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

Abstract

Braid matrices $\hat{R}(θ)$, 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 relation $\hat{R}(-θ)\hat{R}(θ)=I$.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 $θ\to \pm \infty$ 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.