Quasi-Hamiltonian Groupoids and Multiplicative Manin Pairs

Abstract

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids.We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.

Left and right centers in quasi-Poisson geometry of moduli spaces

“…Their integrability follows from the fact that a L , a R , and a are the anchor maps for certain Lie algebroid structures (see [LS3] for a L and a R and [BC1] for a). Notice that…”

“…1 If a is a quasi-Lie bialgebra then a Lie subalgebra b ⊂ a is a quasi-Lie sub-bialgebra if δa In this case left (and right) leaves of M coincide with the big leaves [LS3,Theorem 3]. In particular, if M is quasi-symplectic then σ is non-degenerate.…”

Pseudo-Dirac structures

“…Dirac Lie groups for which multiplication is a morphism of Manin pairs were classified in [36,35] (see also [45,29] for a different setting). There it was shown that the underlying Courant algebroid can be canonically trivialized as an action Courant algebroid A = d × H (see Example 2.8), and the Dirac structure is a constant subbundle E = g × H under this trivialization.…”

“…This example arises in q-Poisson geometry.Ševera and the author [36] showed that given any q-Poisson (g ⊕ḡ, g ∆ )-structure on a manifold M (see Example 5.1 for details), T * M carries the structure of a Lie algebroid. Of significance is that T * M cannot generally be identified with a Dirac structure in any exact Courant algebroid, which contrasts the case of a Poisson structure on M (c.f.…”

Homogeneous spaces of Dirac groupoids

“…Hence, it is natural to ask how to recover the two results above on classification of multiplicative Poisson bivectors and closed 2-forms on a Lie groupoid in terms of data on its algebroid, which are by nature very different, as special cases of a more general result about the infinitesimal description of Dirac groupoids. These objects have been defined in [26] (see also [17]); a Dirac groupoid is a groupoid endowed with a Dirac structure that is a subgroupoid of the Pontryagin groupoid (T G ⊕ T * G) ⇉ (T M ⊕ A * ). This paper is the first part of a series of papers devoted to the study of the infinitesimal description of Dirac groupoids [11,15].…”