Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups GLq(N), SOq(N), and Spq(N)

Abstract

Modified braid equations satisfied by generalized R̂ matrices (for a given set of group relations obeyed by the elements of T matrices) are constructed for q-deformed quantum groups GLq(N), SOq(N), and Spq(N) with arbitrary values of N. The Baxterization of R̂ matrices, treated as an aspect complementary to the 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 SOq(N) and Spq(N). The R̂ matrices of this class, while being distinct from the restrictions of the universal R̂ 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 R̂ matrices. Diagonalization of the generalized R̂ matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the corresponding R̂ matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces.