  First, it is proven that the three main operator-approaches to the quantum
Liouville exponentials --- that is the one of Gervais-Neveu (more recently
developed further by Gervais), Braaten-Curtright-Ghandour-Thorn, and Otto-Weigt
--- are equivalent since they are related by simple basis transformations in
the Fock space of the free field depending upon the zero-mode only. Second, the
GN-G expressions for quantum Liouville exponentials, where the $U_q(sl(2))$
quantum group structure is manifest, are shown to be given by q-binomial sums
over powers of the chiral fields in the $J=1/2$ representation. Third, the
Liouville exponentials are expressed as operator tau-functions, whose chiral
expansion exhibits a q Gauss decomposition, which is the direct quantum
analogue of the classical solution of Leznov and Saveliev. It involves q
exponentials of quantum group generators with group "parameters" equal to
chiral components of the quantum metric. Fourth, we point out that the OPE of
the $J=1/2$ Liouville exponential provides the quantum version of the Hirota
bilinear equation.
