Unital algebras being Morita equivalent to weighted Leavitt path algebras

Abstract

In this article, we describe the endomorphism ring of a finitely generated progenerator module of a weighted Leavitt path algebra $L_K(E,w)$ of a finite vertex weighted graph $(E,w)$. Contrary to the case of Leavitt path algebras, we show that a (full) corner of a weighted Leavitt path algebra is, in general, not isomorphic to a weighted Leavitt path algebra. However, using the above result, we show that for every full idempotent $\epsilon$ in $L_K(E,w)$, there exists a positive integer $n$ such that $M_n(\epsilon L_K(E,w)\epsilon )$ is isomorphic to the weighted Leavitt path algebra of a weighted graph explicitly constructed from $(E,w)$. We then completely describe unital algebras being Morita equivalent to weighted Leavitt path algebras of vertex weighted graphs. In particular, we characterize unital algebras being Morita equivalent to sandpile algebras.