Poisson cohomology and quantization.

By J. Huebschmann. 2013

398 citations recorded by the source corpus.

Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of K\"ahler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for an arbitrary Poisson algebra. A geometric version thereof includes the canonical homology and Poisson cohomology of a Poisson manifold introduced by Brylinski, Koszul, and Lichnerowicz, and absorbes the latter in standard homological algebra by expressing them as Tor and Ext groups, respectively, over a suitable algebra of differential operators. Furthermore, the Poisson structure determines a closed 2-form in the complex computing Poisson cohomology. This 2-form generalizes the 2-form defining a symplectic structure on a smooth manifold; moreover, the class of that 2-form in Poisson cohomology generalizes the class in de Rham cohomology of a symplectic structure on a smooth manifold and appears as a crucial ingredient for the construction of suitable linear representations of A, viewed as a Lie algebra; representations of this kind occur in quantum theory. To describe this class and to construct the representations, we relate formal concepts of connection and curvature generalizing the classical ones with extensions of Lie algebras. We illustrate our results with a number of examples of Poisson algebras and with a quantization procedure for a relativistic particle with zero rest mass and spin zero.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ebdf36a37cc3c9fe869107613b18b0458132b7ad

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Citations in the corpus

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds (2008, background): “The study of that cohomology was carried on by Vaisman [48], Huebschmann [20], Xu [54] and many other authors.”
Grothendieck Groups of Poisson Vector Bundles (2000, methodology): “The construction of pre-quantization generalizes to Poisson algebras associated with closed two-forms or Dirac structures (see [GGK, Chapter 6]) and to some Poisson manifolds, [ Hu , Va].”
Bimodule deformations, picard groups and contravariant connections (2002, background): “16 In fact, one can compute the semiclassical limit of the action Φ, obtaining a firstorder obstruction to Morita equivalence of formal deformations in terms of algebraic Poisson cohomology [20]; the arguments are similar to the case of star products [6].”
Twisted Differential String and Fivebrane Structures (2009, background): “More generally, if A is any commutative associative algebra, we may speak of an ∞-Lie-Rinehart algebra [65] [44] which we may think of as an ∞-Lie algebroid over a noncommutative base space.”
MODULAR VECTOR FIELDS AND BATALIN-VILKOVISKY ALGEBRAS (2000, background): “The main features of the construction are valid in the more general algebraic framework of Lie-Rinehart algebras and their associated Gerstenhaber algebras [9] [10].”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “A deriving operator for the Courant bracket of the double, TM ⊕ T ∗M , of the Lie bialgebroid of a Poisson manifold is d − ∂π, where ∂π = [iπ,d] is the Poisson homology operator, often called the Koszul-Brylinski operator, defined by Koszul in [38] and studied by Huebschmann in [22], in the more gen”
Poisson (co)homology and isolated singularities (2005, background): “{· , ·}φ, which is a 2-cocycle of the Poisson cohomology of (A, {· , ·}φ), is not a 2-coboundary (see [9]).”