Dirac geometry, quasi-Poisson actions and D/G-valued moment maps

TL;DR

This paper explores the relationship between Dirac structures and quasi-Poisson geometry in the context of D/G-valued moment maps, unifying different frameworks for Hamiltonian spaces.

Contribution

It provides a Dirac geometric perspective on Hamiltonian spaces with D/G-valued moment maps and establishes their equivalence with quasi-Poisson structures.

Findings

Categories of Hamiltonian spaces are isomorphic under both frameworks

Connections between Dirac geometry and equivariant differential forms are clarified

Examples include q-Hamiltonian spaces and Poisson-Lie group actions

Abstract

We study Dirac structures associated with Manin pairs (\d,\g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of "double" in each context.