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arXiv · 1306.1684

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents

Alberto De Sole·Victor G. Kac·Daniele Valeri
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Abstract

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.

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Alberto De Sole, Victor G. Kac, Daniele Valeri. 2014-07-23. Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents. https://doi.org/10.1007/s00220-014-2049-2

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