Hamiltonian Reduction and the Construction of q-Deformed Extensions of the Virasoro Algebra

In this paper we employ the construction of Dirac bracket for the remaining current of $sl(2{)}_q$ deformed Kac-Moody algebra when constraints similar to those connecting the $sl(2)$-WZW model and the Liouville theory are imposed and show that it satisfy the q-Virasoro algebra proposed by Frenkel and Reshetikhin. The crucial assumption considered in our calculation is the existence of a classical Poisson bracket algebra induced, in a consistent manner by the correspondence principle, mapping the quantum generators into commuting objects of classical nature preserving their algebra.