Solvable Lie algebras and generalized Cartan matrices arising from isolated singularities
To any isolated hypersurface singularity (V,0) of dim n (i.e. dim V\(=n)\), \textit{J. Mather} and the author [Invent. Math. 69, 243-251 (1982; Zbl 0499.32008)] associated a commutative local Artinian algebra A(V) (called the moduli algebra) and they showed that the correspondence: \(V\mapsto A(V)\) is one-to-one. Let L(V) denote the (finite dimensional) Lie algebra of derivations of A(V) and let \({\mathfrak g}(V)\) denote the nil-radical of L(V). The author proves (in the paper under review) the following: (1) If (V,0) admits a \({\mathbb{C}}^*\)-action, then the Lie algebra L(V) is abelian if and only if (V,0) is either \(A_ 1\) or \(A_ 2\)-singularity. (2) If dim V\(=2\), then L(V) is a solvable Lie algebra. (The author probably would discuss the higher dimensional analogue of (2) in a future paper.) Now, using the 'root systems' for the nilpotent Lie algebras introduced (I believe) by Bratzlavsky and Favre, Santharoubane associated a GCM (generalized Cartan matrix) to any nilpotent Lie algebra. In particular, any isolated hypersurface singularity (V,0) determines a GCM C(V), associated to the (nilpotent) Lie algebra \({\mathfrak g}(V)\). This generalized Cartan matrix is an analytic invariant of isolated hypersurface singularities, which the author feels is, in fact, a topological invariant. The author has included an appendix to tabulate the various invariants associated to the Lie algebras L(V) of rational double points. I caution (to the readers of the paper) against a lot of typographical errors.
scientific article; zbMATH DE number 3946411
scientific article