Leavitt path algebras of labelled graphs

TL;DR

This paper introduces Leavitt labelled path algebras associated with labelled spaces, generalizing existing algebraic structures, and explores their realizations, isomorphisms, simplicity, and connections to partial skew group rings.

Contribution

It generalizes Leavitt path algebras to labelled spaces and establishes their realizations as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras.

Findings

Realization of Leavitt labelled path algebras as various algebraic structures

Generalized uniqueness theorems for these algebras

Characterization of simplicity and isomorphisms

Abstract

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.