Champs de vecteurs invariants sur une algèbre de Lie réductive complexe. (Invariant vector fields on a reductive complex Lie algebra)
type: Item
instance of: scholarly article
title: Champs de vecteurs invariants sur une algèbre de Lie réductive complexe. (Invariant vector fields on a reductive complex Lie algebra)
zbMATH Open document ID: 0679.17004
author: Yanyan Li
published in: Journal of the Mathematical Society of Japan
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a simple complex Lie algebra, \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\), W be the Weyl group of the pair (\({\mathfrak g},{\mathfrak h})\), and G be the adjoint group of \({\mathfrak g}\). A classical result of Chevalley asserts that there exists an isomorphism of the algebra of G-invariant polynomial functions on \({\mathfrak g}\) onto the algebra of W-invariant polynomial functions on \({\mathfrak h}\), induced by restriction. \textit{J. Sekiguchi} [J. Math. Soc. Japan 36, 147-159 (1984; Zbl 0539.17005)] proved that there exists a similar isomorphism between the Lie algebra of G-invariant vector fields on \({\mathfrak g}\) and a certain Lie algebra of vector fields on \({\mathfrak h}/W.\) In the paper under review, one shows that Sekiguchi's result (and some generalizations) can be obtained from a theorem of \textit{G. Schwarz} [Publ. Math., Inst. Hautes Etud. Sci. 51, 37-135 (1980; Zbl 0449.57009)].
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 4112828
zbMATH Keywords: invariant vector field
zbMATH Keywords: simple complex Lie algebra
zbMATH Keywords: Cartan subalgebra
zbMATH Keywords: Weyl group
zbMATH Keywords: G-invariant polynomial functions
reviewed by: A. Verona
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04040615
OpenAlex ID: W2148638123
DOI: 10.2969/JMSJ/04040615
description: scientific article; zbMATH DE number 4112828