Purely infinite simple ultragraph Leavitt path algebras

TL;DR

This paper characterizes when ultragraph Leavitt path algebras are purely infinite simple or von Neumann regular, classifying graded simple cases into three algebraic types.

Contribution

It provides necessary and sufficient conditions for ultragraph Leavitt path algebras to be purely infinite simple and classifies graded simple cases.

Findings

Ultragraph Leavitt path algebras are purely infinite simple under specific conditions.

Graded simple ultragraph Leavitt path algebras are either locally matricial, matrix rings over Laurent polynomials, or purely infinite simple.

The paper establishes criteria linking ultragraph properties to algebraic simplicity and regularity.

Abstract

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $L_K(\mathcal{G})$ of an ultragraph $\mathcal{G}$ over a field $K$ is purely infinite simple and that it is von Neumann regular. Consequently, we obtain that every graded simple ultragraph Leavitt path algebra is either a locally matricial algebra, or a full matrix ring over $K[x, x^{-1}]$, or a purely infinite simple algebra.