Geometric interpretation of the Poisson structure in affine Toda field theories

We express the Poisson brackets of local fields of the affine Toda field theories in terms of the Drinfeld-Sokolov dressing operator. For this, we introduce a larger space of fields, containing ``half screening charges'' and ``half integrals of motions''. In addition to local terms, the Poisson brackets contain nonlocal terms related to trigonometric r-matrices.