ABSTRACT

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

Keywords: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 [1] -[3] , this is equal to giving an antisymmetric contravariant 2-tensor on which satisfies Jacobi identity and the relation

(1)

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

(2)

where are smooth functions vanishing at and

(3)

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.,

(4)

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

(5)

Both properties 1) and 2) imply that the element has to be a constant solution of the modified classical Yang-Baxter equation (mCYBE) [4] -[6] :

(6)

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

(7)

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.

2. The Group and Lie Algebra

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

3. The Lie Bialgebra Structure on and the Poisson Lie Structure on

3.1. Lie Bialgebra Structures on

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

(8)

So any Lie bialgebra structure of can be written as

(9)

3.2. Poisson-Lie Structures on

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

(10)

Let

(11)

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

4. Linearization of Poisson-Lie Structures on

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

(12)

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 [7] [8] :

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

(13)

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

(14)

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

(15)

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

(16)

and will be written

(17)

The Poisson bracket associated to reads

(18)

5. Casimir Functions and Infinitesimal Automorphisms on

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

(19)

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:

(20)

Now we shall clarify the gap between and.

We consider the vector field

(21)

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

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

(22)

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

(23)

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 [9] , 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 [10] [11] stating that for a Poisson structure formally linearizable around a singular point any local Poisson automorphism is Hamiltonian, and not just in the category.

References