Quantum spaces: notes and comments on a lecture by S. L. Woronowicz
Quantum stochastic convolution cocycles -algebraic and C*-algebraic
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...
Quantum-classical interactions and galois type extensions
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...
Quasi-bigèbres jacobiennes
Quasi-commutative polynomial algebras and the power property of 2 × 2 quantum matrices
Quasi-linear algebras and integrability (the Heisenberg picture).
-matrice universelle pour et invariant d’entrelacs associé
En utilisant la méthode du double quantique, nous construisons une -matrice universelle pour la quantification de la superalgèbre de Lie . 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.
Rational smoothness of varieties of representations for quivers of Dynkin type
We study the Zariski closures of orbits of representations of quivers of type , ou . With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.
Regular representation of the quantum Heisenberg double ( is a root of unity)
Regularity of the four dimensional Sklyanin algebra
Représentations de groupes quantiques et permutations
Représentations des groupes quantiques
Representations of quantum algebras.
Representations of quantum algebras (Addendum).
Representations of quantum algebras. The mixed case.
Representations of quantum groups and (conditionally) invariant q-difference equations
We give a systematic discussion of the relation between q-difference equations which are conditionally -invariant and subsingular vectors of Verma modules over (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....
Representations of at the roots of unity
In this paper we study the irreducible finite dimensional representations of the quantized enveloping algebra associated to , at the roots of unity. It is known that these representations are parametrized, up to isomorphisms, by the conjugacy classes of the group . 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 -modules at the roots of in the...
Representations of the quantum algebra and discrete -ultraspherical polynomials.
Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials
Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula.