Advances in Pure Mathematics
Vol.4 No.4(2014), Article ID:44631,5 pages DOI:10.4236/apm.2014.44015
On the Structure of Infinitesimal Automorphisms of the Poisson-Lie Group
Bousselham Ganbouri
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco
Email: g.busslem@gmail.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 6 January 2014; revised 6 February 2014; accepted 15 February 2014
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.
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