On the deformation complex of homotopy affine actions

Abstract

An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, 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 ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtimes {\rm End}(V)$, where $A$ is an $A_\infty$-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 ${\rm SC}_2$ operad. That structure is naturally compatible with the ${\rm E}_2$ structure on the deformation complex of the $A_\infty$-algebra.