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Cluster algebras and skein algebras for surfaces

Hiroaki Karuo, Han-Bom Moon, Helen Wong

TL;DR

This work clarifies the deep connections between generalized cluster algebras for marked surfaces and generalized skein algebras by constructing explicit maps between the Muller-Roger-Yang and Lê-Roger-Yang skein variants and the cluster structure. It proves finite generation for the skein algebras and embeds cluster algebras into skein settings via a boundary-localized, $q=1$ framework, while introducing a robust compatibility via the moving trick and stability under flips. The results reveal that $ ext{A}(oldsymbol{ extSigma})$ sits inside the upper cluster algebra and often inside a skein-constructed subalgebra $ ext{S}^{oxempty}(oldsymbol{ extSigma})$, with nuanced behavior depending on boundary components and interior punctures. The paper also outlines key open problems in the algebraic structure, upper cluster algebra status, and representation theory, highlighting rich interactions with decorated Teichmüller theory and potential applications in quantum topology and geometric representation theory.

Abstract

We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and generalizations of skein algebras that include arcs whose endpoints are marked points on the boundary or in the interior of the surface. We show that the generalizations are closely related by maps that can be explicitly defined, and we explore the structural implications, including (non-)finite generation. We also discuss open questions about the algebraic structure of the algebras.

Cluster algebras and skein algebras for surfaces

TL;DR

This work clarifies the deep connections between generalized cluster algebras for marked surfaces and generalized skein algebras by constructing explicit maps between the Muller-Roger-Yang and Lê-Roger-Yang skein variants and the cluster structure. It proves finite generation for the skein algebras and embeds cluster algebras into skein settings via a boundary-localized, framework, while introducing a robust compatibility via the moving trick and stability under flips. The results reveal that sits inside the upper cluster algebra and often inside a skein-constructed subalgebra , with nuanced behavior depending on boundary components and interior punctures. The paper also outlines key open problems in the algebraic structure, upper cluster algebra status, and representation theory, highlighting rich interactions with decorated Teichmüller theory and potential applications in quantum topology and geometric representation theory.

Abstract

We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and generalizations of skein algebras that include arcs whose endpoints are marked points on the boundary or in the interior of the surface. We show that the generalizations are closely related by maps that can be explicitly defined, and we explore the structural implications, including (non-)finite generation. We also discuss open questions about the algebraic structure of the algebras.

Paper Structure

This paper contains 15 sections, 9 theorems, 23 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Skein algebras
  3. Compatibility between skein algebras
  4. Finite generation of skein algebra
  5. Cluster algebra of surface
  6. Informal description
  7. Formal definition
  8. Upper cluster algebra
  9. Compatibility between skein algebra and cluster algebra
  10. Cluster algebra in skein algebra
  11. Applications of Compatibility
  12. Open questions
  13. Algebraic structure of skein algebras
  14. Upper cluster algebra
  15. Representation theory of skein algebra

Key Result

Theorem A

The generalized skein algebras $\mathcal{S} _{q}^{\mathrm{MRY}}(\Sigma)$ and $\mathcal{S} _{q}^{\mathrm{LRY}}(\Sigma)$ are finitely generated.

Figures (15)

  • Figure 1: A bad arc
  • Figure 2: Moving trick. The numbers under the left diagram describes relative height of three ends at the boundary marked points.
  • Figure 3: A surface $\Sigma$ with a disk $D$ with all interior punctures on it
  • Figure 4: Intersection number reduction -- when $B$ meets a puncture
  • Figure 5: Intersection number reduction -- when $B$ does not meet a puncture
  • ...and 10 more figures

Theorems & Definitions (32)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2: Muller-Roger-Yang skein algebra
  • Definition 2.3: Lê-Roger-Yang skein algebra
  • Remark 2.4
  • Definition 2.5: CL22
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • ...and 22 more