Classification of unital simple Leavitt path algebras of infinite graphs

JOURNAL OF ALGEBRA

卷:384

Article

Ruiz, Efren¹  Tomforde, Mark² 

[1] Univ Hawaii, Dept Math, Hilo, HI 96720 USA

[2] Univ Houston, Dept Math, Houston, TX 77204 USA

关键词: Graph algebras;    Graph C*-algebras;    Leavitt path algebras;    Morita equivalence;    Classification;    Flow equivalence;   

DOI  :  10.1016/j.jalgebra.2013.03.004

来源: Elsevier

【 摘 要 】

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(alg)(L-K(E)) congruent to K-0(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(alg)(L-K(E)) congruent to K-1(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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free