Abstract
A quantum Bäcklund transformation can provide a mapping from one realization of su(1,1), and any of its quantum deformations or Virasoro algebra extensions, to another. Nontrivial realizations can be related to trivial ones. Since the quantum transformation is not necessarily unitary, the two realizations can have different hermiticity properties when acting on arbitrary states. These general features are illustrated in detail using the Liouville field theory in 1+1 dimensions, and a simple potential model.