The commuting algebra

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 a more general setting, where KQ is replaced by KQ/I with I a two-sided ideal in KQ. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper studies the ideal C in the path algebra KQ, proving that KQ/C is always finite dimensional with finite global dimension, and it is Morita equivalent to an incidence algebra.