  A Temperley-Lieb algebra is extracted from the operator structure of a new
class of $N^{2}\times N^{2}$ braid matrices presented and studied in previous
papers and designated as $S\hat{O}_{(q)}(N)$, $S\hat{p}%_{(q)}(N)$ for the
q-deformed orthogonal and symplectic cases respectively. Spin chain
Hamiltonians are derived from such braid matrices and the corresponding chains
are studied. Time evolutions of the chains and the possibility of transition of
data encoded in the parameters of mixed states from one end to the other are
analyzed. The entanglement entropies $% S(q,N)$ of eigenstates of the crucial
operator, namely the q-dependent $% N^{2}\times N^{2}$ projector $P_{0}$
appearing in the corresponding Hamiltonian are obtained. Study of entanglements
generated under the actions of \ $S\hat{O}(N)$, $S\hat{p}(N)$ braid operators,
unitarized with imaginary rapidities is presented as a perspective.
