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

De Sole, A., Kac, V. G. and Valeri, D. (2014) Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents. Communications in Mathematical Physics, 331(2), pp. 623-676. (doi: 10.1007/s00220-014-2049-2)

Abstract

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 Cn , we associate to the minimal nilpotent element of g yet another generalized Drinfeld–Sokolov hierarchy.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Valeri, Dr Daniele
• Authors: De Sole, A., Kac, V. G., and Valeri, D.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Communications in Mathematical Physics
• Publisher: Springer
• ISSN: 0010-3616
• ISSN (Online): 1432-0916
• Published Online: 01 May 2014