 
Summary: GROUP ORBITS AND REGULAR PARTITIONS OF POISSON
MANIFOLDS
JIANGHUA LU AND MILEN YAKIMOV
Abstract. We study a class of Poisson manifolds for which intersections of
certain group orbits give partitions into regular Poisson submanifolds. Examples
are the varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras
with Poisson structures defined by Lagrangian splittings. In the special case of
g # g, where g is a complex semisimple Lie algebra, we explicitly compute the
ranks of the Poisson structures on L defined by arbitrary Lagrangian splittings
of g # g. Such Lagrangian splittings have been classified by Delorme, and they
contain the BelavinDrinfeld splittings as special cases.
1. Introduction
Lie theory provides a rich class of examples of Poisson manifolds/varieties. In
this paper, we study a class of Poisson manifolds of the form (D/Q,# u,u # ), where
D is an even dimensional connected real or complex Lie group whose Lie algebra d
is quadratic, i.e. d is equipped with a nondegenerate invariant symmetric bilinear
form # , #; the closed subgroup Q of D corresponds to a subalgebra q of d that is
coisotropic with respect to # , #, and (u, u # ) is a pair of complementary subalgebras
of d that are maximal isotropic with respect to # , #. Lie subalgebras of d that are
maximal isotropic with respect to # , # will be called Lagrangian, and we will refer
