Abstract:

We give a brief general review of the ADE classification problem. The survey includes simple Kleinian singularities, symmetries of Platonic solids, finite subgroups of SU(2), the Mckay correspondence, integer matrices of norm 2 and Brieskorn’s theory of subregular orbits. We conclude with some joint work with H. Sabourin and P. Vanhaecke on transverse Poisson structures to subregular orbits in semisimple Lie algebras. We show that the structure may be computed by means of a simple Jacobian formula, involving the restriction of the Chevalley invariants on the slice. In addition, using results of Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional Poisson bracket, intimately related to the simple rational singularity that corresponds to the subregular orbit.  Finally we present some recent results on the minimal orbit.