Connections between Abelian sandpile models and the $K$-theory of weighted 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 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 V-monoid of a standard (i.e., unweighted) Leavitt path algebra.

Leavitt path algebras of weighted and separated graphs

“…and e = f this gives the element e i e * i e i − e i , which belongs to I 0 by Remark 3.2(2). When i = j or e = f this gives an element which also belongs toI 0 .…”

“…Some years later, Bergman found in [9] the precise structure of the monoid V(L(m, n)) of isomorphism classes of finitely generated projective L(m, n)-modules. Recently, an interesting connection between the K-theory of Leavitt path algebras of weighted graphs and the theory of abelian sandpile models has been developed in [2]. We refer the reader to [16] for a nice survey on Leavitt path algebras of weighted graphs.…”

“…and we have ω(e (1) ) = ω(e (2) ) = 3 and ω(e (3) ) = ω(e (4) ) = 1. We can also associate to λ the corresponding matrix of the generators (x ij ) of L(E, w), where x ij = e (j) i , which is the matrix obtained from the full matrix    …”

“…Looking at the algebra L 1 (E, ω) of the weighted graph associated to the (m, n)-partition λ, we can interpret the shape of λ as the refinement matrix R which defines the V-monoid M 1 (E, ω) of L 1 (E, ω) (see Theorem 5.5). For instance for the above partition λ = (4, 2, 2), we have the refinement matrix R =   v(e (1) , 1) v(e (2) , 1) v(e (3) , 1) v(e (4) , 1) v(e (1) , 2) v(e (2) , 2) 0 0 v(e (1) , 3) v(e (2) , 3) 0 0   .…”

Leavitt Path Algebras of Weighted and Separated Graphs

“…Some years later, Bergman reported in [9] the precise structure of the monoid V(L(m, n)) of isomorphism classes of finitely generated projective L(m, n)-modules. Recently, an interesting connection between the K-theory of Leavitt path algebras of weighted graphs and the theory of abelian sandpile models has been developed in [2]. We refer the reader to [16] for an excellent survey on Leavitt path algebras of weighted graphs.…”