Classification of Leavitt Path Algebras with Gelfand-Kirillov Dimension <4 up to Morita Equivalence

TL;DR

This paper classifies Leavitt path algebras with Gelfand-Kirillov dimension less than 4 up to Morita equivalence using combinatorial and categorical invariants derived from the associated digraphs.

Contribution

It provides an explicit classification of irreducible representations and introduces a Morita invariant filtration and weighted Hasse diagram for these algebras.

Findings

Complete Morita invariant for Gelfand-Kirillov dimension < 4

Explicit classification of irreducible representations

Introduction of Morita invariant filtration and weighted Hasse diagram

Abstract

Leavitt path algebras are associated to di(rected )graphs and there is a combinatorial procedure (the reduction algorithm) making the digraph smaller while preserving the Morita type. We can recover the vertices and most of the arrows of the completely reduced digraph from the module category of a Leavitt path algebra of polynomial growth. We give an explicit classification of all irreducible representations of when the coefficients are a commutative ring with 1. We define a Morita invariant filtration of the module category by Serre subcategories and as a consequence we obtain a Morita invariant (the weighted Hasse diagram of the digraph) which captures the poset of the sinks and the cycles of $\Gamma$, the Gelfand-Kirillov dimension and more. When the Gelfand-Kirillov dimension of the Leavitt path algebra is less than 4, the weighted Hasse diagram (equivalently, the complete reduction of the digraph) is a complete Morita invariant.