Non-standard Quantum Group in Toda and WZNW Theories

The basic Poisson brackets in the chira.l sectors of the WZNW theory and its Toda reduction are described in terms of a monodromy dependent r-matrix. In the case of the sl(n) Lie algebras, and only then, this monodromy dependence can be ‘gauged away’. The resulting non-trivial solution of the classical Yang-Baxter equation is the classical limit of the quantum R-matrix of the SL(n) Toda theory found recently by Creminer and Gervais. The deformations of SL(n) and U(sl(n)) defined by this R-matrix are studied in the simplest non-trivial case of n = 3. The multiplicative structure of this deformation of U(sl(3)) can be transformed into that of the standard U_q(sl(3)), but the coproduct is different. Possible generalizations for arbitrary n and applications in conformal field theory and in non-commutative differential geometry are briefly indicated. The Cremmer-Gervais R-matrix is ‘Yang-Baxterized’. The resulting spectral parameter dependent R-matrix may give rise to a new series of integrable models