$L_{\infty }$-structure on Barzdell's complex for monomial algebras

Abstract

Let $A$ be a monomial associative finite dimensional algebra over a field $\mathbb{k}$ 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 explict $L_{\infty }$-structure on $B(A)$ that induces a weak equivalence of $L_{\infty }$-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 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.