Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 064, 26 pages      arXiv:0804.4324      http://dx.doi.org/10.3842/SIGMA.2008.064
Contribution to the Special Issue on Deformation Quantization

Hochschild Homology and Cohomology of Klein Surfaces

Frédéric Butin
Université de Lyon, Université Lyon 1, CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France

Received April 09, 2008, in final form September 04, 2008; Published online September 17, 2008

Abstract
Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.

Key words: Hochschild cohomology; Hochschild homology; Klein surfaces; Groebner bases; quantization; star-products.

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