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The symplectic and Poisson structures of the Liouville theory are derived from the symplectic form of the SL(2, R) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of chiral fields are constructed and their * -products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given.

We calculate correlation functions for vertex operators with negative integer exponentials of a periodic Liouville field, and derive the general case by continuing them as distributions. The path-integral based conjectures of Dorn and Otto prove to be conditionally valid only. We formulate integral representations for the generic vertex operators and indicate structures which are related to the Liouville S-matrix.

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