Generalized Drinfel'd-Sokolov hierarchies
During the recent 25 years, the theory of nonlinear partial differential equations of evolution type has undergone a vast development. The crucial discovery that many of the well-known equations in mathematical physics (e.g., the KdV equation, the nonlinear Schrödinger equation, the sine- Gordon equation, and others) are exactly solvable, and can be interpreted in the framework of completely integrable Hamiltonian systems, has led to wide-spread generalizations. In particular, the possibility of exhibiting many of those evolution equations as members of a whole hierarchy defined by equations for matrices of (pseudo-) differential operators of Lax type revealed an evidently deep relationship to algebraic geometry, on the one hand, and to the theory of infinite-dimensional Lie algebras, on the other hand. The approach of constructing integrable hierarchies of evolution equations defined by matrix Lax equations for certain Lie algebras has been studied, in recent years, from various aspects and by several authors, especially with a view towards their applications to conformal quantum field theory and quantum gravity. One of the main contributions, in this regard, was the generalization of the Korteweg-de Vries hierarchy to certain classes of simple Lie algebras by \textit{V. G. Drinfel'd} and \textit{V. V. Sokolov} [Sov. Math. Dokl. 23, 457-462 (1981), transl. from Dokl. Akad. Nauk. SSSR 258, 11-16 (1981; Zbl 0513.35073)], as well as the established association between derived KdV hierarchies and untwisted affine Lie algebras by \textit{G. Wilson} [Ergodic Theory Dyn. Syst. 1, 361-380 (1981; Zbl 0495.58008)]. In the present paper, the authors describe a method of further generalizing the construction of integrable hierarchies of KdV-like equations to (rather) arbitrary untwisted Kac-Moody algebras and their maximal Heisenberg subalgebras. Their main result is that for each positive element of the Heisenberg subalgebra of an untwisted Kac-Moody algebra, there can be associated a whole collection of hierarchies of integrable partial differential equations, including the generalized KdV hierarchies of Drinfel'd-Sokolov and various other partially modified KdV hierarchies. Moreover, the many hierarchies constructed in this way are shown to be completely classified by the inequivalent Heisenberg subalgebras of the given Kac-Moody algebras and, therefore, also by the conjugacy classes of the corresponding Weyl group. The article is fairly self-contained, in that it starts with an introduction to the necessary formalism of (untwisted) graduated Kac- Moody algebras, their Heisenberg subalgebras, and the Weyl group. After the detailed description of their generalized construction method for KdV-like hierarchies via Kac-Moody algebras, together with the above- mentioned classification result, the authors discuss various concrete examples in the context of the preceding general formalism. Some of these examples turn out to be hierarchies already found by other authors, and by different methods, but here they naturally fit into the present authors' general framework. The concluding remarks, at the end of the paper, point to seemingly further possible generalizations of the approach adopted here, and to interesting relationships with those based on the ``tau-function'' approach and the vertex operator method in the representation theory of Lie groups [cf., \textit{V. G. Kac} and \textit{M. Wakimoto}, Proc. Symp. Pure Math. 49, Pt. 1, 191-237 (1989; Zbl 0691.17014)].
scientific article; zbMATH DE number 54768
scientific article