 
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 skewsymmetric solution r # g#g
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 # = T e G by L x and R x and if r =
# r 1
i # r 2
