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Contemporary Mathematics Triangular Poisson Structures on Lie Groups and
 

Summary: Contemporary Mathematics
Triangular Poisson Structures on Lie Groups and
Symplectic Reduction
Timothy J. Hodges and Milen Yakimov
Abstract. We show that each triangular Poisson Lie group can be decom­
posed into Poisson submanifolds each of which is a quotient of a symplectic
manifold. The Marsden--Weinstein--Meyer symplectic reduction technique is
then used to give a complete description of the symplectic foliation of all tri­
angular Poisson structures on Lie groups. The results are illustrated in detail
for the generalized Jordanian Poisson structures on SL(n).
1. Introduction
A Poisson group structure on a Lie group G is given by specifying a Lie bial­
gebra structure on its Lie algebra g. Such a Lie bialgebra structure is said to be
triangular if the cocommutator is induced from a skew­symmetric solution r # g#g
of the classical Yang--Baxter equation. A triangular Poisson Lie group is a Lie
group equipped with such a Poisson group structure, associated to a triangular Lie
bialgebra structure on g. If we denote the left and right invariant vector fields on
G, induced by an element x # g # = T e G by L x and R x and if r =
# r 1
i # r 2

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics