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 $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. In the special case of $\g \oplus \g$, where $\g$ 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.

23 pages, AMS Latex, minor changes in v.2