  Modified braid equations satisfied by generalized ${\hat R}$ matrices (for a
{\em given} set of group relations obeyed by the elements of ${\sf T}$ matrices
) are constructed for q-deformed quantum groups $GL_q (N), SO_q (N)$ and $Sp_q
(N)$ with arbitrary values of $N$. The Baxterization of ${\hat R}$ matrices,
treated as an aspect complementary to the {\em modification} of the braid
equation, is obtained for all these cases in particularly elegant forms. A new
class of braid matrices is discovered for the quantum groups $SO_{q}(N)$ and
$Sp_{q}(N)$. The ${\hat R}$ matrices of this class, while being distinct from
restrictions of the universal ${\hat{\cal R}}$ matrix to the corresponding
vector representations, satisfy the standard braid equation. The modified braid
equation and the Baxterization are obtained for this new class of ${\hat R}$
matrices. Diagonalization of the generalized ${\hat R}$ matrices is studied.
The diagonalizers are obtained explicitly for some lower dimensional cases in a
convenient way, giving directly the eigenvalues of the corresponding ${\hat R}$
matrices. Applications of such diagonalization are then studied in the context
of associated covariantly quantized noncommutative spaces.

Quantum Algebra