Groups of reflections in the theory of singularities. (Russian) Zbl 0753.57020

Givental’, A. B.

Operator theory in function spaces, Rev. Lect. 13th All-Union Sch., Kujbyshev/USSR 1988, 43-81 (1989).

[For the entire collection see Zbl 0714.00017.]
This interesting paper deals with the study of the series of singularities \(A\), \(D\), \(E\) [see for instance V. Arnol’d, A. N. Varchenko and S. M. Gusejn-Zade, Singularities of differentiable mappings. Classification of critical points, caustics and wave fronts (Russian) (1982; Zbl 0513.58001)] and it contains five sections. In sections 1-3 the connections between quaternionic finite groups, singularities of surfaces, simple Lie algebras, Dynkin diagrams and symmetry groups are presented. As application of these constructions one obtains the description of the above-mentioned singularities and of their discriminants as orbits of the symmetry groups of proper regular polyhedra. Sections 4-5 extend some of these results to nonsimple singularities of surfaces. A substantial result for unimodular quasihomogeneous singularities is given.

Reviewer: D.Andrica (Cluj-Napoca)

MSC:

57R45	Singularities of differentiable mappings in differential topology
58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory
22E46	Semisimple Lie groups and their representations

Keywords:

singularities; quaternionic finite groups; singularities of surfaces; simple Lie algebras; Dynkin diagrams; symmetry groups; discriminants; orbits of the symmetry groups of proper regular polyhedra; nonsimple singularities of surfaces; unimodular quasihomogeneous singularities

Citations:

Zbl 0714.00017; Zbl 0513.58001