Abstract
In our main result, we establish that any conical sandpile monoid \(M = \textrm{SP}(E)\) of a directed sandpile graph E can be realised as the \({\mathscr {V}}\)-monoid of a weighted Leavitt path algebra \(L_{\textsf{k}}(F,w)\) (where F is an explicitly constructed subgraph of E), and consequently, the sandpile group \({\mathscr {G}}(E)\) is realised as the Grothendieck group \(K_0(L_{\textsf{k}}(F,w))\). Additionally, we describe the conical sandpile monoids which arise as the \({\mathscr {V}}\)-monoid of a standard (i.e., unweighted) Leavitt path algebra.
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The authors are extremely grateful to the two anonymous referees, both of whom provided valuable and insightful comments that helped us improve the original version of this article.
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A part of this work was done when the second author was an Alexander von Humboldt Fellow at the University of Münster in the Winter of 2021. He would like to thank both institutions for an excellent hospitality.
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Abrams, G., Hazrat, R. Connections between abelian sandpile models and the K-theory of weighted Leavitt path algebras. European Journal of Mathematics 9, 21 (2023). https://doi.org/10.1007/s40879-023-00613-4
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DOI: https://doi.org/10.1007/s40879-023-00613-4