Modified braid equations for SOq(3) and noncommutative spaces

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Modified braid equations satisfied by generalizedR matrices ( for a given set of group relations obeyed by the elements of 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 ofR 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 SO q (N ) and Sp q (N ). TheR matrices of this class, while being distinct from restrictions of the universalR matrix to the corresponding vector representations, satisfy the standard braid equation. The modified braid equation and the Baxterization are obtained for this new class ofR matrices. Diagonalization of the generalizedR matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the correspondingR matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces. *

Section: Baxterizationmentioning

confidence: 99%

“…Therefore these braiding relations have the usual form, given, for example, in Eq. (3.48) of [2]. For the sake of completeness we present them here:…”

Section: Generalized Braid Matrices and Consequent Noncommutative Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…These authors indicate the significance of and the interest in this equation. In the works [1,2] the most general solutions of the quantum inverse scattering equationR…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chakrabarti

2003

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Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications

Chakrabarti

2003

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Braid matricesR(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio f i (θ)/f i (−θ) for the coefficient of each projector P i appearing in the decomposition. This directly yields a factorization (F (−θ)) −1 F (θ) for the braid matrix, implying also the relationR(−θ)R(θ) = I.This is achieved for GL q (n), SO q (2n + 1), SO q (2n), Sp q (2n) for all n and also for various other interesting cases including the 8-vertex matrix.We explain how the limits θ → ±∞ can be interpreted to provide factorizations of the standard (non-Baxterized) braid matrices. A systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for GL q (2), GL q (3), SO q (3), SO q (4), Sp q (4) and various other cases such as the 8-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. In each factor F (θ) our diagonalization again factors out all dependence on the spectral parameter θ as a diagonal matrix. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. Applications of our formalism to the construction of L−operators and transfer matrices are indicated. In an Appendix our type of factorization is compared to another one proposed by other authors. * Laboratoire Propre du CNRS UPR A.0014

Aspects of a new class of braid matrices: Roots of unity and hyperelliptic q for triangularity, L-algebra, link-invariants, noncommutative spaces

Chakrabarti

2005

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Various properties of a class of braid matrices, presented before, are studied considering N 2 × N 2 (N = 3, 4, ...) vector representations for two subclasses. For q = 1 the matrices are nontrivial. Triangularity (R 2 = I) corresponds to polynomial equations for q, the solutions ranging from roots of unity to hyperelliptic functions. The algebras of L− operators are studied. As a crucial feature one obtains 2N central, group-like, homogenous quadratic functions of L ij constrained to equality among themselves by the RLL equations. They are studied in detail for N = 3 and are proportional to I for the fundamental 3 × 3 representation and hence for all iterated coproducts. The implications are analysed through a detailed study of the 9 × 9 representation for N = 3. The Turaev construction for link invariants is adapted to our class. A skein relation is obtained. Noncommutative spaces associated to our class ofR are constructed. The transfer matrix map is implemented, with the N = 3 case as example, for an iterated construction of noncommutative coordinates starting from an (N − 1) dimensional commutative base space. Further possibilities, such as multistate statistical models, are indicated.