We consider a quantum spin chain (one-dimensional array of coupled spins) that is integrable and has quantum group symmetry Uq(sl(2)). The basic problem is to find a complete set of eigenvalues and eigenvectors of the Hamiltonian. We first review the case when q is generic. We then consider the case that q is a root of unity, for which several surprising new features appear. (Based on recent work with Azat Gainutdinov).
