Lie algebras of derivations and affine algebraic geometry over fields of characteristic \(0\)
type: Item
instance of: scholarly article
title: Lie algebras of derivations and affine algebraic geometry over fields of characteristic \(0\)
zbMATH Open document ID: 0858.17018
DOI: 10.1007/BF01444221
published in: Mathematische Annalen
publication date: 1996-11-25T00:00:00Z
full work available at URL: https://eudml.org/doc/182987
review text: The purpose of this paper is the study of derivation Lie algebras \(Der(A)\) of finitely generated commutative algebras \(A\) over an algebraically closed field of characteristic 0. The basic models are `Lie algebras of vector fields' \(Der(A_X)\) for the rings \(A_X\) of coordinates of affine varieties \(X\). The main result states that two normal affine varieties \(X\) and \(X'\) are isomorphic if and only if their Lie algebras \(Der(A_X)\) and \(Der (A_{X'})\) are so, which is an analog of the well-known theorem by \textit{M. E. Shanks} and \textit{L. E. Pursell} [Proc. Am. Math. Soc. 5, 468-472 (1954; Zbl 0055.42105)] in the case of smooth manifolds. The key idea is to relate certain Lie subalgebras, described by purely Lie theoretic properties, to geometric objects like points, subvarieties, or germs of subvarieties. As a further result it is deduced that an affine variety \(X\) is smooth if and only if the Lie algebra \(Der(A_X)\) is simple, which is an extension of a result by \textit{D. A. Jordan} [J. Lond. Math. Soc., II. Ser. 33, No. 2, 33-39 (1986; Zbl 0591.17007)].
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 14A10
zbMATH DE Number: 931957
zbMATH Keywords: ideals
zbMATH Keywords: Lie algebras of vector fields
zbMATH Keywords: derivation Lie algebras
zbMATH Keywords: affine varieties
author: Thomas Siebert
reviewed by: J. Grabowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 931957