Bethe Ansatz and Quantum Groups

The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when $q$ is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact $SU(2)_q$ invariance on the lattice. The algebra of the Yang-Baxter (YB) and $SU(2)_q$ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the $SU(2)_q$ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of $SU(2)_q$. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS($p$) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$, at $q=e^{i\pi/p}$, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J<(p-1)/2$ and no {\bf singular} roots in the solutions of the BAE. We thus give a microscopic derivation of the lattice $S-$matrix of the massive kinks and show that the continuum limit of the RSOS($p+1$) model is the $p-$restricted Sine--Gordon field theory.