Nilpotency criteria for multiplicative Lie algebras
type: Item
instance of: scholarly article
title: Nilpotency criteria for multiplicative Lie algebras
zbMATH Open document ID: 0863.20015
DOI: 10.1016/0022-4049(95)00115-8
author: Françoise Point
author: Patricia Wantiez
published in: Journal of Pure and Applied Algebra
publication date: 1997-06-03T00:00:00Z
review text: The structure of a multiplicative Lie algebra has been introduced by \textit{G. J. Ellis} [J. Aust. Math. Soc., Ser. A 54, No. 1, 1-19 (1993; Zbl 0777.20001)]. A multiplicative Lie algebra is a (possibly nonabelian) group with an extra binary function \(\{.,.\}\) satisfying the identities: \[ \{x,x\}=1,\quad\{x,yy'\}=\{x,y\}^y\{x,y'\},\quad\{xx',y\}={}^x\{x',y\}\{x,y\}, \] \[ \{y,x,{}^xz\}\{x,z,{}^zy\}\{z,y,{}^yx\}=1,\;{}^z\{x,y\}=\{^zx,^zy\},\text{ where }\{x,y,z\}=\{\{x,y\},z\},\;{}^yx=yxy^{-1}. \] The two main examples of multiplicative Lie algebras are a group with the binary function \(\{.,.\}\) defined as the group commutator \([x,y]\equiv xyx^{-1} y^{-1}\) and a Lie ring with \(\{.,.\}\), defined as the Lie bracket. The notions of ideal, nilpotency, solubility can be defined in terms of a new binary function. The authors prove several nilpotency criteria for multiplicative Lie algebras which extend known results about groups and Lie algebras. The main results of the paper are some variations of Gruenberg's theorem, stating nilpotency of a soluble finitely generated Engel group (Lie ring) [\textit{K. W. Gruenberg}, Proc. Camb. Philos. Soc. 49, 377-380 (1953; Zbl 0050.01901), Theorem 1]. The authors also prove an analog of the Hirsch-Plotkin theorem for multiplicative Lie algebras.
reviewed by: N. Yu. Makarenko
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 920032
zbMATH Keywords: multiplicative Lie algebras
zbMATH Keywords: group commutators
zbMATH Keywords: Lie rings
zbMATH Keywords: Lie brackets
zbMATH Keywords: nilpotency criteria
zbMATH Keywords: soluble finitely generated Engel groups
zbMATH Keywords: Hirsch-Plotkin theorem
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00115-8
OpenAlex ID: W2085736064
Wikidata QID: Q115364385
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description: scientific article; zbMATH DE number 920032