Projective Representations of Mapping Class Groups in Combinatorial Quantization

Abstract

Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The graph algebra 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 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. their regular support and their useful remarks. I thank A. Gainutdinov for inviting me to present this work at the algebra seminar of the University of Hamburg. Notations.If A is a C-algebra, V is a finite-dimensional A-module and x ∈ A, we denote by V x ∈ End C (V ) the representation of x on the module V . Similarly, if X ∈ A ⊗n and if V 1 , . . . , V n are A-modules, we denote by V 1 ...Vn X the representation of X on V 1 ⊗ . . . ⊗ V n . Here we consider only finite-dimensional representations, hence H-module implicitly means finite-dimensional H-module. IJ R 12 I T 1 J T 2 = J T 2 I T 1 IJ R 12 . Since H is finite-dimensional, it exists right and left integrals µ r , µ l ∈ O(H) defined by I B or I M , let J V be J A or J B or J M and let i < j. Then, by definition of the right action and by (18):

Unrestricted quantum moduli algebras, II: Noetherianity and simple fraction rings at roots of $1$

“…Besides, the quantum moduli algebras are very interesting objects in themselves. They are now recognized as central objects from the viewpoints of factorization homology [18], (stated) skein theory [19,41,29] and, as already said, the mapping class group representations associated to topological quantum field theories [40].…”

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

“…In [Fai18b], we generalize Theorem 5.8 and Theorem 5.12 to higher genus. The projective representation of the mapping class group is shown to be equivalent to that constructed by Lyubashenko using the coend of a ribbon category [Lyu95].…”

“…As pointed out in Remark 6, the matrices I C corresponding to the boundary circle vanishes on SLF(H). Thus applying ρ SLF amounts to gluing back the disk D. See [Fai18b] for the generalization of this fact to higher genus.…”

“…This isomorphism was stated in [Ale94] for H = U q (g), q generic, but it can be generalized to our assumptions on H (see [Fai18b,Prop. 3.5]).…”

“…a ✷ It can be shown that I C satisfies the fusion relation of L 0,1 (−)−1 . The details in our general setting are given in[Fai18b] for arbitrary genus.…”