The matrix type of purely infinite simple Leavitt path algebras

TL;DR

This paper characterizes when matrix rings over purely infinite simple unital Leavitt path algebras are isomorphic, based on the order of the identity element in the algebra's Grothendieck group.

Contribution

It provides a complete classification of matrix ring isomorphisms over these algebras using K-theoretic invariants.

Findings

Determines all pairs (c,d) with M_c(R) ≅ M_d(R).

Connects matrix ring isomorphisms to the order of [1_R] in K_0(R).

Advances understanding of algebraic structure of Leavitt path algebras.

Abstract

Let $R$ denote the purely infinite simple unital Leavitt path algebra $L(E)$. We completely determine the pairs of positive integers $(c,d)$ for which there is an isomorphism of matrix rings $M_c(R)\cong M_d(R)$, in terms of the order of $[1_R]$ in the Grothendieck group $K_0(R)$.