Courant algebroids and Poisson Geometry

Details Courant algebroids and Poisson Geometry Courant algebroids and Poisson Geometry

2008-11-27

arxiv.org/abs/0811.4554v3

International Mathematics Research Notices 2009, 2106-2145 (2009)

Mathematics

Differential Geometry

Scientific paper

10.1093/imrn/rnp048

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.

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