Deformations of Algebras over Operads and Deligne's Conjecture

摘要

The paper is organized in the following way. Section 1 is devoted to the reviewof operads. It includes a brief introduction to operads, language of trees, polynomial functors etc. The deformation theory of algebras over free operads is discussed in Section 2. In Section 3 we outline the strategy and state the theorem (proved in Section 6) which explain why the GT group appears in the deformation theory of associative algebras. Our approach is based on the notion of a free resolution of an operad. This is the subject o Section 4. In Section 5 we construct an operad M which acts on the Hochschild complex of an A-infinity-algebra. It turns out that M is closely related to the compactifications of configuration spaces of point introduced by Fulton and Macpherson in (FM). Section 6 is devoted to the proof of the theorem from Section 3. Section 7 is devoted to Deligne's conjecture and its proof.