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

Abstract

Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The 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 . Here we focus on the two building blocks L 0,1 (H) and L 1,0 (H) under the assumption that the gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We construct a projective representation of SL 2 (Z), the mapping class group of the torus, using L 1,0 (H) and we study it explicitly for H = U q (sl (2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.and this give the result sinceIt follows that ψ ∈ H * is symmetric if, and only if, J C (±) ⊲ ψ = ψI dim(J) for all J.

Projective Representations of Mapping Class Groups in Combinatorial Quantization

“…Second, for H = U q (sl(2)), our representations of the mapping class group should be associated to logarithmic conformal field theory in arbitrary genus. For the torus Σ 1,0 , this is indeed the case: combining the results of [FGST06] and [Fai18b], the projective representation of SL 2 (Z) obtained via the combinatorial quantization is equivalent to that coming from logarithmic conformal field theory. Hence, a natural problem is to study in depth the representation of the mapping class group obtained for H = U q (sl(2)) (basis of the representation space, explicit formulas for the action on this basis and structure of the representation).…”

“…This has the advantage to show immediately that L g,n (H) is a H-module-algebra and to emphasize the role of the two building blocks of the theory, namely L 0,1 (H) and L 1,0 (H). We quickly recall the main properties of these building blocks, and we refer to [Fai18b] for more details about them under our assumptions on H.…”

“…is an isomorphism of algebras (see [Fai18b] for a proof). It follows that L 1,0 (H) is isomorphic to a matrix algebra, and in particular has trivial center.…”

“…Proof: Denote as usual X i ⊗ Y i = RR ′ , X i ⊗ Y i = (RR ′ ) −1 and let µ l be the left integral on H (unique up to scalar). Using basic facts about integrals and (3), we have (see [Fai18b,Prop. 5.3]):…”

“…In this paper, we consider the algebras L g,n (H) from a purely algebraic viewpoint, under the general assumption that the gauge algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. The algebras L 0,1 (H) and L 1,0 (H), which are the building blocks of the theory (see Definiton 3.3), and the associated projective representation of SL 2 (Z), have already been studied under these assumptions in [Fai18b].…”