Strongly coupled quantum discrete Liouville theory. I: Algebraic   approach and duality
  The quantum discrete Liouville model in the strongly coupled regime, 1<c<25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables. 
L.D. Faddeev, R.M. Kashaev, A.Yu. Volkov