En utilisant la méthode du double quantique, nous construisons une $R$-matrice universelle pour la quantification de la superalgèbre de Lie $D(2,1,x)$. Nous utilisons ce résultat pour construire un invariant d’entrelacs et nous montrons qu’il est égal à une spécialisation du polynôme de Dubrovnik introduit par Kauffman.

A new algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This gives the topological interpretation of the link invariants associated with the Weinstein-Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theories are discussed.

We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if ${}_{\Gamma }$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k{\oplus }_{\Gamma }$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U{(}_{\Gamma })$ is a bialgebra in the category of A-comodules.

We give a systematic discussion of the relation between q-difference equations which are conditionally $U_q\left(\right)$-invariant and subsingular vectors of Verma modules over $U_q\left(\right)$ (the Drinfeld-Jimbo q-deformation of a semisimple Lie algebra over ℂg or ℝ). We treat in detail the cases of the conformal algebra, = su(2,2), and its complexification, = sl(4). The conditionally invariant equations are the q-deformed d’Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel’fand-Gel’fand....

In this paper we study the irreducible finite dimensional representations of the quantized enveloping algebra ${\mathcal{U}}_q\left(g\right)$ associated to $g=sl\left(3\right)$, at the roots of unity. It is known that these representations are parametrized, up to isomorphisms, by the conjugacy classes of the group $G=SL\left(3\right)$. We get a complete classification of the representations corresponding to the submaximal unipotent conjugacy class and therefore a proof of the De Concini-Kac conjecture about the dimension of the ${\mathcal{U}}_q\left(g\right)$-modules at the roots of $1$ in the...

We describe the action of the Kauffman bracket skein algebra on some vector spaces that arise as relative Kauffman bracket skein modules of tangles in the punctured torus. We show how this action determines the Reshetikhin-Turaev representation of the punctured torus. We rephrase our results to describe the quantum group quantization of the moduli space of flat SU(2)-connections on the punctured torus with fixed trace of the holonomy around the boundary.