Hochschild cohomology and characteristic classes for star-products. (English) Zbl 0956.53055

Weinstein, Alan; Xu, Ping

Khovanskij, A. (ed.) et al., Geometry of differential equations. Dedicated to V. I. Arnold on the occasion of his 60th birthday. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 186(39), 177-194 (1998).

Authors’ abstract: We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold, is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation parameter is seen to be closely related to the characteristic class of the quantization. A fundamental role in the analysis is played by “quantum Liouville operators”, which rescale the deformation parameter in the same way in which Liouville vector fields scale the Poisson structure (or the units of action). Several examples are given.
For the entire collection see [Zbl 0896.00014].

MSC:

53D55	Deformation quantization, star products
57R20	Characteristic classes and numbers in differential topology
19D55	\(K\)-theory and homology; cyclic homology and cohomology
22A22	Topological groupoids (including differentiable and Lie groupoids)

Keywords:

quantum Liouville operators; Hochschild cohomology; formal deformation quantization; de Rham cohomology; star-product; Poisson structure