A nested sequence of projectors and corresponding braid matrices $\hat{R}(\theta )$: (1) Odd dimensions

Abstract

A basis of $N^2$ projectors, each an $N^2\times N^2$ matrix with constant elements, is implemented to construct a class of braid matrices $\hat{R}(\theta )$, $\theta$ being the spectral parameter. Only odd values of $N$ are considered here. Our ansatz for the projectors $P_{\alpha }$ appearing in the spectral decomposition of $\hat{R}(\theta )$ leads to exponentials $exp(m_{\alpha }\theta )$ as the coefficient of $P_{\alpha }$. The sums and differences of such exponentials on the diagonal and the antidiagonal respectively provide the $(2N^2-1)$ nonzero elements of $\hat{R}(\theta )$. One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves $1/2(N+3)(N-1)$ free parameters $m_{\alpha }$. The diagonalizer of $\hat{R}(\theta )$ is presented for all $N$. Transfer matrices $t(\theta )$ and $L(\theta )$ operators corresponding to our $\hat{R}(\theta )$ are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of $N^2$ constant $N\times N$ matrices. The $\theta$-dependence factors out for such combinations. $\hat{R}(\theta )$ is developed in a power series in $\theta$. The basic difference arising for even dimensions is made explicit. Some special features of our $\hat{R}(\theta )$ are discussed in a concluding section.

latex file, 32 pages

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