COHOMOLOGIE DE HOCHSCHILD DES SURFACES DE KLEIN

Frédéric Butin (1)

1 ICJ - Institut Camille Jordan

Given a physical 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 to obtain righter results as the ones given by classical mechanics. An answer is provided with the deformation quantization, which consists in constructing a starproduct on the algebra of formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of starproducts 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 plan --- so we rediscover, by a different way, a result proved by Fronsdal and we precise it --- and in the case of Klein surfaces. The use of a complex suggested by Kontsevich and the help of Gröbner bases allow us to solve the problem.

Mots clés: cohomologie; Hochschild; surfaces de Klein; bases de Gröbner; quantification; starproduits

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]; Anneaux et algèbres [math.RA]; Algèbre commutative [math.AC]

Frédéric Butin. COHOMOLOGIE DE HOCHSCHILD DES SURFACES DE KLEIN. 2008. ⟨hal-00266435v1⟩