Dirac geometry and integration of Poisson homogeneous spaces

Abstract

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M\to M/H$, integrations of a Dirac structure on $M/H$ to $H$-admissible integrations of its pullback Dirac structure on $M$ by pre-symplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneous spaces, including their symmetric spaces, conjugacy classes, and flag varieties. In a more general framework, we also show integrability of all homogeneous spaces of ${\mathcal{L}\mathcal{A}}^{\vee }$-Lie groups in the sense of E. Meinrenken.

v4: 46 pages. The paper has been significantly re-written, new material added and results presented in a broader framework

References in corpus (1)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

Cited by in corpus (4)

• Dirac structures and Nijenhuis operators
• Poisson groupoids and moduli spaces of flat bundles over surfaces
• Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
• Quotients of multiplicative forms and Poisson reduction