The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero

TL;DR

This paper explores the Lie module structure of Hochschild cohomology for monomial algebras with radical square zero, linking combinatorial quiver data to Lie algebra classifications.

Contribution

It provides a combinatorial description of the Lie module structure on Hochschild cohomology for a specific class of monomial algebras, connecting to Lie algebra module classification.

Findings

Describes the Lie module structure via quiver combinatorics

Relates the structure to simple Lie algebra modules in specific cases

Provides a framework for understanding Hochschild cohomology in this context

Abstract

We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.