Subsets of Vertices Give Morita Equivalences of Leavitt Path Algebras

Abstract

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in-or out-delaying of a graph, all fit into this setting.

Full idempotents in Leavitt path algebras

“…In [10,Theorem 3], it was shown that, for any directed graph E and for any commutative ring R with identity, there is a subset V ⊂ E 0 by which we can define a Leavitt path algebra such that it is Morita equivalent to L R (E). In this paper, for any directed graph E and any field K, we give necessary and sufficient conditions for which there is subset V ⊂ E 0 by which we can define a full idempotent in L K (E).…”