The talented monoid of a directed graph with applications to graph algebras

Abstract

It is a conjecture that for the class of Leavitt path algebras associated to finite directed graphs, their graded Grothendieck groups $K_0^{\mathrm{gr}}$ are a complete invariant. For a Leavitt path algebra $L_{\mathsf{k}}(E)$, with coefficients in a field $\mathsf{k}$, the monoid of the positive cone of $K_0^{\mathrm{gr}}(L_{\mathsf{k}}(E))$ can be described completely in terms of the graph $E$. In this note we further investigate the structure of this “talented monoid”, showing how it captures intrinsic properties of the graph and hence the structure of its associated Leavitt path algebras. More precisely, we show that the standard graph moves that give graded Morita equivalence of Leavitt path algebras also preserve the associated talented monoids and, for the class of strongly connected graphs, we show that the notion of the period of a graph can be completely described via the talented monoid. As an application, we give a finer characterisation of the purely infinite simple Leavitt path algebras in terms of properties of the associated graph. We show that graded isomorphisms of algebras preserve the period of the graphs, and obtain results giving more evidence to support the graded classification conjecture.