Graded irreducible representations of Leavitt path algebras: a new type and complete classification

TL;DR

This paper introduces a new class of graded irreducible representations for Leavitt path algebras, expanding the classification of simple modules beyond existing Chen modules using graph-based primitive ideal characterization.

Contribution

It presents a novel type of graded simple module for Leavitt path algebras, completing the classification of graded simple modules and analyzing primitive ideals.

Findings

New class of graded simple modules introduced

Complete classification of graded simple modules achieved

Characterization of primitive ideals linked to graph properties

Abstract

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type. Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in [K. M. Rangaswamy, Theory of prime ideals of Leavitt path algebras over arbitrary graphs, J. Algebra 375 (2013), 73 -- 90].