Courant algebroids and Poisson Geometry

TL;DR

This paper explores how Courant algebroids arising from quadratic Lie algebra actions on manifolds can generate Poisson structures, including known examples like Lu-Yakimov and Evens-Lu structures, through Lagrangian splittings.

Contribution

It establishes a link between Courant algebroids and Poisson geometry via Lagrangian splittings and recovers several known Poisson structures within this framework.

Findings

Courant algebroids can be constructed from quadratic Lie algebra actions on manifolds.

Lagrangian splittings in the Lie algebra induce Poisson structures on the manifold.

The framework recovers known Poisson structures such as Lu-Yakimov and Evens-Lu.

Abstract

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.