundary

r with r g denes a Lie bialgebra structure on a Lie algebra g if and

only if the Schouten bracket rr g is invariant under the extended adjoint

action of g on g Such a Lie bialgebra is called cob oundary or exact

Let g and g be two Lie bialgebras A linear map g g is a Lie bialgebra

morphism if is a Lie algebra morphism and its transp ose g g is also a

Lie algebra morphism A bijective morphism is called an isomorphism

Two Lie bialgebra structures and on a Lie algebra g are called

equivalent if there exists a Lie bialgebra isomorphism g g

Received May

Mathematics Subject Classication W B

Key words and phrases Heisenb erg algebra Lie bialgebras central extensions

The rst author would like to thank G Tuynman for discussions The second author would

like to thank G Tuynman for the hospitality at Universite de Lille I and also C Brezinski and

J Mikram principal investigators of the grant Action Integree AI

c

State UniversityofNewYork

ISSN

Miloud Benayed and El Mamoun Souidi

For further details on Lie bialgebras we refer the reader to In all the sequel

the ground eld is R

b

A Lie bialgebra g is called a central extension of a Lie bialgebra g by R if there

exists an exact sequence

i

b

R g g

in which i and are Lie bialgebra morphisms and iR is contained in the center

b b b

of the Lie algebra g Two central extensions g and g of g by R will be called

b b

equivalent if there exists a Lie bialgebra morphism g g inducing the identity

b b

on R and g in the exact sequences dening g and g

The Heisenb erg algebra H is a Lie algebra central extension of the ab elian Lie

n

n

algebra R by R asso ciated with the co cycle which is the canonical symplectic

n n

form of R A Lie bialgebra structure on the ab elian Lie algebra R is equivalent

n

to the data of a Lie algebra structure on R

n

This pap er is organized as follows In Section we describ e the set Ext R R

big

of all inequivalent Lie bialgebra central extensions of a given Lie bialgebra structure

n n

on the ab elian Lie algebra R by R Let Ext R R b e the subset consisting

big

n

of elements in Ext R R whichhave the Heisenb erg algebra as the underlying

big

Lie algebra structure Such Lie bialgebra structures on H will b e called central

n

n n

Weprove that Ext R R is non emptyifandonlyifR is an ab elian Lie

big

n

algebra If the last condition is fullled then the set Ext R R is parametrized

big

n

by the endomorphisms of R Section is devoted to giveacharacterization of

exact central structures on H by means of the asso ciated endomorphims of

n

n

R In Section wegive the orbits of central structures on H under its auto

n

morphisms group action In the last section wegive a motivationofthiswork as a

partial answer of an op en question

 Central Structures on H

n

n

We endow R with a Lie bialgebra structure by taking the ab elian Lie algebra

n

structure on R and giving a Lie algebra structure on its dual vector space

n n

R Let ad b e the adjoint action of R on itself and coad represents the

n n n

coadjoint action of R on its dual vector space R We denote by DerR

n n

the vector space of derivations of the Lie algebra R and DexR stands for

n

the outer derivations of R

n

Denition An element of R is called compatible with the Lie alge

n

bra structure on R if the condition coad y coad x holds for all x y

e x e y

n n n n

in R wheree R R is the linear map induced by We let R

c

n

to denote the subspace of all such compatible elements of R

n

Theorem There is a onetoone correspondence between Ext R R and

big

n n

R DexR

c

n n

Pro of Toany Lie bialgebra central extension R R of R by R one can asso ciate

n n

acouple f R DerR as follows see

c

n

x a y b x y

R R

n n

f f

R R R

Heisenberg Lie Bialgebras

n n

Two elements f and f of R DerR dene equivalent Lie

c

n

bialgebras if and only if and f f ad where lies in R Reversing

the arguments we also get the converse

n n n

For every R we let Ext R R to b e the subspace of Ext R R

big

c

big

n

corresp onding to f g DexR Weareinterested in the case where is the

n

canonical symplectic form of R

n n

R R R

x y x y x y y x

n

The dots stand here for the inner pro duct in R In order to restrict ourselves to

n

Ext R R we must verify the compatibility of with the given Lie algebra

big

n

structure on R

eequivalent Lemma The fol lowing conditions ar

n

i R

c

n

ii R is an abelian Lie algebra

Pro of It is enough to prove the implication i ii The fact that lies in

n

R implies the following condition

c

n n

x R R e coad x e x

n

Writing this condition for e y with y R and using the fact that

n

R we obtain

c

n

x y R e x e y e y e x

n

which implies that e x e y for all x y in R The nondegeneracy of

n n

means that e R R isavector space isomorphism So we conclude that

n

R is an ab elian Lie algebra

n

Remark The condition R implies that the range of e is an ab elian

c

n

Lie subalgebra of R The converse is false in general as one can see in the

following example We endowR with the Lie algebra structure dened by the

only non vanishing bracket X X X where X is the dual basis of the

k k

canonical ordered basis X of R Let b e the elementof R given by

k

k

X X and vanishing elsewhere The range of e is ab elian but R

c

for example coad X X coad X

e X e X

 

As a consequence of the previous lemma and theorem we get the following

n n

Prop osition Ext R R is non empty if and only if R is an abelian

big

n

Lie algebra If the last condition is full led then Ext R R is parametrizedby

big

n n

the space EndR of al l endomorphisms of R

n n n

Henceforth we assume that R is an ab elian Lie bialgebra ie R and R

n

are endowed with the zero Lie brackets Otherwise as wehave seen Ext R R

big

will b e empty

n

The set Ext R R is viewed as the Heisenb erg algebra H endowed with

n

big

n

Lie bialgebra structures parametrized byEndR suchthateach structure makes

n

H as a central extension of the ab elian Lie bialgebra R by R Such a structure

n

on H will b e called central Let us sp ecify the central structures on H in

n n

Miloud Benayed and El Mamoun Souidi

terms of the corresp onding co cycles H H the transp oses of Lie

n n

brackets in H See the pro of of Theorem

n

The center Z H of the Lie algebra H is one dimensional let Z be a

n n

n n

non zero elementofZ H Let X b e the canonical ordered basis of R

n k

k

n

then X Zis the canonical basis of H The only non vanishing Lie

k n

k

brackets on H are given by

n

For all i j n X X Z

i nj ij

where is the Kroneckers symbol

ij

Denition A central structure on H is the data of a co cycle

n f

n

H H where f EndR satisfying

n n

i Z

f

n

ii For all X R X Z f X

f

 Exact Central Structures on H

n

An arbitrary element r of H can b e written as

n

n

X X

r X X X Z

ij i j i i

i

ij n

where is an antisymmetric matrix and the are in R

ij ij n i

P

n

X Z vanishes Lemma The couboundary r of r

i i

i

Pro of For all H H wehave

n

n

X

r H H X Z X H Z

i i i

i

Since H H Z H and Z is central in the Lie algebra H then

n n n n

r H for all H H

n

P

Henceforth we assume that r X X The matrix determines

ij i j

ij n

r completely and viceversa

Lemma Every element r of H denes an exact Lie bialgebr astruc

n

ture r on H ie the Schouten bracket rr is adinvariant for al l r in

n

H

n

Let us remark that an exact Lie bialgebra r on H is necessarily central

n

Our aim is to giveacharacterization of central structures on H that are exact

n

n

Prop osition An endomorphism f of R denes an exact central structureon

A B

n

H if and only if the matrix of f in the basis X has the form

n k

t

k

C A

where A B C are n n matrices with B and C antisymmetric The corresponding

B A

r matrix has the form

t

A C

Heisenberg Lie Bialgebras

P

Pro of Let r X X b e an elementof H Set

ij i j n ij

ij n

n

Weverify there exists an f EndR with matrix f in the canonical basis

ij

suchthat r if and only if f f for all k n and

f ik ink nkk

n Using these relations and the f f for all n k

ik k ni k nk

antisymmetry of the matrix we obtain the following conditions on f for all

ij

p q n

f f f f f f

pnq qnp npnq qp npq nqp

A B

In other words the matrix of f is necessarily of the form f where

ij

t

C A

B and C are n n antisymmetric matrices Hence the corresp onding r matrix

can b e written as

B A

t

A C

Reversing the arguments we also get the converse

Remark If there exists k n such that f or f for

nkk k nk

n k n then the central structure dened by f on H is not exact

n

We denote byI the identity matrix p p

p

Example The exact central structures on H are all of typ e a I wherea R

The corresp onding r matrix is given by r aX X

 Equivalence of Central Structures on H

n 

n

Twocentral structures and on H dened by f and g in EndR re

f g n

sp ectively will b e called equivalent if they dene equivalent Lie bialgebra structures

ie if there exists a Lie algebra automorphism of H say AutH

n n

such that the following diagram commutes

H H

n n

g

f

y y

H H

n n

n

We dene the extended symplectic group of R by

I

n

t

S n RfA GLn R js R AJ A sJ g where J

I

n

n

Prop osition Let f and g be two endomorphisms of R with the matrixes

n

M and M in the canonical basis of R respectively The central structures

f g f

and on H are equivalent if and only if there exists A S n R such that

g n

t

M sAM A where s is denedby AJ A sJ

g f

The pro of is an immediate consequence of the following realization of Aut H

n

Lemma The automorphisms group of H is given by

n

t

AJ A sJ with s R and A

GLn R G

n

v v v R arbitrary v s

n

Miloud Benayed and El Mamoun Souidi

Pro of We distinguish the following classes of automorphisms of H that we

n

n

representby their matrices in the canonical basis X Z

k

k

A

i The symplectic automorphisms where A is a n n symplectic

t

matrix ie AJ A J

I

n

n

ii The inner automorphisms where v R

v

r I

n

where r iii The dilatations

r

I

n

A

I

iv The inversion

n

Every automorphism of H can be written as a pro duct of these automor

n

phisms It is easy to see that Aut H G Recipro cally consider an element

n

A

of G By multiplying this matrix byaninversion we can assume that

v s

p

s By multiplying by the dilatation of co ecient followed byaconvenient

s

B

inner automorphism we obtain a matrix of the form where B is a n n

symplectic matrix Thus is an automorphism of H ie G AutH

n n

hence G AutH

n

 Conclusion

This work is motivated by the op en question ab out the classication of all Lie

bialgebra structures on a given nilp otent Lie algebra

Every nilp otent Lie algebra can be obtained by successives central extensions

from an ab elian Lie algebra Our hop e was to use the notion of Lie bialgebra central

extensions in order to classify all Lie bialgebra structures on a xed nilp otent Lie

algebra

Our test Lie bilagebra was the Heisenb erg algebra where all Lie bialgebra struc

tures are known Let us recall this result in terms of our previous notations

H H on H Theorem Each Lie bialgebra structure

n n n

is one of the fol lowing two forms

n n

i Z and for al l X R X Z f X with f EndR

n

ii Z Z A and X R X Z N X X A X Ae with

A

n n

A R fg and N is the endomorphism of R given by

A

n

N A U I U e A A e U A e A U R R

A n

Following our study the rst family i is the central structures on H realized

n

by the notion of Lie bialgebra central extensions The second family ii cannot

b e obtained by this notion One can see that for central extensions Z vanishes

necessarily This is not the case for the second family ii

References

M Benayed Central extensions of Lie bialgebras and Poisson Lie groups Journal of Geometry

and Physics  MR d

Heisenberg Lie Bialgebras

M Cahen and C Ohn Bialgebra structures on the Heisenberg algebra Bulletin de la Classe

des Sciences de lAcademie Royale de Belgique  MR d

V Chari and A Pressley A Guide to Quantum GroupsCambridge University Press Cam

bridge MR j

I Szymczak and S Zakrzewski Quantum deformations of the Heisenberggroup obtainedby

geometric quantization Journal of Geometry and Physics  MR i

Universite des Sciences et Technologies de Lille UFR de MathematiquesURA au

CNRS D Villeneuve dAscq Cedex France

b enayedgatunivlillefr

Universite Mohammed V Faculte des Sciences Departement de Mathematiques et

Informatique BP Rabat Maroc

souidimaghrebnetnetma

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