Braid-group approach to the derivation of universal matrices

A new method for deriving universal matrices from braid-group representation is discussed. In this case, universal operators can be defined and expressed in terms of products of braid-group generators. The advantage of this method is that matrix elements of are rank independent, and leaves multiplicity problem-concerning coproducts of the corresponding quantum groups untouched. As examples, -matrix elements of , , , and with multiplicity two for -type and for -type, -type, and -type quantum groups, which are related to Hecke algebra and Birman - Wenzl algebra, respectively, are derived by using this method.