Comparison morphisms between two projective resolutions of monomial algebras

TL;DR

This paper constructs explicit comparison morphisms between two projective resolutions of monomial algebras, enabling simplified descriptions of Hochschild cohomology structures and Lie actions, with implications for computational efficiency.

Contribution

It introduces explicit comparison morphisms between the bar and Bardzell's resolutions, facilitating better understanding of Hochschild cohomology operations for monomial algebras.

Findings

Simplified description of the cup product in even degrees.

Explicit description of the Lie algebra action on Hochschild cohomology.

Applicable to monomial algebras with one-dimensional hom spaces.

Abstract

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution and Bardzell's resolution; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $HH^*(A)$ and the second one has been shown to be an efficient tool for computations of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A=k Q/I$ a monomial algebra such that $dim_k \ e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $HH^1(A)$ on $HH^{\ast}(A)$.