Deformations of algebras over operads and Deligne's conjecture

By M. Kontsevich, Y. Soibelman. 2000

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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-Teichm\"uller 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).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb4e0a6ec4a358cefdfdd0131c4da4154afaec49

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Citations in the corpus

M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra (2010, methodology): “ry to [41] we will use a formality morphism of the operad of chains of the little disks operad as in [39]. The latter operad is quasi-isomorphic to the braces operad Br, which naturally acts on Dpoly [28]. One may write down a chain of quasi-isomorphisms of operads Br→···→CNPaCD →BUt←e2 ←hoe2. The o”
On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra (2003, background): “In the following trees will play a key role for indexing purposes and in the definition of operads and operadic actions.”
On the cyclic Deligne conjecture (2004, unknown): “Q A ] 2 7 O ct 2 00 4”
DG quotients of DG categories (2002, background): “, they prove in [32, 34] that endomorphisms of the identity 1-morphism of an object of a homotopy 2-category form an algebra over the chain complex of the little squares operad (Deligne’s conjecture).”
Poisson deformations of symplectic quotient singularities (2002, background): “ew and extend it below (relations with deformation theory were not discussed in [Fr]). For a more general formalism of deformation theory of an algebra over an arbitrary operad the reader may consult [KS]. Let Abe a ﬁnitely generatedcommutative algebra. The standard Lie bracket of vector ﬁelds exten”
THE BOARDMAN-VOGT RESOLUTION OF OPERADS IN MONOIDAL MODEL CATEGORIES (2005, unknown): “W e callthe underlying algebraic object a segm ent (i.e. an augm ented associative m onoid with absorbing elem ent),and use the term intervalforsegm entswhich,in addition,induce cylinder-objectsin Q uillen’sm odeltheoretic sense [17].”