Modified braid equations, Baxterizations and noncommutative spaces for   the quantum groups $GL_{q}(N), SO_{q}(N)$ and $Sp_{q}(N)$
  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. 
A. Chakrabarti and R. Chakrabarti