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arXiv · 2312.15704

Unital algebras being Morita equivalent to weighted Leavitt path algebras

Roozbeh Hazrat·Tran Giang Nam
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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.

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Roozbeh Hazrat, Tran Giang Nam. 2023-12-25. Unital algebras being Morita equivalent to weighted Leavitt path algebras. https://arxiv.org/abs/2312.15704

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