A classification of weighted homogeneous Saito free divisors

Abstract

We describe an approach to classification of weighted homogeneous Saito free divisors in ${\mathit{C}}^3$. This approach is mainly based on properties of Lie algebras of vector fields tangent to reduced hypersurfaces at their non-singular points. In fact we also obtain a classification of such Lie algebras having similar properties as ones for discriminants associated with irreducible real reflection groups of rank 3. Among other things we briefly discuss some applications to the theory of discriminants of irreducible reflection groups of rank 3, some interesting relationships with root systems of types $E_6$, $E_7$, $E_8$, and few examples in higher dimensional cases.