191 research outputs found
Coupling symmetries with Poisson structures
We study local normal forms for completely integrable systems on Poisson
manifolds in the presence of additional symmetries. The symmetries that we
consider are encoded in actions of compact Lie groups. The existence of
Weinstein's splitting theorem for the integrable system is also studied giving
some examples in which such a splitting does not exist, i.e. when the
integrable system is not, locally, a product of an integrable system on the
symplectic leaf and an integrable system on a transversal. The problem of
splitting for integrable systems with additional symmetries is also considered.Comment: 14 page
A cohomological framework for homotopy moment maps
Given a Lie group acting on a manifold preserving a closed -form
, the notion of homotopy moment map for this action was introduced in
Callies-Fregier-Rogers-Zambon [6], in terms of -algebra morphisms.
In this note we describe homotopy moment maps as coboundaries of a certain
complex. This description simplifies greatly computations, and we use it to
study various properties of homotopy moment maps: their relation to equivariant
cohomology, their obstruction theory, how they induce new ones on mapping
spaces, and their equivalences. The results we obtain extend some of the
results of [6].Comment: 18 pages, final version. Added reference [16] by L. Ryvkin and T.
Wurzbacher, who obtain independently results similar to ours putting an
emphasis on the differential geometry of multisymplectic form
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