We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\widehat{\Lambda }\subset G$. In the matrix case $G\subset U_n^{+}$ the embedding condition is equivalent to having a quotient map ${\Gamma }_U\to \Lambda$, where $F=\{{\Gamma }_U\mid U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon’s classification of group dual subgroups $\widehat{\Lambda }\subset S_n^{+}$. These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian...

We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group...

We summarise recent results concerning quantum stochastic convolution cocycles in two contexts-purely algebraic and C*-algebraic. In each case the class of cocycles arising as the solution of a quantum stochastic differential equation is characterised and the form taken by the stochastic generator of a *-homomorphic cocycle is described. Throughout the paper a common viewpoint on the algebraic and C*-algebraic situations is emphasised; the final section treats the unifying example of convolution...

We introduce the notion of a completely quantum C*-system (A,G,α), i.e. a C*-algebra A with an action α of a compact quantum group G. Spectral properties of completely quantum systems are investigated. In particular, it is shown that G-finite elements form the dense *-subalgebra of A. Furthermore, properties of ergodic systems are studied. We prove that there exists a unique α-invariant state ω on A. Its properties are described by a family of modular operators ${{\sigma }_z}_{z\in ℂ}$ acting on . It turns out that ω...

An algebraic model for the relation between a certain classical particle system and the quantum environment is proposed. The quantum environment is described by the category of possible quantum states. The initial particle system is represented by an associative algebra in the category of states. The key new observation is that particle interactions with the quantum environment can be described in terms of Hopf-Galois theory. This opens up a possibility to use quantum groups in our model of particle...

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.