Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

TL;DR

This paper links Abelian sandpile models to the K-theory of weighted Leavitt path algebras, showing how sandpile monoids and groups can be realized within algebraic structures derived from directed graphs.

Contribution

It establishes a correspondence between sandpile monoids and the $ ext{V}$-monoids of weighted Leavitt path algebras, providing explicit constructions from graphs.

Findings

Sandpile monoids can be realized as $ ext{V}$-monoids of weighted Leavitt path algebras.

Sandpile groups correspond to Grothendieck groups $K_0$ of these algebras.

Characterization of conical sandpile monoids arising from unweighted Leavitt path algebras.

Abstract

In our main result, we establish that any conical sandpile monoid $M = SP(G)$ of a directed sandpile graph $G$ can be realised as the $\mathcal{V}$-monoid of a weighted Leavitt path algebra $L_K(E,w)$, and consequently, the sandpile group as the Grothendieck group $K_0(L_K(E,w))$. We show how to explicitly construct $(E,w)$ from $G$. Additionally, we describe the conical sandpile monoids which arise as the $\mathcal{V}$-monoid of a standard (i.e., unweighted) Leavitt path algebra.