On the deformation complex of homotopy affine actions

2016/12/19 by Eduardo Hoefel, Hoefel, Eduardo, Muriel Livernet +3
Mathematics · #18D50 #18G55 #Algebraic Topology (math.AT) #FOS: Mathematics #Quantum Algebra (math.QA) #math.AT #math.QA #msc:18D50 #msc:18G55

paper · pdf · doi:10.48550/arxiv.1612.06363

40 pages, 5 figures

arxiv created 2016/12/19 · arxiv updated 2016/12/20

Abstract

An affine action of an associative algebra A on a vector space V is an algebra morphism A → V \rtimes \rm End(V), where V is a vector space and V \rtimes \rm End(V) is the algebra of affine transformations of V. The one dimensional version of the Swiss-Cheese operad, denoted \mathrm\bfsc₁, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by (\mathrm\bfsc₁)_∞. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an A_∞-morphism A → V \rtimes \rm End(V), where A is an A_∞-algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an \rm SC₂ operad. That structure is naturally compatible with the \rm E₂ structure on the deformation complex of the A_∞-algebra.

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