Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

Renaud Detcherry, Ramanujan Santharoubane

Abstract

We construct embeddings of Kauffman bracket skein algebras of surfaces (either closed or with boundary) into localized quantum tori using the action of the skein algebra on the skein module of the handlebody. We use those embeddings to study representations of Kauffman skein algebras at roots of unity and get a new proof of Bonahon-Wong's unicity conjecture. Our method allows one to explicitly reconstruct the unique representation with fixed classical shadow, as long as the classical shadow is irreducible with image not conjuguate to the quaternion group.

An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

Abstract

We construct embeddings of Kauffman bracket skein algebras of surfaces (either closed or with boundary) into localized quantum tori using the action of the skein algebra on the skein module of the handlebody. We use those embeddings to study representations of Kauffman skein algebras at roots of unity and get a new proof of Bonahon-Wong's unicity conjecture. Our method allows one to explicitly reconstruct the unique representation with fixed classical shadow, as long as the classical shadow is irreducible with image not conjuguate to the quaternion group.
Paper Structure (17 sections, 28 theorems, 128 equations, 7 figures)

This paper contains 17 sections, 28 theorems, 128 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Embeddings of skein algebras of surfaces into localized quantum tori
  3. Localized quantum tori
  4. An embedding of skein algebras through curve operators
  5. Injectivity of $\sigma$
  6. Proof of Proposition \ref{['prop:expand']}
  7. Localized quantum torus associated to a sausage graph
  8. Representation of the skein algebra
  9. Action of the $Q$ operators
  10. Action of a one-cycle edge shift
  11. Action of a two-cycle edge shift
  12. Action of the square of a separating edge shift
  13. Commutation between the square of an edge and a one-cycle
  14. Commutation between the square of an edge and a two-cycle
  15. Proof of Theorem \ref{['thm:lift_rep']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

There is an injective $\mathbb{Z}[A^{\pm 1}]$-algebra homomorphism that factors through the natural action $S(\Sigma) \longrightarrow \mathrm{End}(S(H,\mathbb{Q}(A)))$ where $H$ is a handlebody with boundary $\hat{\Sigma},$ such that curves in $\mathcal{P}$ bound a disk in $H.$ Moreover for any curve $\alpha \in \mathcal{P}$ there exists $Q \in \{Q_1,\ldots,Q_n,C_1

Figures (7)

  • Figure 1: Fusion rules for computing curve operators in the basis $\varphi_c.$ Thick edges represent edges of the trivalent graph $\Gamma,$ which are colored by integers, while dashed arcs are colored by 1. We let $\lbrace n \rbrace=A^{2n}-A^{-2n}.$
  • Figure 2: A multicurve in Dehn-Thurston position follows one of the above patterns in the pants and annuli of the decomposition.
  • Figure 3: On the top the different patterns of the intersection of a multicurve (in black) with an annulus or pants piece of the decomposition. The trivalent graph $\Gamma$ is shown in red. On the bottom the remaining patterns after fusion.
  • Figure 4: When $\partial \Sigma = \emptyset$, the two rightmost curves coincide.
  • Figure 5: The $\beta$ and $\gamma$ curves
  • ...and 2 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 43 more