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Projective representations of mapping class groups in combinatorial quantization

Matthieu Faitg

Abstract

Let $Σ_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of the moduli space of flat connections on $Σ_{g,n}$. We construct a projective representation of the mapping class group of $Σ_{g,n}$ using $\mathcal{L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.

Projective representations of mapping class groups in combinatorial quantization

Abstract

Let be a compact oriented surface of genus with open disks removed. The graph algebra was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of the moduli space of flat connections on . We construct a projective representation of the mapping class group of using and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.

Paper Structure

This paper contains 20 sections, 21 theorems, 155 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Factorizable ribbon Hopf algebras
  4. Heinsenberg double of $\mathcal{O}(H)$
  5. Definition of $\mathcal{L}_{g,n}(H)$ and the Alekseev isomorphism
  6. Definition and properties of $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$
  7. Braided tensor product and definition of $\mathcal{L}_{g,n}(H)$
  8. The Alekseev isomorphism
  9. Representation of $\mathcal{L}^{\mathrm{inv}}_{g,n}(H)$
  10. The matrices $\overset{I}{C}_{g,n}$
  11. Determination and representation of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$
  12. Projective representations of mapping class groups
  13. Mapping class group of $\Sigma_g$
  14. Dehn twists as automorphisms of $\mathcal{L}_{g,0}(H)$
  15. Representation of the mapping class group
  16. ...and 5 more sections

Key Result

Lemma 3.4

If $N$ satisfies the fusion relation of $\mathcal{L}_{0,1}(H)$, $\overset{I \otimes J}{N}\!\!_{12} = \overset{I}{N}(i)_1\,\overset{IJ}{(R')}_{12}\,\overset{J}{N}(i)_2\, \overset{IJ}{(R')}{_{12}^{-1}}$, then $j_N$ is a morphism of $H$-module-algebras: $(xy)_N = x_N y_N$.

Figures (5)

  • Figure 1: To each generating loop in $\pi_1(\Sigma_{g,n}\!\!\setminus\! D, \bullet)$, we associate a matrix.
  • Figure 2: Surface $\Sigma_g^{\mathrm{o}}$ and generators of $\pi_1(\Sigma_g^{\mathrm{o}})$.
  • Figure 3: Surface $\Sigma_g^{\mathrm{o}}$ viewed as a ribbon graph.
  • Figure 4: A canonical set of curves on the surface $\Sigma_g^{\mathrm{o}}$.
  • Figure 5: A positively oriented loop in the neighborhood of the basepoint fixed in Figure \ref{['figureSurfaceRuban']}.

Theorems & Definitions (28)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Lemma 4.1
  • Definition 4.2
  • Proposition 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 18 more