Henrique Bursztyn, Marius Crainic
Dirac Geometry, Quasi-Poisson Actions and D/G-Valued Moment Maps
Preprint series:
ESI preprints
- MSC:
- 58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}
- 70H05 Hamilton's equations
- 70H33 Symmetries
- 58H05 Pseudogroups and differentiable groupoids, See also {22A22,
Abstract: We study Dirac structures associated with Manin pairs
$(\frakd,\frakg)$ and give a Dirac geometric approach to
Hamiltonian spaces with $D/G$-valued moment maps, originally
introduced by Alekseev and Kosmann-Schwarzbach \cite{AK} 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.
Keywords: Dirac structures, Manin pairs, Moment maps, quasi-Poisson spaces