Cohomologie De Hochschild Des Surfaces De Klein

Details Cohomologie De Hochschild Des Surfaces De Klein Cohomologie De Hochschild Des Surfaces De Klein

2008-03-23

arxiv.org/abs/0803.3347v2

Physics

Mathematical Physics

19 pages

Scientific paper

Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is important to be able to quantize it, in order to obtain more precise results than through classical mechanics. An available method is the deformation quantization, which consists in constructing a star-product on the algebra of formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of star-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$. The aim of this article is to determine this Hochschild cohomology in the case of singular curves of the plane -- so we rediscover, by a different way, a result proved by Fronsdal and make it more precise -- and in the case of Klein surfaces. The use of a complex suggested by Kontsevich and the help of Gr\"obner bases allow us to solve the problem.

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