On the semisimplicity of the outer derivations of monomial algebras

TL;DR

This paper investigates the structure of Hochschild cohomology and Lie algebra properties of monomial algebras, revealing conditions under which the cohomology vanishes and the cohomology Lie algebra is reductive or semisimple.

Contribution

It establishes new criteria for the vanishing of Hochschild cohomology and characterizes the Lie algebra structure of Hochschild cohomology for specific classes of monomial algebras.

Findings

Hochschild cohomology vanishes from degree two if the first cohomology is semisimple.

First Hochschild cohomology of radical square zero algebras is reductive.

For multiple loops quivers, the Lie algebra is isomorphic to square matrices of corresponding size.

Abstract

We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.