We consider a skew-symmetric n -ary bracket on the polynomial algebra K [ x 1 , … + ] ( ≥ 2 ) over field of characteristic zero defined by { } = Jac C where is fixed element and Jacobian. If then this Poisson if 3 it an -Lie-Poisson . describe center corresponding show that quotient / − λ ideal generated 0 ≠ ∈ simple central homogeneous not proper power any nonzero polynomial. This construction...

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a ...

The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C∗-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.

This encyclopedia article briefly reviews without proofs some of the main results in Poisson reduction. The article recalls most the necessary prerequisites to understand the main results. The Poisson reduction techniques allow the construction of new Poisson structures out of a given one by combination of two operations: restriction to submanifolds that satisfy certain compatibility assumption...

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers ...

In a recent paper, S.-E. Takahasi defined the notion of a BSE Banach module over a commutative Banach algebra A with bounded approximate identity. We show that the multiplier space &f(X) of X can be represented as a space of sections in a bundle of Banach spaces, and we use bundle techniques to obtain shorter proofs of various of Takahasi’s results on P-algebra modules and to answer several que...

Poisson algebras are usually defined as structures with two operations, a commutative associative one and an anticommutative one satisfying the Jacobi identity. These operations are tied up by a distributive law, the Leibniz law. We present Poisson algebras as algebras with one operation which enables one to study them as part of nonassociative algebras. We study the algebraic and cohomological...

In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of ...

We construct operators t(z) in the elliptic algebra Aq,p(ŝl(2)c). They close an exchange algebra when pm = qc+2 for m ∈ Z. In addition they commute when p = q2k for k integer non-zero, and they belong to the center of Aq,p(ŝl(2)c) when k is odd. The Poisson structures obtained for t(z) in these classical limits are identical to the q-deformed Virasoro Poisson algebra, characterizing the exchang...