## Classical W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

##### Author(s)

Valeri, Daniele; Sole, Alberto De; Kac, Victor
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We derive explicit formulas for λ-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 integrable 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 ˇ−3 functions, where h ˇ is the dual Coxeter number of g. In the case when g is sl2 both these equations coincide with the KdV equation. In the case when g is not of type C[subscript n], we associate to the minimal nilpotent element of g yet another generalized Drinfeld–Sokolov hierarchy.

##### Date issued

2014-05##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Communications in Mathematical Physics

##### Publisher

Springer Berlin Heidelberg

##### Citation

Sole, Alberto De, Victor G. Kac, and Daniele Valeri. “Classical W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents.” Communications in Mathematical Physics 331.2 (2014): 623–676.

Version: Author's final manuscript

##### ISSN

0010-3616

1432-0916