On varieties of Lie algebras and varieties of restricted Lie algebras
type: Item
instance of: scholarly article
title: On varieties of Lie algebras and varieties of restricted Lie algebras
zbMATH Open document ID: 0840.17005
author: A. Yu. Stepanov
published in: Moscow University Mathematics Bulletin
publication date: 1996-03-27T00:00:00Z
review text: Let \(K\) be a field of positive characteristic \(p\), let \(L(X)\) be the free restricted Lie algebra over \(K\) freely generated by a countable set \(X\) and let \(L(X)\) be the free Lie algebra canonically embedded into \(L(X)\). The author constructs a natural correspondence between varieties of restricted Lie algebras \(U\) and varieties of Lie algebras \(U\) as follows: \(U= \tau(U)\) is determined by all Lie polynomial identities in \(U\) and \(U= \eta (U)\) consists of all restricted Lie algebras satisfying the same Lie identities as \(U\). The properties of the mappings \(\tau\) and \(\eta\) are investigated. As a consequence the author establishes that over a finite field \(K\) there exist two solvable just-non-Cross varieties of restricted Lie algebras only, i.e. these varieties are not generated by a finite algebra and all their proper subvarieties are generated by such an algebra. An analogue for restricted Lie algebras of a result of E. V. Ritter on the skeleton of locally finite varieties of Lie algebras is obtained as well. Reviewer's remark. In the title of the English translation of the paper under review, ``bounded Lie algebras'' should be replaced by ``restricted Lie algebras''.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 848911
zbMATH Keywords: solvable Lie algebras
zbMATH Keywords: Lie algebras with polynomial identity
zbMATH Keywords: modular Lie algebras
zbMATH Keywords: free restricted Lie algebra
zbMATH Keywords: varieties of restricted Lie algebras
zbMATH Keywords: varieties of Lie algebras
zbMATH Keywords: Lie polynomial identities
zbMATH Keywords: solvable just-non-Cross varieties
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 848911