Graphs with disjoint cycles, classification via the talented monoid

2021/10/04 by Hazrat, Roozbeh, Sebandal, Alfilgen N., Vilela, Jocelyn P.
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2110.01180

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-Hölder 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 K₀^gr(L_K (E)), we conclude that for graphs E and F, if K₀^gr(L_K (E))≅ K₀^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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