The notions of the crossed product and the braided category are discussed within the theory of C∗-algebras. Like in the purely algebraic approach of S. Majid these notions are used to generalize the quantum group theory. We give an example of a braided group. It coincides with the closed subset C = {z : |z| = . . . , q−2, q−1, 1, q, q2, . . . , 0} of the complex plane equipped with the q2braidi...

In the first part we recall two famous sources of solutions to the Yang-Baxter equation – R-matrices and Yetter-Drinfeld (=YD) modules – and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exh...

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present ...

When a quantum hyperboloid is realized, as a three parameter algebra Ah,q, in the usual manner, the following problem arises: what is a ”representation theory” of this algebra? We construct the series of all spin representations of Ah,q, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in g∗ and g-modules. A braided trace and a braided involution are di...

We construct a representation theory of a “quantum hyperboloid” in terms of so-called braided modules. We treat these objects in the framework of twisted Quantum Mechanics. Résumé Nous construisons une théorie de représentations pour « l’hyperboloïde quantique » en termes de modules tressés. Nous traitons ces objets dans le cadre de la mécanique quantique tordue.

Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebr...