Classification of unital simple Leavitt path algebras of infinite graphs

TL;DR

This paper characterizes when simple Leavitt path algebras of infinite graphs are Morita equivalent, using algebraic K-theory invariants and graph transformations, advancing the classification of these algebras.

Contribution

It provides a complete Morita equivalence classification for unital simple Leavitt path algebras of infinite graphs based on K-theory and graph moves.

Findings

Morita equivalence characterized by K_0^alg and graph singular vertices

K_1^alg equivalence replaces singular vertices condition for certain fields

Graph moves preserve Morita equivalence and aid classification

Abstract

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only if K_0^\textnormal{alg} (L_K(E)) \cong K_0^\textnormal{alg} (L_K(F)) and the graphs $E$ and $F$ have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K_1^\textnormal{alg} (L_K(E)) \cong K_1^\textnormal{alg} (L_K(F)), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras - the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.