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

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.