Modular Double of Quantum Group

As it is clear from the title, I shall deal with some question connected with the theory of Quantum Groups. If I remember right, Moshe did not like Quantum Groups after this notion was cristallized by Drinfeld [1] in pure algebraic manner. However his own attraction to the deformations (as well as the pressure of authors in LMP) made him to change his mind. So when I presented the subject described below at St. Petersburg meeting on May 1998, he did not express any bad feelings. So I decided to publish it in this memorial volume. There are several sources of my proposal. I shall give just two, one ”mathematical” and another ”physical”, as it is appropriate for a paper on Mathematical Physics. 1. In the definition of Quantum Group one uses the deformation of the Chevalley generatorsK, f , e, whereas for the construction of the universal Rmatrix one needs nonpolinomial elements like H = lnK. Explicite formulas will be reminded below. This unfortunate obstacle can be circumvented in several ways: one, à-la Lusztig [2] is just not to use explicite formula of Drinfeld; another, followed in the most of texts on Quantum Groups (see i.e. [3]), is to employ formal series in ln q. However the value of R-matrix as a genuine operator is too high and deserves more friendly attitude. 2. In the applications of Quantum Groups to Conformal Field Theory one explicitely sees, that together with the attributes of Quantum Groups (i.e. eigenvalues of Laplacians) for q = e there enter analogous objects, corresponding to q̃ = e . This modular duality is a well known ”experimental” fact, which goes without satisfactory explanation.