Centers of path algebras, Cohn and Leavitt path algebras

TL;DR

This paper investigates the centers of various path algebras, revealing conditions under which the center is trivial, a field, or a polynomial algebra, and extends results to prime Cohn and Leavitt path algebras.

Contribution

It provides a comprehensive analysis of the centers of path, Cohn, and Leavitt path algebras, including new bounds for Leavitt path algebra centers using the graded Baer radical.

Findings

Center of path algebra is zero for infinite vertices.

Center is field K for finite non-cycle graphs.

Center is polynomial algebra K[x] for cycle graphs.

Abstract

We study the center of several types of path algebras. We start with the path algebra $KE$ and prove that if the number of vertices is infinite then the center is zero. Otherwise, it coincides with the field $K$ except when the graph $E$ is a cycle in which case the center is $K[x]$, the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras.