Dirac geometry and integration of Poisson homogeneous spaces

Abstract

Using tools from Dirac geometry, we show through an explicit construction that any 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 → B, integrations of a Poisson structure on B to 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. Contents 1. Introduction 2. Preliminaries on Dirac geometry 2.1. Dirac structures and pre-symplectic groupoids 2.2. Invariant Dirac structures and Poisson quotients 2.3. Integrability of Poisson structures on quotients 3. Integration of homogeneous Dirac structures on Poisson Lie groups 3.1. Review of definitions 3.2. Explicit integrations 4. Integration of Poisson homogeneous spaces 4.1. Explicit integrations 4.2. Proof of the main theorem 5. The holomorphic setting 6. Examples: old and new 6.1. Special cases and previous examples 6.2. Examples related to semi-simple Lie groups Appendix A. The pre-symplectic groupoid (G(L), ω) A.1. Deriving the formula for ω A.2. Proof of Theorem 3.12

Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson–Lie groups

“…Let (G, Π) be a connected Poisson-Lie group of dimension n such that the dual Lie algebra g * is unimodular (that is, the Poisson-Lie structure Π is unimodular). If ν ∈ ∧ n g * with ν = 0, then the volume form √ f 0 ν l is preserved by all Hamiltonian vector fields, where f 0 : G → R is the real function on G given in (8).…”

Dirac structures and Nijenhuis operators

“…An interesting class of holomorphic Dirac structures is given by the affine Dirac structures on complex Poisson Lie groups studied in [9], where an explicit description of their holomorphic presymplectic groupoids is presented together with an application to the construction of holomorphic symplectic groupoids integrating holomorphic Poisson homogeneous spaces.…”

“…By means of these general results, we derive in § 6 the infinitesimal-global correspondence between Dirac-Nijenhuis structures and presymplectic-Nijenhuis groupoids in Theorem 6.3 (see also Remark 6.4), as well as the special case of this correspondence relating holomorphic Dirac structures and holomorphic presymplectic groupoids, see Theorems 6.8 and 6.9. (Concrete constructions of holomorphic presymplectic groupoids integrating holomorphic Dirac structures arising in Poisson-Lie theory can be found in [9].) Our differentiation-integration results are schematically illustrated in the next diagram.…”

“…A Dirac structure on a manifold M is a subbundle L ⊂ TM := T M ⊕ T * M that is lagrangian (for the canonical symmetric pairing on TM ) and involutive with respect to the Courant-Dorfman bracket; any Poisson structure π on M may be regarded as a Dirac structure L π given by the graph of π ♯ : T * M → T M , α → i α π. Dirac structures have also been considered in the holomorphic category, e.g. in connection with generalized Kähler geometry [25] and Poisson-Lie theory [9].…”

Quotients of multiplicative forms and Poisson reduction

“…((G, ω) is called a "q M -admissible presymplectic integration" in [1].) This refined criterion can be used proceduce to integrations of a variety of Poisson manifolds obtained as quotients, including Poisson homogeneous spaces, see [1,7]. We stress that their criterion to produce the groupoid simple quotient q integrating the pullback type quotient q is non-trivial (see Example 3.4 and the cited references) and is independent of the topics covered in the present paper.…”