Deformations of algebras over operads and Deligne's conjecture

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In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace operad (the latter was used in the previous approaches to the Deligne's conjecture). It follows from our results that the Grothendieck-Teichmüller group acts (homotopically) on the moduli space of structures of 2-algebras on the Hochschild complex. In the Appendix we develop a theory of piecewise algebraic chains and forms. It is suitable for real semialgebraic manifolds with corners (like Fulton-Macpherson compactifications of the configuration spaces of points).
67 pages, many figures, Abstract is modified

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Quantum Algebra, High Energy Physics - Theory, Algebraic Geometry, Algebraic Topology, Category Theory, Rings and Algebras, 18D50; 55P48; 16S80

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http://salesiana.dossiersoluciones.com/handle/123456789/58575

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0001151
 http://arxiv.org/abs/math/0001151

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arXiv