Two-Parameter Quantum Groups Of Type E-Series & Restricted Two-Parameter Quantum Groups Of Type D

We investigate two-parameter quantum groups corresponding to Lie algebras of type E-series and restricted two-parameter quantum groups corresponding to Lie algebras of type D in this paper.1. Two-parameter quantum groups of type E-seriesWe define two-parameter quantum groups of type E-series by generators and relations. Motivated by Kharchenko's work we describe the convex PBW-type basis of them. We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf skew-pairing. We give the dual basis for U_r,s(n) and U_r,s(n^-). Using these dual basis we construct the R-matrix. We also discuss isomorphisms between these quantum groups and the one-parameter quantum Drinfel'd doubles.2. Restricted two-parameter quantum groups of type DWe restrict the parameter r and s to be roots of unity and describe some central elements under these conditions. We construct a family of finite-dimensional Hopf algebras. We show that these Hopf algebras are pointed and are Drinfel'd doubles. We give the explicit forms of the left and right integrals and determine the necessary and sufficient conditions for these Hopf algebras to possess ribbon elements.