Some thoughts on the current state of the theory of identical relations in Lie algebras
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 construction due to Golod that one cannot drop the requirement of finite dimensionality. There are other examples, briefly discussed in the paper, which can be found in the paper [\textit{V. M. Petrogradsky} et al., Groups Geom. Dyn. 4, No. 4, 873--900 (2010; Zbl 1229.17025)], and its references. One may ask, like in the case of groups, what happens if for every \(x\in L\), the linear transformation \(ad \,x\) is nilpotent of bounded degree \(n\). Such algebras are called \(n\)-Engel Lie algebras, and they tie up the theory of Lie algebras with Group theory. A well known example due to P. Cohn shows that an \(n\)-Engel Lie algebra needs not be nilpotent if the base field is of positive characteristic. On the other hand, \textit{E. I. Zel'manov} [Sov. Math., Dokl. 35, 44--47 (1987; Zbl 0625.17006); translation from Dokl. Akad. Nauk SSSR 292, 265--268 (1987)], proved the fundamental theorem that an \(n\)-Engel Lie algebra is nilpotent if the base field is of characteristic 0. The author also discusses the local version of the above question, and describes the important achievements due to Kostrikin and to Zelmanov. In Section 4, The isomorphism problem, the author starts with the well known theorem of \textit{A. Kh. Kushkuleĭ} and \textit{Yu. P. Razmyslov} [Vestn. Mosk. Univ., Ser. I 1983, No. 5, 4--7 (1983; Zbl 0516.17007)], which states that non-isomorphic simple finite dimensional Lie algebras satisfy different identities. The author shows how one can deduce this result from a theorem by \textit{Yu. P. Razmyslov} [Identities of algebras and their representations. Transl. from the Russian. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)], and provides several important corollaries and developments the theorem of Razmyslov. Then the author proceeds with several concrete types of identities in Lie algebras. He discusses the following conjecture due to Razmyslov, see the above cited monograph: If \(L\) is a simple Lie algebra satisfying an identity then \(L\) is of Cartan type. He also discusses several advances to the solution of the conjecture. The section ends with a discussion of the Lie standard and of the Capelli identities. Section 5 is devoted to a topic that has attracted the attention of very many specialists: the codimension growth of the identities of Lie algebras. In Section 6 the author considers graded Lie algebras and their graded identities. Then he proceeds with actions on Lie algebras, with a special attention to Lie algebras acted on by Hopf algebras. Several generalizations of the results discussed in Section 4 are also stated. The book provides the readers with a wealth of information concerning recent developments in the theory of Lie algebras with identical relations. The list of references includes 55 titles, and it will be also helpful to the reader who wants to know the detailed proofs of the statements. Let me mention that the interested readers can also consider consulting the very recent monograph by the same author, [Identical relations in Lie algebras. 2nd edition. Berlin: De Gruyter (2021; Zbl 1487.17001)]. There one can find a detailed and extended account of the material in this paper, and much more information about the topic. For the entire collection see [Zbl 1461.16003].
scientific article; zbMATH DE number 7445603
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