Unital algebras being Morita equivalent to weighted Leavitt path algebras

In this article, we describe the endomorphism ring of a finitely generated progenerator module of a weighted Leavitt path algebra Lk(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 ϵ in Lk(E,w), there exists a positive integer n such that Mn(ϵLk(E,w)ϵ) 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.