Group Orbits and Regular Partitions of Poisson Manifolds

Abstract

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 ${\mathcal {L}}$ of Lagrangian subalgebras of reductive quadratic Lie algebras ${\mathfrak {d}}$ with Poisson structures defined by Lagrangian splittings of ${\mathfrak {d}}$ . In the special case of ${\mathfrak {g} \oplus \mathfrak {g}}$ , where ${\mathfrak {g}}$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal {L}}$ defined by arbitrary Lagrangian splittings of ${\mathfrak {g} \oplus \mathfrak {g}}$ . Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.