APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2014.44015APM-44631ArticlesPhysics&Mathematics On the Structure of Infinitesimal Automorphisms of the Poisson-Lie Group <i>SU</i>(2,R) ousselhamGanbouri1*Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco* E-mail:g.busslem@gmail.com09042014040493976 January 20146 February 2014 15 February 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

We study the Poisson-Lie structures on the group SU(2,R). We calculate all Poisson-Lie structures on SU(2,R) through the correspondence with Lie bialgebra structures on its Lie algebra su(2,R). We show that all these structures are linearizable in the neighborhood of the unity of the group SU(2,R). Finally, we show that the Lie algebra consisting of all infinitesimal automorphisms is strictly contained in the Lie algebra consisting of Hamiltonian vector fields.

Poisson-Lie Structure Lie Bialgebra Hamiltonian Poisson Automorphism Linearization
1. Introduction

Let be a Lie group. A Poisson-Lie structure on is a Poisson structure on for which the group multiplication is a Poisson map. Then as is usual in  - , this is equal to giving an antisymmetric contravariant 2-tensor on which satisfies Jacobi identity and the relation

where and respectively denote the left and right translations in by and. We note that a Poisson-Lie structure has rank zero at a neutral element of, i.e.,.

If we choose local coordinates in a neighborhood of neutral element of, the Poisson-Lie structure reads

where are smooth functions vanishing at and

where is the Poisson bracket associated to. By this Poisson bracket, becomes a Lie algebra.

Let be a Lie algebra of. The derivative of at defines a skewsymetric co-commutator map such that:

1) The map is a 1-cocycle, i.e.,

2) The dual map is a Lie bracket on.

The map is said a Lie bialgebra structure associated to. Conversely, if is simply connected, any Lie bialgebra structure on the Lie algebra can be integrated to define a unique Poisson-Lie structure on such that.

The bialgebra structure is called a coboundary one when there exists an skewsymmetric element of (the classical r-matrix) such that

Both properties 1) and 2) imply that the element has to be a constant solution of the modified classical Yang-Baxter equation (mCYBE)  - :

Therefore, a constant solution of mCYBE on a given Lie algebra provide a coboundary Poisson-Lie structure on (connected and simply connected) group given by

where and denote respectively the left and right translations in by.

Finally, recall that for semisimple Lie algebras, all Lie bialgebra structures are coboundaries, and the corresponding Poisson-Lie structures can be fully solved through the classical r-matrices.

In this work, We shall treat the case of the Poisson-Lie group. We will calculate, firstly, all Poisson-Lie structures through the correspondence with Lie bialgebra; secondly, we will show that these Poisson-Lie structures are linearizable in a neighborhood of the unity of the group and, finally, we shall study infinitesimal automorphism of with a linear Poisson-Lie structure, and show that the Lie algebra, consisting of all infinitesimal automorphisms is strictly contained in the Lie algebra consisting of Hamiltonian vector fields.

The special unitary group is defined by Let and. can be identified with the unit sphere in with the unity.

The Lie algebra of group is defined by Let a basis of. The Lie bracket on is defined by Through a straightforward computation, the left invariant fields associated to this basis had this local expression Recall that the Lie algebra is semisimple. Then, all Lie bialgebra structures on are coboundaries, there exists an skew symmetric element r of such that the cocommutator is given by We stress that the element satisfies the classical Yang-Baxter Equation (CYBE) (6). Through a long but straightforward computation, we show that these solutions are of the form

So any Lie bialgebra structure of can be written as

Since the Lie bialgebra structures on are coboundaries, the Poisson-Lie structures on corresponding to are given by where is the solution of Yang-Baxter equation given by (8) and and respectively denote the right and left translations in by. Then, using, and one gets

Let

be the components of in the basis of the bivector field.

By taking back the formula (2), The Taylor series of the functions reads

where are the structure constants of a Lie algebra, dual of a Lie algebra, and the are smooth functions vanishing at.

The term of (12) definines a linear Poisson structure, called the linear part of. The linearization problem for a structure around is the following   :

Linearization problem. Are there new coordinates where the functions vanish identically, so that the Poisson structure is linear in these coordinates?

Let us notice that the Lie bialgebra structure associated to defines a linear Poisson-Lie structure on the additive group that can be expressed as follows

where is the canonical basis of.

Let us notice that (13) coincides with the linear part of, so, the linearization problem would be the following:

There is a local Poisson diffeomorphism of a neighborhood in of G to a neighborhood of 0 in?

If are the components of, then is solution of the system of equations

For the Poisson-Lie structure on given by (10), the system of equations (14) would be

With the identification of the subgroups of the singular points and the symplectic leaves of and, we have:

Proposition 1. The map: is a diffeomorphism in the neighborhood of such that and So, the Poisson-Lie structure on is linear in the new variables

and will be written

The Poisson bracket associated to reads

Recall that for, defines a derivation of. Hence there corresponds a vector field, which we call the Hamiltonian vector field. We denote by the Lie algebra of Hamiltonian vector fields.

A Casimir function on is a function such that for all function. On the other words, is an element of the center of the Lie algebra. By simple consideration, we know that for each Casimir function there exists a function of one variable such that.

Each symplectic leaf is the common level manifold of casimir functions. So, these have for equation: and hence are spheres.

By an automorphism of, we mean a smooth vector field on such that

where denotes the Lie derivative along.

If we denote by the Lie algebra consisting of all infinitesimal automorphism, it is easy to see that is an ideal of. Let be a vector field of. Then three function and must satisfy:

Now we shall clarify the gap between and.

We consider the vector field

where are the components of the structure in the basis given by (11).

In the local coordinates given by (14), this vector field reads

A simple check shows that the components of satisfy the relations (20). So, the vector field belongs to. In other hand, is locally Hamiltonian if and only if there exist a smooth function in a neighborhood of the unity of the group such that, this is translated by the fact that is a solution of the following system of equations

It is easy to see that (23) does not admit solutions. Hence does not belong. Thus we have proved:

Proposition 2. The ideal is strictly contained in the Lie algebra.

In terms of Poisson cohomology  , recall that the first Poisson cohomology group is the quotient of the Lie algebra by its ideal. Then, by Proposition 2, we show that the vector field defines a non trivial class. On the other hand, this result shows that the classical result due to Conn   stating that for a Poisson structure formally linearizable around a singular point any local Poisson automorphism is Hamiltonian, and not just in the category.

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