Generalized Drinfel'd-Sokolov hierarchies

Abstract

A general approach is adopted to the construction of integrable hierarchies of partial differential equations. A series of hierarchies associated to untwisted Kac-Moody algebras, and conjugacy classes of the Weyl group of the underlying finite Lie algebra, is obtained. The generalized KdV hierarchies of V.G. DrinfeΓd and V.V. Sokolov are obtained as the special case for the Coxeter element. Various examples of the general formalism are treated in some detail; including the fractional KdV hierarchies.

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

“…In conclusion, both the canonical and integrability formalisms agree and lead to the W (2) 3 symmetry algebra at any finite λ. The preservation of the W (2) 3 integrability structure under deformation was also noticed in [75,76]. stimulating discussions and useful comments.…”

Generalized integrability and two-dimensional gravitation

“…In this section we review the construction of a general class of integrable hierarchies [11], [12], [13], [10]. This class includes the Drinfel'd-Sokolov hierarchies as special cases.…”

“…For each regular element b k ∈ s (k > 0), so that g admits the decomposition 5) we may regard (3.4) as an integrable hierarchy of partial differential equations on q(k), modulo a gauge symmetry we discuss below. (In the language of [11] these are the "type-I" hierarchies.) In this case, q(k) are the dynamical variables of the hierarchy, and one can express all the other functions q(j) for j = k in terms of q(k) and its t k -derivatives.…”

The soliton equations associated with the affine Kac–Moody Lie algebra G2(1)

“…This relation first studied and described under different points of view in a sequel of seminal papers by Sato [18,19] Date, Jimbo, Kashiwara and Miwa [11], Hirota [13] Drinfeld and Sokolov [12] and Kac and Wakimoto [16] has inspired almost innumerable further investigations and generalizations (see for example the quite interesting papers of Burroughs, de Groot, Hollowood, Miramontes [1] [2]). Nevertheless, as far as we know, it seems that no explicit description of the hierarchy corresponding in the scheme of Drinfeld and Sokolov to the affine Kac-Moody Lie algebra G (1) 2 (even of the first non trivial equations) can be found in the literature, fact probably related to the size of standard realization of G 2 (namely by 7 × 7 matrices). The aim of this letter is to fill this gap and to show how the bihamiltonian formulation of the Drinfeld-Sokolov reduction [8] [9] [4] makes the computations involved more reasonable.…”