Generalization and Deformations of Quantum Groups; Quantization of All Simple Lie Bi-Algebras

A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Special values of the parameters are characterized by the appearance of certain ideals; in this case the universal R-matrix exists on the associated algebraic quotient. In special cases the quotient is a "standard" quantum group; all familiar quantum groups including twisted ones are obtained in this way. In other special cases one finds new types of coboundary bi-algebras. A large class of first order deformations of all these standard bi-algebras is investigated and the associated deformed universal R-matrices have been calculated. One obtains, in particular, universal R-matrices associated with all simple, complex Lie algebras (classification by Belavin and Drinfeld) to first order in the deformation parameter.