Graphs with disjoint cycles, classification via the talented monoid

TL;DR

This paper characterizes directed graphs with disjoint cycles using their talented monoids and relates this to the finite Gelfand-Kirillov dimension of their associated Leavitt path algebras, supporting the graded classification conjecture.

Contribution

It provides a characterization of graphs with disjoint cycles via Jordan-Hölder series of talented monoids and links this to the Gelfand-Kirillov dimension of Leavitt path algebras.

Findings

Graphs with disjoint cycles have talented monoids with specific Jordan-Hölder series.

The Gelfand-Kirillov dimension of Leavitt path algebras can be determined by ideal series of talented monoids.

Isomorphic graded Grothendieck groups imply equal Gelfand-Kirillov dimensions, supporting the classification conjecture.

Abstract

We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph $E$ consists of disjoint cycles precisely when its talented monoid $T_E$ has a certain Jordan-H\"older composition series. These are graphs whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K (E))$, we conclude that for graphs $E$ and $F$, if $K_0^{gr}(L_K (E))\cong K_0^{gr}(L_K (F))$ then $GKdim L_K(E) = GKdim L_K(F)$, thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.