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arXiv · 2110.01180

Graphs with disjoint cycles, classification via the talented monoid

Roozbeh Hazrat·Alfilgen N. Sebandal·Jocelyn P. Vilela
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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.

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Roozbeh Hazrat, Alfilgen N. Sebandal, Jocelyn P. Vilela. 2021-10-04. Graphs with disjoint cycles, classification via the talented monoid. https://arxiv.org/abs/2110.01180

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