Closed star products and cyclic cohomology. (English) Zbl 0767.55005

Connes, Alain; Flato, Moshé; Sternheimer, Daniel

Lett. Math. Phys. 24, No. 1, 1-12 (1992).

From the authors’ abstract: We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold \(W\)) is closed iff integration over \(W\) is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in the usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well- defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.

Reviewer: P.J.Kahn (Ithaca)

MSC:

55N35	Other homology theories in algebraic topology
19D55	\(K\)-theory and homology; cyclic homology and cohomology
19K56	Index theory
17B37	Quantum groups (quantized enveloping algebras) and related deformations
81S30	Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
58J40	Pseudodifferential and Fourier integral operators on manifolds
57R20	Characteristic classes and numbers in differential topology

Keywords:

deformation of the associative product of functions on a symplectic manifold; closed star product; trace; cyclic cohomology; Hochschild cohomology; character of a closed star product; pseudodifferential operators; Todd class

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