ON UNIVERSAL R-MATRICES AT ROOTS OF UNITY

It is well-known that quantum algebras at roots of unity are not quasi-triangular. They indeed do not possess,an invertible universal a-matrix. They have, however, families of quotients, on which no obstruction a priori forbids the existence an universal R-matrix. In particular, the universal R-matrix of the so-called finite dimensional quotient is already known. We try here to answer the following questions: are most of these quotients equivalent (or Hopf equivalent)? Can the universal R-matrix of one be transformed to the universal R-matrix of another using isomorphisms?