Structure of moduli algebras and application to skein algebras

• Matthieu Faitg

Room: Salle Johnson The moduli algebra of a compact oriented surface with n punctures (n>0) is a "twisted tensor product" of several copies of the quantized coordinate algebra O_q(G). I will first explain the definition. Then I will present results on the structure of these algebras, namely that they are finitely generated, Noetherian and do not contain zero divisors. If time permits, the ingredients of the proofs will be discussed. Finally I will define an isomorphism between moduli algebras and skein algebras. In this talk we only consider quantum groups at generic parameter (no roots of unity). Joint work with S. Baseilhac and P. Roche.

Classification of representations of reduced stated skein algebras

• Julien Korinman

Room: Salle Johnson In this talk, I will introduce a family of algebras named reduced stated skein algebras and present a classification of their finite dimensional (semi-weight) representations. These representations are conjectured to be the building blocks of some SL_2 TQFT which extend some constructions of Blanchet-Costantino-Geer-Patureau Mirand and Baseilhac-Benedetti. If time permits, I will explain how we can deduce from this classification some projective representations of the mapping class groups and some new links invariants. This is a joint work with H.Karuo.

On the SL_n stated skein algebra of the triangle

• Thang Le

Room: Salle Johnson

Quantum moduli algebras at roots of unity

• Philippe ROCHE (IMAG)

Room: Salle Johnson We prove that the graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mc(q)$. Moreover, we show that these two properties still hold on $\mc[q,q^{-1}]$ for the integral version of the graph algebra. We also study the specializations $\Ll_{0,n}^\e$ of the graph algebra at a root of unity $\e$ of odd order, and show that $\Ll_{0,n}^\e$ and its invariant algebra under the quantum group $U_\e(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degree that we compute.