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Maximal green sequences for quantum and Poisson CGL extensions

Milen Yakimov

Abstract

We prove that the quantum and classical cluster algebras for all members of the axiomatically defined classes of symmetric quantum and Poisson Cauchon-Goodearl-Letzter extensions possess maximal green sequences in the sense of Keller. Previously, maximal green sequences were constructed for explicit families of cluster algebras; many of those can be recovered from the general result for CGL extensions.

Maximal green sequences for quantum and Poisson CGL extensions

Abstract

We prove that the quantum and classical cluster algebras for all members of the axiomatically defined classes of symmetric quantum and Poisson Cauchon-Goodearl-Letzter extensions possess maximal green sequences in the sense of Keller. Previously, maximal green sequences were constructed for explicit families of cluster algebras; many of those can be recovered from the general result for CGL extensions.
Paper Structure (16 sections, 13 theorems, 83 equations, 2 figures)

This paper contains 16 sections, 13 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. CGL extensions
  4. Preliminaries on cluster algebras and maximal green sequences
  5. Combinatorial background on cluster algebras
  6. Maximal green sequences
  7. Quantum and Poisson CGL extensions
  8. Quantum CGL extensions
  9. Poisson CGL extensions
  10. Layered $T$-systems and maximal green sequences
  11. Layered $T$-systems
  12. Mutations of the $A_n$ quiver
  13. Proof of Theorem \ref{['tT-syst-green']}
  14. Maximal green sequences for quantum and Poisson CGL extensions
  15. Paths in $\Xi_N$
  16. ...and 1 more sections

Key Result

Theorem A

Each quantum (classical) cluster algebra on a symmetric quantum (Poisson) CGL extension of dimension $N$ possesses maximal green sequences parametrized by all reduced expressions $\boldsymbol{w}$ of the longest element of the symmetric group $S_N$ with the property that its initial subwords are such

Figures (2)

  • Figure 1: Layered $T$-systems
  • Figure 2: Local picture at the mutation

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Definition 3.6
  • Theorem 3.7
  • ...and 24 more