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Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

Tran Quang Do, Roozbeh Hazrat, Tran Giang Nam

TL;DR

The paper proves Williams' conjecture for graphs of Gelfand-Kirillov dimension $3$, showing that shift equivalence and strong shift equivalence coincide for this class and that these dynamical relations are equivalent to graded Morita equivalence of the associated Leavitt path algebras and to isomorphisms of graded $K$-theory. It does so by connecting symbolic dynamics to Leavitt path algebras via the talented monoid and Krieger’s dimension group, reducing GK-$3$ graphs to a normal form, and deriving a concrete, number-theoretic SSE criterion. The main result yields a chain of equivalences among adjacency-matrix relations, graded Morita theory, and invariant $K_0^{ ext{gr}}$, with further consequences for graph $C^*$-algebras and singularity categories. The work thus extends Williams' program to a new algebraic setting, providing tools to classify graphs and their algebras through combinatorial invariants.

Abstract

A graph of Gelfand-Kirillov dimension three is a connected finite essential graph such that its Leavitt path algebra has Gelfand-Kirillov dimension three. We provide number-theoretic criteria for graphs of Gelfand-Kirillov dimension three to be strong shift equivalent. We then prove that two graphs of Gelfand-Kirillov dimension three are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K^{\text{gr}}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, we obtain that the Leavitt path algebras of graphs of Gelfand-Kirillov dimension three are graded Morita equivalent if and only if their graph $C^*$-algebras are equivariant Morita equivalent, and two graphs $E$ and $F$ of Gelfand-Kirillov dimension three are shift equivalent if and only if the singularity categories $\text{D}_{\text{sg}}(KE/J_E^2)$ and $\text{D}_{\text{sg}}(KF/J_F^2)$ are triangulated equivalent.

Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

TL;DR

The paper proves Williams' conjecture for graphs of Gelfand-Kirillov dimension , showing that shift equivalence and strong shift equivalence coincide for this class and that these dynamical relations are equivalent to graded Morita equivalence of the associated Leavitt path algebras and to isomorphisms of graded -theory. It does so by connecting symbolic dynamics to Leavitt path algebras via the talented monoid and Krieger’s dimension group, reducing GK- graphs to a normal form, and deriving a concrete, number-theoretic SSE criterion. The main result yields a chain of equivalences among adjacency-matrix relations, graded Morita theory, and invariant , with further consequences for graph -algebras and singularity categories. The work thus extends Williams' program to a new algebraic setting, providing tools to classify graphs and their algebras through combinatorial invariants.

Abstract

A graph of Gelfand-Kirillov dimension three is a connected finite essential graph such that its Leavitt path algebra has Gelfand-Kirillov dimension three. We provide number-theoretic criteria for graphs of Gelfand-Kirillov dimension three to be strong shift equivalent. We then prove that two graphs of Gelfand-Kirillov dimension three are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded -theories, , are order-preserving -module isomorphic. As a consequence, we obtain that the Leavitt path algebras of graphs of Gelfand-Kirillov dimension three are graded Morita equivalent if and only if their graph -algebras are equivariant Morita equivalent, and two graphs and of Gelfand-Kirillov dimension three are shift equivalent if and only if the singularity categories and are triangulated equivalent.

Paper Structure

This paper contains 9 sections, 19 theorems, 87 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Monoids, Graphs and Algebras
  3. Monoids, ℤ-monoids and order ideals
  4. Graphs and associated monoids
  5. Leavitt path algebras
  6. Symbolic Dynamics
  7. The dynamics of graphs of Gelfand-Kirillov dimension 3
  8. Williams' Conjecture for graphs of Gelfand-Kirillov dimension $3$
  9. Acknowledgements

Key Result

Lemma 2.3

For every row-finite graph $E$, the following statements hold: $(1)$ If $a,b\in F_E\setminus\left\{0\right\}$, then $a=b$ in $M_E$ if and only if there exists $c\in F_E$ such that $a\to c$ and $b\to c$. (Note that, in this case, $a=b=c$ in $M_E$.) $(2)$ If $a=a_1+a_2$ and $a\to b$ in $F_E$, then the

Figures (5)

  • Figure 1: out-splitting at $v$ makes new multi-source vertices $u_1, u_2, u_3$ instead of $v$ which $d(u_1),d(u_2),d(u_3) > d(v)$.
  • Figure 2: out-splitting at $v$ makes new multi-range vertices $w_1, w_2$ instead of $v$ which $\ell(w_1), \ell(w_2) > \ell(v)$.
  • Figure 3: Out-splitting at a vertex on a source cycle. The dashed arrows represent trails that may or may not exist, but nevertheless are not modified by the graph move.
  • Figure 4: Out-splitting at $v$ or out-amalgamation.
  • Figure 5: In-splitting at $v$ or in-amalgamation.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: amp
  • Definition 2.4: hazli
  • Theorem 2.5: hazli
  • Definition 2.6
  • Conjecture 2.7: The Graded Classification Conjecture
  • Definition 2.8: lpabook
  • Definition 2.9: lpabook
  • Theorem 2.10: Williams williams
  • ...and 34 more