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A note on n-ary Poisson brackets

Michor, Peter W.Vaisman, Izu — 2000

Proceedings of the 19th Winter School "Geometry and Physics"

An n -ary Poisson bracket (or generalized Poisson bracket) on the manifold M is a skew-symmetric n -linear bracket { , , } of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order n , i.e., σ S 2 n - 1 ( sign σ ) { { f σ 1 , , f σ n } , f σ n + 1 , , f σ 2 n - 1 } = 0 , S 2 n - 1 being the symmetric group. The notion of generalized Poisson bracket was introduced by et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]. They established...

The BV-algebra of a Jacobi manifold

Izu Vaisman — 2000

Annales Polonici Mathematici

We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give new examples of Lie bialgebroids over Poisson manifolds.

Aspects of Geometric Quantization Theory in Poisson Geometry

Izu Vaisman — 2000

Banach Center Publications

This is a survey exposition of the results of [14] on the relationship between the geometric quantization of a Poisson manifold, of its symplectic leaves and its symplectic realizations, and of the results of [13] on a certain kind of super-geometric quantization. A general formulation of the geometric quantization problem is given at the beginning.

Remarks on the Lichnerowicz-Poisson cohomology

Izu Vaisman — 1990

Annales de l'institut Fourier

The paper begins with some general remarks which include the Mayer-Vietoris exact sequence, a covariant version of the Lichnerowicz-Poisson cohomology, and the definition of an associated Serre-Hochshild spectral sequence. Then we consider the regular case, and we discuss the Poisson cohomology by using a natural bigrading of the Lichnerowicz cochain complex. Furthermore, if the symplectic foliation of the Poisson manifold is either transversally Riemannian or transversally symplectic, the spectral...

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