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Stated Skeins and DAHAs

Raymond Matson, Peter Samuelson

Abstract

Skein algebras of surfaces quantize character varieties of topological surfaces, and in low genus, these quantizations are often related to algebras arising in representation theory. For example, Terwilliger defined a universal $SL_2$ spherical double affine Hecke algebra $A$; a combination of results in the literature shows $A$ is isomorphic to the $SL_2$ skein algebra of the punctured torus. Stated skein algebras are a generalization which quantize decorated character varieties. In this paper, we used stated skein algebras to construct a new embedding of $A$ into a rank 6 quantum torus, and we show that each marked 3-manifold with a torus boundary produces a module over $A$. We also determine a generating set for the stated skein algebra of $T^2\setminus D^2$, and we find many relations; however, finding a complete list of relations is still an open problem.

Stated Skeins and DAHAs

Abstract

Skein algebras of surfaces quantize character varieties of topological surfaces, and in low genus, these quantizations are often related to algebras arising in representation theory. For example, Terwilliger defined a universal spherical double affine Hecke algebra ; a combination of results in the literature shows is isomorphic to the skein algebra of the punctured torus. Stated skein algebras are a generalization which quantize decorated character varieties. In this paper, we used stated skein algebras to construct a new embedding of into a rank 6 quantum torus, and we show that each marked 3-manifold with a torus boundary produces a module over . We also determine a generating set for the stated skein algebra of , and we find many relations; however, finding a complete list of relations is still an open problem.

Paper Structure

This paper contains 24 sections, 13 theorems, 72 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Classical skein theory -- Kauffman bracket skein modules
  4. Kauffman bracket skein modules
  5. The Kauffman bracket skein algebra in genus 1
  6. The $\mathfrak{sl}_2$ double affine Hecke algebra
  7. The Poincarè-Birkhoff-Witt property
  8. The spherical subalgebra
  9. Previous Relationships
  10. The Stated Skein Algebra of the Torus with Boundary
  11. Stated Skein Algebras
  12. Generators of $\mathscr{S}(T^2 \setminus D^2)$
  13. DAHA modules from quantum tori
  14. General surfaces
  15. The punctured torus embedding
  16. ...and 9 more sections

Key Result

Theorem 2.6

The map $f:K_q(T^2) \to A_q^{\mathbb{Z}_2}$ given by $f((m,l)_T) = e_{m,l}+e_{-m,-l}$ is an isomorphism of algebras.

Figures (7)

  • Figure 1: Left: When $(p,q) = (0,1)$ we get $y_1 = x_1$ and $y_2 = x_2$ and so each $\gamma_i$ is trivial. Middle: When $\gamma_0$ has a positive slope, we trace out a geodesic path clockwise from $y_i$ to $x_i$. Right: When $\gamma_0$ has a negative slope we move counterclockwise instead.
  • Figure 2: Example of $f_{A}\left(0,1,\frac{1}{2}\right)$ and $f_{A}\left(1,1,\frac{1}{2}\right)$
  • Figure 3: Turning the tangle, $\alpha$, in the complement of $A$ into a closed curve on the torus via $\Psi$. This example becomes a curve in $T^2$ with classification $(2,3)$
  • Figure 4: Left: Example of a curve with a positive slope. Right: Example with a negative slope.
  • Figure 5: A quasitriangulation of $T^2 \setminus D^2$
  • ...and 2 more figures

Theorems & Definitions (36)

  • Definition 2.1: Prz91
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 2.6: FG00
  • Theorem 2.7: BP00
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 26 more