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

Abstract

Braid matrices $\hat{R}(\theta )$, corresponding to vector representations, are spectrally decomposed obtaining a ratio $f_i(\theta )/f_i(-\theta )$ for the coefficient of each projector $P_i$ appearing in the decomposition. This directly yields a factorization $(F(-\theta ){)}^{-1}F(\theta )$ for the braid matrix, implying also the relation $\hat{R}(-\theta )\hat{R}(\theta )=I$.This is achieved for $G{L}_q(n),S{O}_q(2n+1),S{O}_q(2n),S{p}_q(2n)$ for all $n$ and also for various other interesting cases including the 8-vertex matrix.We explain how the limits $\theta \to \pm \infty$ 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 $G{L}_q(2),G{L}_q(3),S{O}_q(3),S{O}_q(4),S{p}_q(4)$ 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 $F(\theta )$ our diagonalization again factors out all dependence on the spectral parameter $\theta$ 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.

38 pages, no figures

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• A new eight vertex model and higher dimensional, multiparameter generalizations
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• New classes of spin chains from $(S{\hat{O}}_{(q)}(N)$, $S{\hat{p}}_{(q)}(N))$ Temperley-Lieb algebras: Data transmission and (q, N) parametrized entanglement entropies
• A new class ${\hat{o}}_N$ of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable $S$-matrix