Graded chain conditions and Leavitt path algebras of no-exit graphs

TL;DR

This paper provides a comprehensive structural analysis of Leavitt path algebras over no-exit graphs, introducing graded conditions and characterizations that extend existing theories and relax previous restrictions.

Contribution

It offers a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras and introduces non-unital graded analogues of noetherian and artinian rings.

Findings

Characterization of Leavitt path algebras as graded involutive algebras

Graph-theoretic conditions for algebraic properties

Relaxation of assumptions in existing classifications

Abstract

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams-Aranda-Perera-Siles characterization of locally noetherian and locally artinian Leavitt path algebras.