Quantum groups and cylinder braiding

Abstract

The purpose of this paper is to introduce a new structure into the representation theory of quantum groups. The structure is motivated by braid and knot theory. Representations of quantum groups associated to classical Lie algebras have an additional symmetry which cannot be seen in the classical limit. We first explain the general formalism of these symmetries (called cylinder forms ) in the context of comodules. Basic ingredients are tensor representations of braid groups of type B derived from standard R -matrices associated to socalled four braid pairs. These are applied to the Faddeev-Reshetikhin-Takhtadjian construction of bialgebras from R -matrices. As a consequence one obtains four braid pairs on all representations of the quantum group. In the second part of the paper we study in detail the dual situation of modules over the quantum enveloping algebra U q ( sl 2 ). The main result here is the computation of the universal cylinder twist .