 
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
