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 not equal +/-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(m, n). Finally we show how the quantized Ladder representations and their analogues for other Lie algebras play an important role.