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 gr 0 are a complete invariant. For a Leavitt path algebra L k .E/, with coefficients in a field k, the monoid of the positive cone of K gr 0 .L 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.

On Gamma-Ideals, Gamma-Submonoids and Isomorphism Theorems of Gamma-Monoids via Gamma-Submonoids

“…The talented monoid of a row-finite directed graph E = (E 0 , E 1 , r, s), denoted by T E , is the commutative monoid generated by {v(i) : v ∈ E 0 , i ∈ Z} such that v(i) = e∈s −1 (v) r(e)(i + 1) for every i ∈ Z and every v ∈ E 0 that is not a sink. The additive group Z of integers acts on T E by monoid automorphisms by shifting indices: for each n, i ∈ Z and v ∈ E 0 , define n v(i) = v(i + n), which extends to an action of Z on T E [3]. Monoids with a group Γ acting (by monoid automorphisms) on it, called Γ-monoids, was first introduced in the paper of Hazrat and Li [1] as a tool in the study of talented monoids.…”

“…Definition 8. [3] Let M be a monoid and Γ a group. M is said to be a Γ-monoid if there is an action ϕ : Γ × M → M of Γ on M via monoid automorphism, that is, ϕ is an action which satisfies: for all α ∈ Γ and x, y ∈ M , ϕ((α, x * y)) = ϕ((α, x)) * ϕ((α, y)).…”

A Talented Monoid View on Lie Bracket Algebras over Leavitt Path Algebras

“…Proposition 2.3. [9] Let Γ be a group and T a Γ-monoid. For x ∈ T , the Γ-order-ideal generated by x, denoted by x , is given by the set…”

Graphs with disjoint cycles, classification via the talented monoid

“…In this form the talented monoids were introduced in [8] and further studied in [6,9]. It was shown that the talented monoid captures certain geometry of its associated graph, and hence the algebraic properties of the corresponding Leavitt path algebra.…”

“…For instance, a graph has Condition (L), i.e., any cycle has an exit, if and only if the group Z acts freely on T E [8]. Or, the period of a graph, i.e., the greatest common divisor of the lengths of all closed paths based a vertex, can be described completely via its associated talented monoid [6]. Using these results one can give finer descriptions of purely infinite simple Leavitt path algebras associated to graphs.…”