ar X iv : m at h / 06 09 73 2 v 2 [ m at h . SG ] 1 5 Fe b 20 07 GROUP ORBITS AND REGULAR PARTITIONS OF POISSON MANIFOLDS

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras d with Poisson structures defined by Lagrangian splittings of d. In the special case of g ⊕ g, where g is a complex semi-simple 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 P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.