Quivers with Polynomial Identities

TL;DR

This paper characterizes quivers whose path algebras satisfy polynomial identities, introduces locally A-graded algebras, and provides examples of PI quiver algebras beyond path algebras.

Contribution

It offers a topological characterization of PI quiver path algebras and introduces the concept of locally A-graded algebras, expanding understanding of polynomial identities in quiver algebras.

Findings

Characterization of quivers with PI path algebras including cycles and acyclic cases

Introduction of locally A-graded algebras that are also PI

Existence of quiver algebras satisfying polynomial identities beyond path algebras

Abstract

We provide a topological characterization of quivers whose path algebra satisfies a polynomial identity. This class includes the oriented cycle and acyclic quivers and, in the latter case, we describe the associated T-ideal. We introduce a generalization of Arnold's A-graded algebras, which we call locally A-graded algebras, and prove that they are also PI. We give an example of a quiver algebra satisfying a polynomial identity, even if the path algebra of the quiver does not.