Algebraic Structure of Multiparameter Quantum Groups

Abstract

Let G be a connected semi-simple complex Lie group. We define and study the multi-parameter quantum group Cq, p[G ] in the case where q is a complex parameter that is not a root of unity. Using a method of twisting bigraded Hopf algebras by a cocycle, [2], we develop a unified approach to the construction of Cq, p [G ] and of the multi-parameter Drinfeld double Dq, p . Using a general method of deforming bigraded pairs of Hopf algebras, we construct a Hopf pairing between these algebras from which we deduce a PeterWeyl-type theorem for Cq, p[G ]. We then describe the prime and primitive spectra of Cq, p[G ], generalizing a result of Joseph. In the one-parameter case this description was conjectured, and established in the SL(n)-case, by the first and second authors in [15, 16]. It was proved in the general case by Joseph in [18, 19]. In particular the orbits in Prim Cq, p[G] under the natural action of the maximal torus H are indexed, as in the one-parameter case by the elements of the double Weyl group W_W. Unlike the one-parameter case there is not in general a bijection between Symp G and Prim Cq, p[G ]. However in the case when