The Quantum R-Matrix via Canonical Quantization in the Liouville Theory

Starting from the classical Liouville theory, we study its quantum theory through canonical quantization. We find that if the Poisson bracket relations between two vectors are,dominated by the classical r-matrix in the classical case, their quantum analogue is replaced by the exchange relations dominated by the quantum R-matrix. The quantum group structure in the quantum LiouviUe theory is studied and the central charge of the quantum Liouville theory is also obtained.