The Commuting Algebra

TL;DR

This paper investigates a class of algebras derived from path algebras of finite quivers, showing they are finite dimensional, have finite global dimension, and are Morita equivalent to incidence algebras.

Contribution

It establishes that the algebra $KQ/C$ is finite dimensional, has finite global dimension, and is Morita equivalent to an incidence algebra, extending the understanding of these algebraic structures.

Findings

$KQ/C$ is always finite dimensional.

$KQ/C$ has finite global dimension.

$KQ/C$ is Morita equivalent to an incidence algebra.

Abstract

Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all differences of parallel paths in $Q$. We show that $KQ/C$ is always finite dimensional and its global dimension is finite. Furthermore, we prove that $KQ/C$ is Morita equivalent to an incidence algebra. The paper starts with the more general setting, where $KQ$ is replaced by $KQ/I$ with $I$ a two-sided ideal in $KQ$.