Unitarity of highest weight modules for quantum groups

We determine the highest weights that give rise to unitarity when q is real. We further show that when q is on the unit circle and q ± 1, then unitary highest-weight representations must be finite-dimensional and q must be a root of unity. We analyze the special case of the 'Ladder' representations for su . Finally we show how the quantized Ladder representations and their analogues for other Lie algebras play an important role.