Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups $G{L}_q(N),S{O}_q(N)$ and $S{p}_q(N)$

Abstract

Modified braid equations satisfied by generalized $\hat{R}$ matrices (for a {\em given} set of group relations obeyed by the elements of $\mathsf{T}$ matrices ) are constructed for q-deformed quantum groups $G{L}_q(N),S{O}_q(N)$ and $S{p}_q(N)$ 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 $S{O}_q(N)$ and $S{p}_q(N)$. The $\hat{R}$ matrices of this class, while being distinct from restrictions of the universal $\hat{\mathcal{R}}$ 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.

33 pages

References in corpus (3)

• Duality for Exotic Bialgebras
• Duality and Representations for New Exotic Bialgebras
• Modified Braid Equations for SO_q (3) and noncommutative spaces

Cited by in corpus (3)

• A nested sequence of projectors and corresponding braid matrices $\hat{R}(\theta )$: (1) Odd dimensions
• Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications
• Aspects of a new class of braid matrices: roots of unity and hyperelliptic $q$ for triangularity, L-algebra,link-invariants, noncommutative spaces