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

Abstract

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

Higher dimensional multiparameter unitary and nonunitary braid matrices: Even dimensions

“…This particularly evident form the respective canonical forms ((6.5) for 6-vertex and (7.2) with (7.6), (7.7) for 8-vertex of Ref. [6]) where the normalization factors are also suitably adapted. The coefficient of each real symmetric projector is evidently inverted under conjugation (iθ −→ −iθ).…”

“…Already for the 4 × 4 case the intensively studied 6-and 8-vertex solutions can be canonically expressed on our basis (sections 6 and 7 of Ref. [6]). The multidimensional generalization of the 6-vertex matrix presented in Ref.…”

Eigenstates and Eigenvalues of Chain Hamiltonians Based on Multiparameter Braid Matrices for All Dimensions

“…We intend to generalize in another direction, the nested sequence of projectors (2) that can be generalized by including parameters. A 4 × 4 example can be found in the paper where nested sequences were introduced [13]. It appeared in the context of U (2) (Section 5, [13]).…”

“…A 4 × 4 example can be found in the paper where nested sequences were introduced [13]. It appeared in the context of U (2) (Section 5, [13]). We aim to present a systematic generalization of higher dimensions and corresponding Baxterized braid matrices.…”

A new eight vertex model and higher dimensional, multiparameter generalizations

“…Whereas, choosing adequately the normalizing factor (see sec. 7 of [9] and in particular eq. (7.13)) for the standard case the corresponding limiting form is R = 1, q −1 , q −1 , 1 diag.…”