  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.
