Complex Quantum Groups and Their Dual Hopf Algebras

Abstract

The aim of this paper is to realize the complexification of theoriginal quantum groups. A quantum group is described as a Hopfalgebra. This Hopf algebra plays the role of the algebra offunctions on the group. So, it is the Faddeev ReshetikhinTakhtajan (FRT) formalism which is followed here. Once thequantum matrix R has been chosen the relations of the algebraare written down. Besides the usual generators t\sp i\sb j, theauthors introduce the conjugates t\sp {*i}\sb j, with somerelations needed to complete the RTT relation. All theserelations can also be gathered into a unique RTT relation whichinvolves a bigger quantum matrix \scr R. The expression of\scr R, depending on R but also on some complex parametersα\sb i, is given. After that, the regular functionalsL\sp {\pm} are introduced along the lines initiated by FRT.Naturally, the quantum enveloping algebra U\sb R is defined asbeing the algebra generated by the L\sp {\pm}, and U\sb R isa * Hopf algebra. Some formulas involving the parametersα\sb i are derived. If R is the matrix associated to{\rm sl}(2,\bold C) and if the previous formulas are thesimplest ones (thus imposing a constraint on the parametersα\sb i), a * Hopf algebra U\sb q({\rm sl} (2,\bold C))is obtained with explicit relations. The classical limit isperformed in order to recover the * Hopf algebraU({\rm sl}(2,\bold C)).\par {For the entire collection see MR\Cite{Gielerak92:Quantum:Kluwer}[93h:00021].}