Courant Algebroids and Poisson Geometry

By David Li-Bland, E. Meinrenken. 2008

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Given a manifold M with an action of a quadratic Lie algebra , such that all stabilizer algebras are coisotropic in , we show that the product becomes a Courant algebroid over M. If the bilinear form on is split, the choice of transverse Lagrangian subspaces of defines a bivector field π on M, which is Poisson if 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 behavior of Lagrangian splittings under Courant morphisms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a602b5749200e5b123a1e8e233912ed4babc0f56