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Classical shadows of stated skein representations at roots of unity

Julien Korinman, Alexandre Quesney

Abstract

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at +1 and the algebra of regular function of a generalization of the SL2-character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra, an invariant which is a point in the character variety.

Classical shadows of stated skein representations at roots of unity

Abstract

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at +1 and the algebra of regular function of a generalization of the SL2-character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra, an invariant which is a point in the character variety.

Paper Structure

This paper contains 42 sections, 50 theorems, 146 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Stated skein algebras
  4. Definitions and general properties of the stated skein algebras
  5. Alternative bases
  6. Removing a puncture
  7. Hopf comodule maps
  8. The image of the splitting morphism
  9. The center of stated skein algebras at odd roots of unity
  10. The case of the bigon
  11. The case of the triangle
  12. The general case: proof of Theorem \ref{['theorem2']}
  13. Step 1: formal definition.
  14. Step 2: arcs with endpoints in distinct boundary arcs.
  15. Step 3: closed curves that intersect $\Delta$ nicely.
  16. ...and 27 more sections

Key Result

Theorem 1.1

The following sequence is exact: where $\sigma(x\otimes y)= y\otimes x$.

Figures (7)

  • Figure 1: A stated diagram $[D,s]$ in the triangle and its associated stated tangle $[T(D), s]$. Here, we use the order $\gamma \prec \beta \prec \alpha$. Here $s$ is $\mathfrak{o}$-increasing so $[T(D),s]\in \mathcal{TB}^{\mathfrak{o}}$.
  • Figure 2: On the top: the coproduct in $\mathcal{S}_{\omega}(\mathbb{B})$. On the bottom: the comodule map.
  • Figure 3: $(1)$ The three diagrams $\alpha, \beta, \gamma$, $(2)$ the stated diagram representing $\alpha_{\varepsilon \varepsilon'}$ and $(3)$ the diagram $\theta^{(2,1,1)}$.
  • Figure 4: On the top: the element $X_{\eta, \eta'}$. On the bottom: an illustration of Equation \ref{['eq_bidon0']}.
  • Figure 5: Instance of tangles $T_{\mathbb{T}}$ and $T^{(N)}_{\mathbb{T}}$.
  • ...and 2 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 106 more