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 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 SP(E) and the lattice of all ideals of the weighted Leavitt path algebra L_k(E,ω) generated by vertices. Also, we describe the sandpile group of a sandpile graph E via archimedean classes of SP(E), and prove that all maximal subgroups of 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 SP(E) is a chain. Consequently, we completely describe the structure of the weighted Leavitt path algebra of a sandpile graph E such that SP(E) has exactly two idempotents.