Group orbits and regular partitions of Poisson manifolds

Details Group orbits and regular partitions of Poisson manifolds Group orbits and regular partitions of Poisson manifolds

2006-09-26

arxiv.org/abs/math/0609732v2

Mathematics

Symplectic Geometry

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

Scientific paper

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.

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