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Classification of semi-weight representations of reduced stated skein algebras

H. Karuo, J. Korinman

Abstract

We classify the finite dimensional semi-weight representations of the reduced stated skein algebras at odd roots of unity of connected marked surfaces which either have a boundary component with at least two boundary edges or which do not have any unmarked boundary component. We deduce computations of the PI-degrees and Azumaya loci of unreduced stated skein algebras of essential surfaces having at most one boundary arc per boundary component and of the unrestricted quantum moduli algebras of lattice gauge field theory.

Classification of semi-weight representations of reduced stated skein algebras

Abstract

We classify the finite dimensional semi-weight representations of the reduced stated skein algebras at odd roots of unity of connected marked surfaces which either have a boundary component with at least two boundary edges or which do not have any unmarked boundary component. We deduce computations of the PI-degrees and Azumaya loci of unreduced stated skein algebras of essential surfaces having at most one boundary arc per boundary component and of the unrestricted quantum moduli algebras of lattice gauge field theory.
Paper Structure (28 sections, 65 theorems, 126 equations, 13 figures)

This paper contains 28 sections, 65 theorems, 126 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background on reduced stated skein algebras and their representations
  3. Main results
  4. Plan of the paper
  5. Reduced stated skein algebras
  6. Definition of reduced stated skein algebras
  7. Bases
  8. Center, Azumaya locus and shadows
  9. Classical moduli spaces
  10. Geometric interpretation of $X(\mathbf{\Sigma})$
  11. Smooth loci of moduli spaces
  12. Behavior for the gluing operation
  13. Representations of $\overline{\mathcal{S}}_{A}(\mathbb{D}_1)$
  14. Classification of semi-weight $D_qB$ modules
  15. Representations in the fully Azumaya locus
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ be a connected essential marked surface which either has a boundary component with at least two boundary edges or which does not have any inner puncture. Let $\widehat{x}=(x, h_p, h_{\partial}) \in \widehat{X}(\mathbf{\Sigma})$.

Figures (13)

  • Figure 1: On the left: a stated tangle. On the right: its associated diagram. The arrows represent the height orders.
  • Figure 2: An illustration of the product in stated skein algebras.
  • Figure 3: A bad arc.
  • Figure 4: An illustration of the splitting morphism $\theta_{a\#b}$.
  • Figure 5: An illustration of the proof of Lemma \ref{['lemma_Muller_basis']}.
  • ...and 8 more figures

Theorems & Definitions (140)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1: Marked surfaces
  • Definition 2.2: Tangles and diagrams
  • Definition 2.3: Stated skein algebras
  • Definition 2.4: Reduced stated skein algebras
  • Remark 2.5
  • Definition 2.6: Splitting morphism
  • ...and 130 more