77 Avenue Denfert Rochereau, Paris, 75014, France
Department of Mathematics-UPC and BGSMath, Barcelona, Spain; CEREMADE (Universite de Paris Dauphine), IMCCE (Observatoire de Paris), and IMJ (Universite de Paris Diderot)
Martínez-Torres D., Miranda E.
Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds
2018, vol. 23, no. 1, pp. 47-53
We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.
Miranda E., Kiesenhofer A.
Noncommutative Integrable Systems on $b$-symplectic Manifolds
2016, vol. 21, no. 6, pp. 643-659
In this paper we study noncommutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a $b$-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure.