Exchange algebra and the Drinfel'd-Sokolov theorem
The relationship between the exchange algebra (Gervais, Neveu; Babelon), encoding the conformal structure of classical Toda field theory, and the separation of the two chiralities in this integrable field model is analyzed. The chiral halves are described as the solutions of the Drinfel'd-Sokolov linear equations [\textit{V. G. Drinfel'd} and \textit{V. V. Sokolov}, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 81-180 (1984; Zbl 0558.58027); see also \textit{A. N. Leznov} and \textit{M. V. Saveliev}, Lett. Math. Phys. 3, 489-494 (1979; Zbl 0415.35017)]. The sets of relations (Poisson brackets) defining the normalized chiral exchange algebras are formulated by taking into account the matrix solutions \((r^ +\) and \(r^ -)\) of the Yang-Baxter equations in classical Toda field theory [\textit{J. Balog}, \textit{L. Dabrowski} and \textit{L. Feher}, Phys. Lett., B. 244, 227-234 (1991; Zbl 0724.58069)]. In the framework of the developed construction, by enlarging the phase space and considering the zero modes of the left and right movers as well as their conjugate variables as independent dynamical variables, it is shown that one can introduce a conformal invariant Bloch wave basis such that the two chiralities commute. As a result, the periodic and local solution of Toda field theory is reconstructed in terms of this basis.
scientific article; zbMATH DE number 25865