On structural connections between sandpile monoids and weighted Leavitt path algebras

TL;DR

This paper explores the deep algebraic connections between sandpile monoids, their associated graphs, and weighted Leavitt path algebras, revealing lattice isomorphisms, group structures, and algebraic invariants.

Contribution

It establishes isomorphisms between lattices of idempotents, ideals, and hereditary subsets, and characterizes the structure of Leavitt path algebras related to sandpile graphs.

Findings

Lattice of idempotents of SP(E) is isomorphic to hereditary subsets

Sandpile group described via archimedean classes

Structure of Leavitt path algebra characterized by graded ideals

Abstract

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 $\text{SP}(E)$ 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 $\text{SP}(E)$ 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 $\text{SP}(E)$, and prove that all maximal subgroups of $\text{SP}(E)$ are exactly the Grothendieck groups of these archimedean classes. Finally, we give the structure of the Leavitt path algebra $L_{K}(E)$ of a sandpile graph $E$ via a finite chain of graded ideals being invariant under every graded automorphism of $L_{K}(E)$, and completely describe the structure of $L_{K}(E)$ such that the lattice of all idempotents of $\text{SP}(E)$ is a chain. Consequently, we completely describe the structure of the weighted Leavitt path algebra of a sandpile graph $E$ such that $\text{SP}(E)$ has exactly two idempotents.