On the deformation complex of homotopy affine actions

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An 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.