Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous singularities
type: Item
instance of: scholarly article
title: Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous singularities
zbMATH Open document ID: 0839.32015
DOI: 10.1016/S0021-8693(05)80001-8
author: Hao Chen
author: Yi-Jing Xu
author: Stephen Yau
published in: Journal of Algebra
publication date: 1995-06-13T00:00:00Z
review text: Let \(f(x_0, \dots, x_n) = 0\) be a germ of analytic hypersurface in \(\mathbb{C}^{ n + 1}\) with isolated singularity at the origin. The finite dimensional \(\mathbb{C}\)-algebra \(A = \mathbb{C} \{z_0, \dots, z_n\}/(f, \partial f/z_0, \dots, \partial f/z_n)\) is called the moduli algebra of the singularity: by a result of \textit{J. N. Mather} and \textit{S. Yau} [Invent. Math. 69, 243-251 (1992; Zbl 0499.32008)] two isolated singularities are analytically isomorphic if and only if their moduli algebras are isomorphic. To \(A\) there is associated the Lie algebra \(L\) of its derivations. If \(f\) is quasihomogeneous, then both \(A\) and \(L\) are naturally graded. In the present paper it is proven that in such a case all homogeneous pieces of negative degree of \(L\) are zero. This leads to a criterion characterizing quasihomogeneous singularities.
reviewed by: Eduardo Casas-Alvero
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S10
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 763192
zbMATH Keywords: isolated singularity
zbMATH Keywords: hypersurface singularity
zbMATH Keywords: germ of analytic hypersurface
zbMATH Keywords: moduli algebra
zbMATH Keywords: quasihomogeneous singularities
MaRDI profile type: Publication
cites work: Classification of isolated hypersurface singularities by their moduli algebras
cites work: Quasihomogeneous isolated singularities of hyperplanes.
cites work: Variation of complex structures and variation of Lie algebras
cites work: Continuous family of finite-dimensional representations of a solvable Lie algebra arising from singularities
cites work: Solvable Lie algebras and generalized Cartan matrices arising from isolated singularities
cites work: Singularities Defined by sℓ(2, C) Invariant Polynomials and Solvability of Lie Algebras Arising From Isolated Singularities
cites work: Solvability of Lie Algebras Arising from Isolated Singularities and Nonisolatedness of Singularities Defined by sℓ(2, C) Invariant Polynomials
full work available at URL: s0021-8693(05)80001-8
OpenAlex ID: W2081689929
description: scientific article; zbMATH DE number 763192