Some thoughts on the current state of the theory of identical relations in Lie algebras
type: Item
instance of: scholarly article
title: Some thoughts on the current state of the theory of identical relations in Lie algebras
zbMATH Open document ID: 1489.17005
DOI: 10.1007/978-3-030-63111-6_1
publication date: 2021-12-15T00:00:00Z
review text: The paper is a survey of results obtained in the theory of Lie algebras satisfying identical relations. The author is one of the leading researchers in the area, and a good part of the advances in it has been obtained with his participation. The area is tied up with the theory of PI algebras; they share a lot of common methods and tools. On the other hand there are reasons to believe that the existence of an identity in a Lie algebra is not that strong a restriction on the algebra and its properties as in the associative case. This implies that sometimes counter-examples are ``easier'' to produce in the case of Lie algebras; but it also implies that positive results may be much harder to obtain in the Lie case. The paper starts with a brief introduction where the author recalls some of the basic notions needed later on. Then he discusses the Finite basis problem: is every ideal of identities in the free Lie algebra finitely generated as an ideal of identities? The answer to it is negative: [\textit{M. R. Vaughan-Lee}, Q. J. Math., Oxf. II. Ser. 21, 297--308 (1970; Zbl 0204.35901)], and then [\textit{V. S. Drensky}, Algebra Logika 13, 265--290 (1974; Zbl 0298.17011)] exhibited counter-examples over fields of characteristic 2, and of arbitrary positive characteristic, respectively. Note that these examples also included finite dimensional Lie algebras. Note also that the corresponding examples in the case of associative algebras were obtained in 1999 only. The section ends with the theorem of [\textit{A. V. Il'tyakov}, Nova J. Algebra Geom. 1, No. 3, 207--259 (1992; Zbl 0892.17007)] which states that in characteristic 0, the identities of finite dimensional representations of a Lie algebra admit a finite basis. Section 3 is entitled Engel Lie algebras. A Lie algebra \(L\) is nil (or Engel) whenever every element of it is ad-nilpotent. The classical Engel theorem states that in case \(\dim L<\infty\) nil implies nilpotency of \(L\). It follows from the well known const…
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 17-02
zbMATH DE Number: 7445603
zbMATH Keywords: Lie algebras
zbMATH Keywords: identical relations
author: Yuri Bahturin
MaRDI profile type: Publication
full work available at URL: 978-3-030-63111-6 1
OpenAlex ID: W3136643202
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description: scientific article; zbMATH DE number 7445603