  We consider the Temperley-Lieb (TL) open quantum spin chain with "free"
boundary conditions associated with the spin-$s$ representation of
quantum-deformed $sl(2)$. We construct the transfer matrix, and determine its
eigenvalues and the corresponding Bethe equations using analytical Bethe
ansatz. We show that the transfer matrix has quantum group symmetry, and we
propose explicit formulas for the number of solutions of the Bethe equations
and the degeneracies of the transfer-matrix eigenvalues. We propose an
algebraic Bethe ansatz construction of the off-shell Bethe states, and we
conjecture that the on-shell Bethe states are highest-weight states of the
quantum group. We also propose a determinant formula for the scalar product
between an off-shell Bethe state and its on-shell dual, as well as for the
square of the norm. We find that all of these results, except for the
degeneracies and a constant factor in the scalar product, are universal in the
sense that they do not depend on the value of the spin. In an appendix, we
briefly consider the closed TL spin chain with periodic boundary conditions,
and show how a previously-proposed solution can be improved so as to obtain the
complete (albeit non-universal) spectrum.
