CLASSICAL R-MATRIX AND EXCHANGE ALGEBRA IN WZNW AND TODA THEORIES

It is shown that the fundamental Poisson brackets in the chiral sectors of WZNW theory and its Liouville-Toda reduction are of the r-matrix form. In general, the r-matrix is monodromy dependent, but in the case of Al Lie algebras, and only then, this monodromy dependence can be "gauged away" by choosing appropriate representatives in the conjugacy classes of the monodromy. The resulting non-trivial solution of the classical Yang-Baxter equation is the classical limit of the quantum R-matrix of Al Toda theory found recently by Cremmer and Gervais. © 1990.