The talented monoid of a Leavitt path algebra

TL;DR

This paper establishes a deep connection between the geometry of directed graphs and the algebraic structure of Leavitt path algebras through an associated monoid, which serves as a complete invariant for algebra classification.

Contribution

It introduces a monoid linked to the graph that reflects the algebraic properties of Leavitt path algebras and demonstrates its role as a complete invariant for classification.

Findings

The monoid is isomorphic to the positive cone of the graded K0-group.

A graph has a cycle without an exit iff the monoid has a periodic element.

The algebraic properties of Leavitt path algebras can be fully described via this monoid.

Abstract

There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts freely on the monoid. We go on to show that the algebraic structure of Leavitt path algebras (such as simplicity, purely infinite simplicity, or the lattice of ideals) can be described completely via this monoid. Therefore an isomorphism between the monoids (or graded K0's) of two Leavitt path algebras implies that the algebras have similar algebraic structures. These all confirm that the graded Grothendieck group could be a sought-after complete invariant for the classification of Leavitt path algebras.