Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications
A. Chakrabarti · Journal of Mathematical Physics · 2003
Braid matrices R̂(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio fi(θ)/fi(−θ) for the coefficient of each projector Pi appearing in the decomposition. This directly yields a factorization (F(−θ))−1F(θ) for the braid matrix, implying also the relation R̂(−θ)R̂(θ)=I. This is achieved for GLq(n), SOq(2n+1), SOq(2n), Spq(2n) for all n and also for various other interesting cases including the eight-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 GLq(2), GLq(3), SOq(3), SOq(4), Spq(4) and various other cases such as the eight-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. In each factor F(θ) our diagonalizer again factors out all dependence on the spectral parameter θ as a diagonal matrix. 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.