Torus embeddings and deformation of simple singularities of space curves (Q1097321)

A very beautiful connection was found by Brieskorn and Slodowy between simple singularities (rational double points) and simple Lie algebras. In the paper under review the author seeks an analogue of this connection in the class of 1-dimensional simple singularities which are complete intersection in \({\mathbb{C}}^ 3.\) These singularities are classified by \textit{M. Giusti} [Singularities, Summer Inst., Arcata/Calif. 1987, Proc. Symp. Pure Math. 40, Part 1, 457-494 (1983; Zbl 0525.32006)] and they are labelled \(S_{\mu} (\mu =5,6,7,...)\), \(T_ 7,T_ 8,T_ 9,U_ 7,U_ 8,U_ 9,W_ 9,W_{10},Z_ 9,Z_{10}\). The author associate to \(S_{\mu}^ a \)diagram called \(D=D_ k[*]\), \(k=\mu -1\) which is constructed from the Dynkin diagram \(D_ k\) by adding one distinguished vertex. They are connected as follows: The diagram determines a torus embedding \({\mathcal X}(D)\) and the parameter space of the semi-universal deformation of the singularity is identified with \({\mathcal X}(D)/W_ 2(D)\), where \(W_ 2(D)\) is an analogue of the Weyl group, and this identification respects the discriminants of the both spaces. A similar result is proved for \(T_{\mu}\) and \(E_{\mu}[*]\) \((\mu =6,7,8)\).