Centers of partly (anti-)commutative quiver algebras and finite generation of the Hochschild cohomology ring

TL;DR

This paper characterizes the centers of partly (anti-)commutative quiver algebras and provides criteria for their Hochschild cohomology rings to be finitely generated, combining combinatorial and algebraic methods.

Contribution

It offers a combinatorial approach to determine the centers and finite generation conditions of Hochschild cohomology for these algebras.

Findings

A combinatorial description of ideals and generator graphs.

Necessary and sufficient conditions for the center to be finitely generated.

Criteria for finite generation of Hochschild cohomology modulo nilpotent elements.

Abstract

A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and sufficient conditions for the center to be finitely generated as a K-algebra. As an application, necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.