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Quantum cluster variables via canonical submodules

Fan Xu, Yutong Yu

Abstract

We study quantum cluster algebras from marked surfaces without punctures. We express the quantum cluster variables in terms of the canonical submodules. As a byproduct, we obtain the positivity for this class of quantum cluster algebra.

Quantum cluster variables via canonical submodules

Abstract

We study quantum cluster algebras from marked surfaces without punctures. We express the quantum cluster variables in terms of the canonical submodules. As a byproduct, we obtain the positivity for this class of quantum cluster algebra.

Paper Structure

This paper contains 14 sections, 31 theorems, 129 equations.

Table of Contents

  1. INTRODUCTION
  2. 2-CALABI-YAU TRIANGULATED CATEGORY
  3. QUANTUM CLUSTER ALGEBRA
  4. CLUSTER CATEGORY OF SURFACE
  5. Gentle algebra
  6. Marked surface
  7. Cluster category of surface
  8. Canonical submodule
  9. PROOF OF THE MAIN THEOREM
  10. Notations
  11. When $\tau_{k-1}\neq \tau_{k+1}$ and $\tau_{k-2}\neq \tau_{k+2}$
  12. When $\tau_{k-1}=\tau_{k+1}$ and $\tau_{k-2}\neq \tau_{k+2}$
  13. Summary
  14. Example

Key Result

Theorem 1.1

There exist functions $v$ and $v'$ valued on $\mathbb{Z}$ satisfying the following polynomial property: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Lemma 4.4: Assem_2010
  • ...and 52 more