Simplicity of Leavitt path algebras via graded ring theory

TL;DR

This paper characterizes when Leavitt path algebras are simple using graded ring theory, linking algebraic simplicity to graph properties and ring simplicity, and describes their centers.

Contribution

It provides a new characterization of simplicity for Leavitt path algebras via graded ring theory, connecting algebraic and graph-theoretic conditions.

Findings

Leavitt path algebra is simple iff R is simple, the graph has no nontrivial hereditary and saturated subsets, and every cycle has an exit.

Complete description of the center of a simple Leavitt path algebra.

Bridges algebraic simplicity with graph properties using graded ring theory.

Abstract

Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in $E$ has an exit. We also give a complete description of the center of a simple Leavitt path algebra.