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Deformations of algebras over operads and the Deligne conjecture. (English) Zbl 0972.18005

Dito, Giuseppe (ed.) et al., Conférence Moshé Flato 1999: Quantization, deformation, and symmetries, Dijon, France, September 5-8, 1999. Volume I. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 21, 255-307 (2000).
The paper is concerned with the deformation theory of operads and algebras over operads. It stresses once more the universality of associative algebras and operads. A remarkable fact is the relationship of the deformation theory of associative algebras to the geometry of configuration spaces of points on surfaces. This is related to Deligne’s conjecture which says that the Hochschild complex of an associative algebra admits a canonical structure of a differential graded (dg) algebra over the chain operad of the little discs operad. A (new) proof of this conjecture is included. It turns out that the Grothendieck-Teichmüller group acts (homotopically) on the moduli space of structures of \(2\)-algebras (i.e., algebras over the operad of chains of the little discs operad) on the Hochschild complex.
The paper consists of an introduction with motivation and eight sections. The first section introduces operads as monoids in the monoidal category of polynomial functors. The basic properties of operads, free operads, algebras over operads as well as trees are given. In the second section the deformation theory of algebras over free operads is discussed. It is shown how to construct a formal pointed dg-manifold that controls the deformation theory of an algebra over an operad. As an example the \(A_{\infty}\)-operad and \(A_{\infty}\)-algebras are considered. It leads to the (homotopy) action of the Grothendieck-Teichmüller group on the dg-manifold controlling the deformation theory of the Hochschild complex of an \(A_{\infty}\)-algebra.
The third section gives the main result of the paper and sets out the strategy to prove it. For an \(A_{\infty}\)-algebra \(A\) one has moduli spaces \(\mathcal M(\mathcal A_{\infty}, A)\) and \(\mathcal M_{\text{cat}}(\mathcal A_{\infty},A)\). One also has a moduli space \(\mathcal M(P,C)\), where \(C=C^{\bullet}(A,A)\) is the Hochschild complex. The principal aim of the paper is to construct an explicit dg-map \(\mathcal M(\mathcal A_{\infty},A)\rightarrow\mathcal M(P,C)\) as well as a dg-map \(\mathcal M_{\text{cat}}(\mathcal A_{\infty},A)\rightarrow\mathcal M(P,C)\) such that the one is obtained from the other by restriction from the moduli space of algebras to the moduli space of categories. The fourth section deals with free resolutions of operads. This is essential in the approach to prove the result of the previous section.
In section five the notion of a minimal operad acting on the Hochschild complex of an \(A_{\infty}\)-algebra is introduced. This operad turns out to be closely related to the compactification of the configuration spaces of points as introduced by W. Fulton and R. MacPherson in [“A compactification of configuration spaces”, Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)]. The sixth section gives the proof of the main result. In section seven the following form of Deligne’s conjecture is proved: Let \(P\) be the free resolution of the minimal operad \(M\) as constructed in previous sections. Then there is a quasi-isomorphism of dg-operads \(P\rightarrow\text{Chains}(FM_2)\), where \(\text{Chains}(FM_2)\) denotes the chain operad for the Fulton-MacPherson operad of configurations of points in \(\mathbb R^2\). Several conjectures that generalize Deligne’s conjecture in some senses are presented.
The last section is an appendix on singular chains and differential forms. The paper closes with a list of references.
For the entire collection see [Zbl 0947.00049].

MSC:

18D50 Operads (MSC2010)
14H99 Curves in algebraic geometry
16S80 Deformations of associative rings

Citations:

Zbl 0820.14037
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