An approach to the tangential Poisson cohomology based on examples in duals of Lie algebras

Abstract

In this paper, we are interested in the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, a cohomology which was first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M , the TP-cohomology coincides with the leafwise de Rham (orČech) cohomology of the symplectic foliation of M . Computing the spaces of such a cohomology leads actually to open and quite nontrivial problems. To get a better understanding of these difficulties, we study explicitly many examples coming from nilpotent and 3-dimensional (real) Lie algebras. For the latter, we compare the TP-cohomology and the usual Poisson cohomology (P-cohomology).

On the splitting of infinitesimal Poisson automorphisms around symplectic leaves

“…Notice that this component coincides also with the image under the homomorphism H 1 Ψ (M ) → Γ(ν(S)) induced by the projection T M → ν(S) along T S and hence its non-triviality implies the existence of non-vanishing Poisson vector fields of Ψ transversal to the symplectic foliation. Formula (1.1) can be used to compute the first Poisson cohomology in some particular cases [7,10,18,17,21,25].…”

Lifts of Poisson structures to Weil bundle

“…are defined. In the general situation this cohomology is very difficult to compute (see, e.g., [6,7,21,23,27,30,42,50]). A map w can be extended to a map Ω k (M) → V k (M) defined by the formula wθ(α 1 , .…”

“…Let (M, w) be a Poisson manifold, and let w C be the complete lift of w to T A M. By virtue of (6) and (63),…”

“…are defined. In the general situation this cohomology is very difficult to compute (see, e.g., [6,7,21,23,27,30,42,50]).…”

Stronglyr-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology

“…Problems studied include the cohomology of regular Poisson manifolds [Vaisman 1990;Xu 1992], (co)homology and resolutions [Huebschmann 1990], duality [Huebschmann 1999;Xu 1999;Evens et al 1999], cohomology in low dimensions or specific cases [Nakanishi 1997;Ginzburg 1999;Gammella 2002;Monnier 2002b;2002a;Roger and Vanhaecke 2002;Roytenberg 2002;Pichereau 2005], and various extensions of Poisson cohomology -for example, the cohomologies Lie algebroid, Jacobi, Nambu-Poisson, double Poisson, and graded Jacobi [de León et al 1997;Ibáñez et al 2001;Monnier 2001;Grabowski and Marmo 2003;de León et al 2003;Nakanishi 2006;Pichereau and Van de Weyer 2008]. In [Masmoudi and Poncin 2007;Ammar and Poncin 2008], we suggest an approach to the cohomology of the Poisson tensors of the Dufour-Haraki classification (DHC).…”