Homotopy Gerstenhaber structure on deformation complex of a morphism

Abstract

$G_{\infty }$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_{\infty }$-algebra by an associative algebra. Actions of $B_{\infty }$-algebras on associative and $B_{\infty }$-algebras are analyzed, extensions of $B_{\infty }$-algebras by associative and $B_{\infty }$-algebras, that they act upon, are constructed. The resulting $G_{\infty }$-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the $G_{\infty }$-algebra on deformation complex of the corresponding diagram algebra.

24 pages