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

Ayten Koç, Murad Özaydın

TL;DR

The work develops a categorical framework to classify Leavitt path algebras with polynomial growth up to Morita equivalence by translating graph reductions into Morita-invariant filtrations of the module category. It provides explicit bases and a height-based GKdim formula for graphs with disjoint cycles, and it identifies the weighted Hasse diagram as a Morita invariant that, for GKdim$<4$, completely determines the Morita type through complete reductions. The results yield concrete classifications of simple modules and their extensions and establish a suite of invariants (Serre filtrations, $G$-polynomials, and reduction data) that can be read off from the module category. This bridges graph-theoretic reduction procedures with algebraic Morita theory, offering practical criteria for distinguishing non-Morita-equivalent LPAs in the low GKdim regime.

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 $Γ$, 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.

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

TL;DR

The work develops a categorical framework to classify Leavitt path algebras with polynomial growth up to Morita equivalence by translating graph reductions into Morita-invariant filtrations of the module category. It provides explicit bases and a height-based GKdim formula for graphs with disjoint cycles, and it identifies the weighted Hasse diagram as a Morita invariant that, for GKdim, completely determines the Morita type through complete reductions. The results yield concrete classifications of simple modules and their extensions and establish a suite of invariants (Serre filtrations, -polynomials, and reduction data) that can be read off from the module category. This bridges graph-theoretic reduction procedures with algebraic Morita theory, offering practical criteria for distinguishing non-Morita-equivalent LPAs in the low GKdim regime.

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 , 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.
Paper Structure (18 sections, 32 theorems, 72 equations)

This paper contains 18 sections, 32 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Digraphs
  4. Leavitt Path Algebras
  5. $L(\Gamma)$-modules and Quiver Representations
  6. The Reduction Algorithm
  7. Normalization/Pincushion
  8. Jellyfish
  9. Gelfand-Kirillov Dimension
  10. A Basis for $L_{\bf k}(\Gamma)$ when $\Gamma$ is a finite digraph whose cycles are pairwise disjoint
  11. Gelfand-Kirillov Dimension of $L_{\bf k}(\Gamma)$
  12. Simple $L(\Gamma)$-modules and Extensions
  13. Simple $L(\Gamma)$-modules
  14. Extensions
  15. Morita Invariants of $L_{\mathbb{F}}(\Gamma)$
  16. ...and 3 more sections

Key Result

Proposition 2

Let $G:= F_{E}$ be the free group on the set of arrows. The $G$-grading defined by $\vert v\vert_{_G}=1$, $\vert e\vert_{_G}= e$ and $\vert e^*\vert_{_G}=e^{-1}$ is an initial object in the category of $G$-gradings of $L(\Gamma)$ with the generators $V \sqcup E \sqcup E^*$ being homogeneous.

Theorems & Definitions (46)

  • Proposition 2
  • Proposition 3
  • Example 5
  • Proposition 6
  • Example 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • Lemma 11
  • Proposition 12
  • ...and 36 more