The Lie algebra of polynomial vector fields and the Jacobian conjecture
type: Item
instance of: scholarly article
title: The Lie algebra of polynomial vector fields and the Jacobian conjecture
zbMATH Open document ID: 0932.17025
DOI: 10.1007/BF01367763
author: Herwig Hauser
author: Gerd Müller
published in: Monatshefte für Mathematik
publication date: 2000-03-12T00:00:00Z
full work available at URL: https://eudml.org/doc/178832
review text: The authors discuss polynomial maps \(\varphi:K^n\to K^n\) in relation to the Lie algebra of vector fields. Namely, let \(P\subset D\) be polynomial and formal vector fields of \(n\) variables. Probably the main result of the article in the following: every endomorphism of \(K[x_1,\dots, x_n]\) with invertible Jacobian is an automorphism (Jacobian conjecture) is true if every automorphism \(\varphi\) of the Lie algebra \(D\) with \(\varphi(P)\subset P\) satisfies \(\varphi(P)= P\). The authors generalize this also by changing \(P\) to a subalgebra \(A= \bigoplus_{\lambda\in H^*} H_\lambda\) where \(H_\lambda\) are root spaces for a Cartan subalgebra \(H\) in \(D\) (which has to be appropriately defined) \(P\) appears for \(H\) spanned by \(x_i\partial_{x_i}\).
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 1236107
zbMATH Keywords: polynomial vector fields
zbMATH Keywords: invertible Jacobian automorphism
reviewed by: Alexei Rudakov
MaRDI profile type: Publication
Wikidata QID: Q115393653
cites work: Q4317259
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cites work: Subalgebras and automorphisms of Lie algebras of Cartan type
cites work: Lie algebras of derivations and affine algebraic geometry over fields of characteristic \(0\)
cites work: Q5340158
description: scientific article; zbMATH DE number 1236107