On the deformation complex of homotopy affine actions

International audienceAn affine action of an associative algebra A on a vector space V is an algebra morphism A→V⋊End(V) , where V is a vector space and V⋊End(V) is the algebra of affine transformations of V . The one dimensional version of the Swiss-cheese operad, denoted sc1 , is the operad whose algebras are affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by (sc1)∞ . Algebras over this minimal model are called Homotopy Affine Actions , they consist of an A∞ -morphism A→V⋊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 SC2 operad. That structure is naturally compatible with the E2 structure on the deformation complex of the A∞ -algebra