Les variétés de Poisson et leurs algèbres de Lie associées

On the geometric quantization of Jacobi manifolds

“…The reason is very clear, Poisson structures are defined by contravariant two-vectors instead of twoforms. So, Vaisman 3 characterized the existence of prequantization bundles of a Poisson manifold M by means of a natural cohomology of multivectors defined by Lichnerowicz, 9 and termed Lichnerowicz-Poisson cohomology of M . This cohomology can be obtained as the cohomology of a subcomplex of the Chevalley-Eilenberg complex associated to the Lie algebra of functions C ϱ (M ,R).…”

“…Theorem II.2: Let (M, ⌳, E) be a Jacobi manifold and ͕,͖ the bracket on ⍀ 1 (M )ϫC ϱ (M ,R) defined by (12). Then, the triple (T*M ϫR,͕,͖,(#,E)) is a Lie algebroid over M, where…”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

“…, for X ∈ ΓA, and π : A * → A is defined by π α = ι α π, and set [88]. The Poisson homology H • (A, ∂ π ) of (A, π) is that of the complex (Γ(∧ • A * ), ∂ π ) which has been studied by Huebschmann [55], and which generalizes the Poisson homology of Poisson manifolds [80,12].…”

“…Since their introduction by Lichnerowicz in 1977 [88], Poisson geometry and the cohomology of Poisson manifolds have developed into a wide field of research. Lie algebroids are vector bundles with a Lie bracket on their space of sections (see [93,94,16] and Section 1.2).…”

Geometric reduction of Hamiltonian systems

“…a bivector with vanishing Schouten bracket, see [6]). This operator induces the Poisson bracket {F, G} Π = dF, ΠdG on the algebra of smooth functions on M, where ·, · is the dual map between T * M and T M. A smooth real-valued function F on M is called Z-invariant if the Lie derivative L Z F = 0 for any vector field Z ⊂ Z.…”