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

TL;DR

This paper constructs an explicit $L_ty$-structure on Bardzell's complex for monomial algebras, enabling better understanding of Hochschild cohomology and Maurer-Cartan elements, with concrete computations for specific algebra types.

Contribution

It introduces an explicit $L_ty$-structure on Bardzell's complex that induces a weak equivalence with the Hochschild complex, facilitating computations and structural insights.

Findings

Bardzell's complex can be endowed with an $L_ty$-structure.

The $L_ty$-structure induces a weak equivalence with the Hochschild complex.

For radical square zero algebras, Bardzell's complex is a dg-Lie algebra.

Abstract

Let $A$ be a monomial associative finite dimensional algebra over a field $\Bbbk$ 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.