On structural connections between sandpile monoids and weighted Leavitt path algebras

In this article, we establish the relations between a sandpile graph, its sandpile monoid and the weighted Leavitt path algebra associated with it. Namely, we show that the lattice of all idempotents of the sandpile monoid $\text{SP}\left(E\right)$ of a sandpile graph E is both isomorphic to the lattice of all nonempty saturated hereditary subsets of E, the lattice of all order-ideals of $\text{SP}\left(E\right)$ and the lattice of all ideals of the weighted Leavitt path algebra $L_k(E,\omega )$ generated by vertices. Also, we describe the sandpile group of a sandpile graph E via archimedean classes of $\text{SP}\left(E\right)$, and prove that all maximal subgroups of $\text{SP}\left(E\right)$ are exactly the Grothendieck groups of these archimedean classes. Finally, we give the structure of the Leavitt path algebra $L_k\left(E\right)$ of a sandpile graph E via a finite chain of graded ideals being invariant under every graded automorphism of $L_k\left(E\right)$, and completely describe the structure of $L_k\left(E\right)$ such that the lattice of all idempotents of $\text{SP}\left(E\right)$ is a chain. Consequently, we completely describe the structure of the weighted Leavitt path algebra of a sandpile graph E such that $\text{SP}\left(E\right)$ has exactly two idempotents.