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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 Marsden­Weinstein­Meyer 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 skew-symmetric solution r gg
of the classical Yang­Baxter 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 = TeG by Lx and Rx and if r = r1
i r2
i , then


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


Collections: Mathematics