2019/10/09 by Matthieu Faitg, Faitg, Matthieu
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1910.04110
openalex publication_date 2019/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The algebras Lg,n(H) have been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the middle of the 1990's, in the program of combinatorial quantization of the moduli space of flat connections over the surface Σg,n of genus g with n open disks removed. In this thesis we apply these algebras Lg,n(H) to low-dimensional topology (mapping class groups and skein algebras of surfaces), under the assumption that the gauge algebra H is a finite dimensional factorizable ribbon Hopf algebra which is not necessarily semisimple, the guiding example being the restricted quantum group Uq(\mathfraksl2) (where q is a 2p-th root of unity). First, we construct from Lg,n(H) a projective representation of the mapping class group of Σg,0. We provide formulas for the representations of Dehn twists generating the mapping class group and we use these formulas to show that our representation is equivalent to the one constructed by Lyubashenko--Majid and Lyubashenko via categorical methods. For the torus Σ1,0 with the gauge algebra Uq(\mathfraksl2), we compute explicitly the representation of SL2(ℤ) and we determine its structure. Second, we introduce a diagrammatic description of Lg,n(H) which enables us to define in a very natural way the Wilson loop map W. This map associates an element of Lg,n(H) to any link in (Σg,n ∖ D) × [0,1] which is framed, oriented and colored by H-modules. When the gauge algebra is H = Uq(\mathfraksl2), we use W and the representations of Lg,n(H) to construct representations of the skein algebras Sq(Σg,n) for q a 2p-th root of unity. For the torus Σ1,0 we explicitly study this representation.