On Engel-Lie algebras
type: Item
instance of: scholarly article
title: On Engel-Lie algebras
zbMATH Open document ID: 0673.17006
DOI: 10.1007/BF00970273
published in: Siberian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The author proves the long-standing conjecture: the nilpotency of an n- Engel Lie algebra over a field of characteristic zero. Earlier the local nilpotency of such an algebra was proved by \textit{A. I. Kostrikin} [Izv. Akad. Nauk SSSR, Ser. Mat. 23, No.1, 3-34 (1959; Zbl 0090.245)] - as well as for characteristic \(p>n\). (The latter case also gave the positive solution to the restricted Burnside problem for groups of exponent p.) \textit{Yu. P. Razmyslov} [Algebra Logika 10, No.1, 33-44 (1971; Zbl 0253.17005)] showed that for positive characteristic p there exist non- solvable (p-2)-Engel Lie algebras. The proof is based on Kostrikin's theorem stating that any n-Engel Lie algebra of characteristic 0 (or \(p>n)\) has a non-trivial abelian ideal. The proof also exploits \({\mathbb{Z}}/2{\mathbb{Z}}\)-graded Lie algebras and the technique of the representation theory of symmetric groups. The earlier version of this theorem of Zel'manov has already been incorporated into \textit{A.I. Kostrikin}'s book ``Around Burnside'' (Nauka 1986; Zbl 0624.17001) but the present one is much shorter. The author points out the following Corollary: An n-Engel Lie algebra of characteristic \(p\gg n\) (sufficiently large with respect to n) is nilpotent. One certainly may also expect some corollaries in group theory.
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 4101416
zbMATH Keywords: verbal ideals
zbMATH Keywords: \(Z_ 2\)-graded Lie algebras
zbMATH Keywords: nilpotency
zbMATH Keywords: n-Engel Lie algebra
zbMATH Keywords: characteristic zero
zbMATH Keywords: local nilpotency
zbMATH Keywords: restricted Burnside problem
zbMATH Keywords: Kostrikin's theorem
author: E. Zelmanov
MaRDI profile type: Publication
cites work: Q3263741
cites work: Q5825742
cites work: Q3338407
cites work: Q3337615
cites work: Q3688524
cites work: Q5340151
description: scientific article; zbMATH DE number 4101416