Braid group approach to the derivation of universal \v{R} matrices

A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this method is that matrix elements of \v{R} are rank independent, and leaves multiplicity problem concerning coproducts of the corresponding quantum groups untouched. As examples, \v{R} matrix elements of $[1]\times [1]$, $[2]\times [2]$, $[1^{2}]\times [1^{2}]$, and $[21]\times [21]$ with multiplicity two for $A_{n}$, and $[1]\times [1]$ for $B_{n}$, $C_{n}$, and $D_{n}$ type quantum groups, which are related to Hecke algebra and Birman-Wenzl algebra, respectively, are derived by using this method.