L infty-structure on Barzdell's complex for monomial algebras

Abstract

Let A be a monomial associative finite dimensional algebra over a field k of characteristic zero. It is well known that the Hochschild cohomology of A can be computed using Bardzell´s complex B(A). The aim of this article is to describe an explicit L∞-structure on B(A) that induces a weak equivalence of L∞-algebras between B(A) and the Hochschild complex C(A) of A. This allows us to describe the Maurer-Cartan equation in terms of elements of degree 2 in B(A). Finally, we make<br />concrete computations when A is a truncated algebra, and we prove that Bardzell´s complex for radical square zero algebras is in fact a dg-Lie algebra.