%0 Journal Article
%T Courant algebroids and Poisson Geometry
%A David Li-Bland
%A Eckhard Meinrenken
%J Mathematics
%D 2008
%I arXiv
%R 10.1093/imrn/rnp048
%X Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behaviour of Lagrangian splittings under Courant morphisms.
%U http://arxiv.org/abs/0811.4554v3