  We study the connection between Zamolodchikov operator-valued relations in
Liouville field theory and in the SL(2,R)_k WZNW model. In particular, the
classical relations in SL(2,R)_k can be formulated as a classical Liouville
hierarchy in terms of the isotopic coordinates, and their covariance is easily
understood in the framework of the AdS_3/CFT_2 correspondence. Conversely, we
find a closed expression for the classical Liouville decoupling operators in
terms of the so called uniformizing Schwarzian operators and show that the
associated uniformizing parameter plays the same role as the isotopic
coordinates in SL(2,R)_k. The solutions of the j-th classical decoupling
equation in the WZNW model span a spin j reducible representation of SL(2,R).
Likewise, we show that in Liouville theory solutions of the classical
decoupling equations span spin j representations of SL(2,R), which is
interpreted as the isometry group of the hyperbolic upper half-plane. We also
discuss the connection with the Hamiltonian reduction of SL(2,R)_k WZNW model
to Liouville theory.
