A Class of Quantum Doubles Which Are Ribbon Algebras

Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras. Introduction Quasitriangular Hopf algebras have an universal R-matrix which is a solution of the Yang-Baxter equation and their modules can be used to determine quasiinvariants of braids, knots and links. Drinfeld’s quantum double construction gives a method to produce a quasitriangular Hopf algebra from a Hopf algebra and its dual. The concept of ribbon categories was introduced by Joyal and Street. Their definition requires the notion of duality and provides isotopy invariants of framed links. Through their representations, ribbon Hopf algebras give rise to ribbon categories. They were introduced by Turaev and Reshetikhin in [13] who also showed that the quantum groups of Drinfeld and Jimbo are ribbon algebras. A ribbon Hopf algebra is a quasitriangular Hopf algebra which possesses an invertible central element known as the ribbon element. Kauffman and Radford [9] have shown that the Drinfeld double D(Al) of a Taft algebra Al (of dimension l ) has a ribbon element if and only if l is odd. The ribbon element of D(Al) for l odd, provides an important invariant of 3-manifolds (see [7]). In [9] the authors also gave a criterion for a general quantum double to possess a ribbon element. Benkart and Witherspoon investigated the structure of two parameter quantum groups of sln and gln [5]. In [6] they have shown that the restricted two parameter quantum groups ur, s(sln) are quantum doubles of certain pointed Hopf algebras and possess ribbon elements under certain compatibility conditions between the parameters r and s. In this paper we provide a new class of quantum doubles which possess ribbon elements. They are the quantum doubles of a family of pointed Hopf algebras Date: September 17, 2007. MSC (2000): 16W35, 16W40. The research was supported by CEx05-D11-11/04.10.05. 1