Classification of unital simple Leavitt path algebras of infinite graphs

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 LK(E) and LK(F) are simple Leavitt path algebras, then LK(E) is Morita equivalent to LK(F) if and only if K0alg(LK(E))≅K0alg(LK(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 K1alg(LK(E))≅K1alg(LK(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.