Classical Exchange Algebra Research Articles

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 A l 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 A l Toda theory found recently by Cremmer and Gervais.

Classical exchange algebras in the Wess-Zumino-Witten model

We show that for the Wess-Zumino-Witten model it is possible to define a classical exchange algebra whose structure follows from the classical equations of motion of this model. This algebra is defined for every positive integer k. We conjecture that this algebra could be considered as a starting point for the independent quantization of the WZW model.