Search

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., Simons Foundation (Simons Fellowship)

The aim of these notes is to provide an elementary introduction to some of the basic elements of exact S-matrix theory. This is a large subject, and only the foothills will be explored here. A particular omission is any serious discussion of the Yang-Baxter equation; instead, I will focus on questions of analytic structure, and the bootstrap equations. Even then, what I have to say will only be a sketch of the simpler aspects. The hope is to give a hint of the many unexpected features of scattering theories in 1+1 dimensions.

In this chapter we study Coxeter systems and Tits systems and certain graphs derived from these. In the first few sections finiteness is not assumed. In the later sections almost all known infinite families of distance-transitive graphs are described in this framework. The chapter ends with a determination of all distance-transitive graphs which naturally arise from a Tits system in a finite Chevalley group. Much more information on Tits systems, Chevalley groups and buildings can be found in Bourbaki [113], Carter [187, 188] and Tits [747, 753].