Identities in Lie algebras.
(Тождества в алгебрах Ли.)

*(Russian)*Zbl 0571.17001
Moskva: ”Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 448 p. R. 4.10 (1985).

Let \(f(x_ 1,...,x_ n)\) be an element of the free Lie algebra. The Lie algebra G satisfies the polynomial identity \(f=0\) if \(f(g_ 1,...,g_ n)=0\) for every \(g_ 1,...,g_ n\in G\). The class of all algebras satisfying a given system of identities is called a variety. The investigation of the varieties of Lie algebras began in the mid-fifties, although many classical results such as the Engel and Lie theorems can formally be restated in the language of identities. Among the achievements of that period are the fundamental results of Kostrikin on the Burnside problem, where one of the main steps is the investigation of Lie rings with the Engel condition, and the series of papers by Shirshov on the free Lie algebras.

The next stage of the development of the theory of Lie algebras with polynomial identities was influenced by the theory of varieties of groups. At the start the study of Lie algebras followed that of groups. But it soon turned out that the theory of varieties of Lie algebras had its considerable peculiarities.

Now this theory is a separate branch of algebra. It follows its own way of development and exercises an influence on neighbouring fields such as the theory of groups and of associative, Jordan and other algebras.

The book under review is written by one of the main contributors to the theory of Lie algebras with polynomial identity. It expands essentially the material of the book by the same author published several years ago [Yu. A. Bakhturin, Lectures on Lie algebras (1978; Zbl 0396.17006)]. For the first time in the theory of Lie algebras a systematic exposition is given of the methods and results on varieties of Lie algebras. Although some aspects of the theory are not included in the book, it is a comprehensive account of the work done during the last 30 years. The exposition is self-contained and proofs are given in all detail. The results considered, both classical and recent ones, and published mainly in research papers, are restated from a unique point of view. Not much special algebraic knowledge is necessary for the understanding of the text, but the difficulties for the reader increase with the numbering of the chapters. The large collection of exercises is useful for the understanding of the material and forms a supplement to the main text.

We give a short summary of the 8 chapters of the book. Chapter 1 contains basic definitions and theorems from the general theory of Lie algebras over commutative rings. Chapter 2 is devoted to the free Lie algebras, their bases and their subalgebras. Here the main subject of the book - the algebras with a polynomial identity - appears. Chapter 3 introduces the representation theory of the symmetric groups and the general linear groups and its application to the investigation of identities. Chapter 4 considers an approach to the theory of varieties of algebras in the spirit of group theory. Some of the problems discussed in this chapter are related to the operations in the set of all varieties, to the basic rank of the varieties and to the subalgebras of the relatively free algebras. Shmel’kin’s wreath product is introduced.

Chapter 5 deals with the finite basis problem. Recall that a variety \({\mathfrak M}\) has the Specht property, if every subvariety of \({\mathfrak M}\) can be defined by a finite set of polynomial identities. The technique of partially well-ordered sets due to Higman and Cohen is applied to establish the Specht property of some classes of varieties. For algebras over a field of positive characteristic, examples of infinitely based varieties are given. Ideas from commutative algebra are exploited in order to prove that every finite dimensional solvable algebra in characteristic 0 has a finite basis of identities. The polynomial identities of the algebra of \(2\times 2\) traceless matrices are studied and a finite basis for them is exhibited.

Chapter 6 is devoted to the theory of SPI-Lie algebras. By definition these are the algebras which can be embedded in associative PI-algebras. The main purpose is to establish that many classical properties of finite dimensional algebras hold for SPI-algebras as well. The technique of critical algebras is introduced in Chapter 7 where the identities of finite Lie rings are studied. The main result states that the identities of these rings follow from a finite number of them. Locally finite varieties are considered as well. In particular, the almost Cross varieties of solvable Lie rings are described. (A variety is Cross if it is generated by a finite ring.)

Chapter 8 is an application of Lie algebra methods to group theory. It contains the classical results due to P. Hall, Magnus, Witt and Mal’tsev on the connection between the free (respectively nilpotent) groups and Lie algebras. A correspondence of varieties of groups to varieties of Lie algebras is exposed. A powerful method of Razmyslov is applied for solving some problems of the theory of varieties. In particular, his theorem on the existence of non-solvable groups of exponent 4 is proved.

This book is interesting both for specialists and for beginners. Undoubtedly it will be useful to mathematicians working on other classes of algebras with polynomial identities. It is the opinion of the reviewer that the translation into English of ”Identities in Lie algebras” will allow a wider algebraic audience to get access to an interesting approach to the study of Lie algebras, which can have implications for problems arising from a more traditional set-up.

The next stage of the development of the theory of Lie algebras with polynomial identities was influenced by the theory of varieties of groups. At the start the study of Lie algebras followed that of groups. But it soon turned out that the theory of varieties of Lie algebras had its considerable peculiarities.

Now this theory is a separate branch of algebra. It follows its own way of development and exercises an influence on neighbouring fields such as the theory of groups and of associative, Jordan and other algebras.

The book under review is written by one of the main contributors to the theory of Lie algebras with polynomial identity. It expands essentially the material of the book by the same author published several years ago [Yu. A. Bakhturin, Lectures on Lie algebras (1978; Zbl 0396.17006)]. For the first time in the theory of Lie algebras a systematic exposition is given of the methods and results on varieties of Lie algebras. Although some aspects of the theory are not included in the book, it is a comprehensive account of the work done during the last 30 years. The exposition is self-contained and proofs are given in all detail. The results considered, both classical and recent ones, and published mainly in research papers, are restated from a unique point of view. Not much special algebraic knowledge is necessary for the understanding of the text, but the difficulties for the reader increase with the numbering of the chapters. The large collection of exercises is useful for the understanding of the material and forms a supplement to the main text.

We give a short summary of the 8 chapters of the book. Chapter 1 contains basic definitions and theorems from the general theory of Lie algebras over commutative rings. Chapter 2 is devoted to the free Lie algebras, their bases and their subalgebras. Here the main subject of the book - the algebras with a polynomial identity - appears. Chapter 3 introduces the representation theory of the symmetric groups and the general linear groups and its application to the investigation of identities. Chapter 4 considers an approach to the theory of varieties of algebras in the spirit of group theory. Some of the problems discussed in this chapter are related to the operations in the set of all varieties, to the basic rank of the varieties and to the subalgebras of the relatively free algebras. Shmel’kin’s wreath product is introduced.

Chapter 5 deals with the finite basis problem. Recall that a variety \({\mathfrak M}\) has the Specht property, if every subvariety of \({\mathfrak M}\) can be defined by a finite set of polynomial identities. The technique of partially well-ordered sets due to Higman and Cohen is applied to establish the Specht property of some classes of varieties. For algebras over a field of positive characteristic, examples of infinitely based varieties are given. Ideas from commutative algebra are exploited in order to prove that every finite dimensional solvable algebra in characteristic 0 has a finite basis of identities. The polynomial identities of the algebra of \(2\times 2\) traceless matrices are studied and a finite basis for them is exhibited.

Chapter 6 is devoted to the theory of SPI-Lie algebras. By definition these are the algebras which can be embedded in associative PI-algebras. The main purpose is to establish that many classical properties of finite dimensional algebras hold for SPI-algebras as well. The technique of critical algebras is introduced in Chapter 7 where the identities of finite Lie rings are studied. The main result states that the identities of these rings follow from a finite number of them. Locally finite varieties are considered as well. In particular, the almost Cross varieties of solvable Lie rings are described. (A variety is Cross if it is generated by a finite ring.)

Chapter 8 is an application of Lie algebra methods to group theory. It contains the classical results due to P. Hall, Magnus, Witt and Mal’tsev on the connection between the free (respectively nilpotent) groups and Lie algebras. A correspondence of varieties of groups to varieties of Lie algebras is exposed. A powerful method of Razmyslov is applied for solving some problems of the theory of varieties. In particular, his theorem on the existence of non-solvable groups of exponent 4 is proved.

This book is interesting both for specialists and for beginners. Undoubtedly it will be useful to mathematicians working on other classes of algebras with polynomial identities. It is the opinion of the reviewer that the translation into English of ”Identities in Lie algebras” will allow a wider algebraic audience to get access to an interesting approach to the study of Lie algebras, which can have implications for problems arising from a more traditional set-up.

Reviewer: Vesselin Drensky (Sofia)

##### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17B01 | Identities, free Lie (super)algebras |

16Rxx | Rings with polynomial identity |

16P10 | Finite rings and finite-dimensional associative algebras |

17A70 | Superalgebras |

17B05 | Structure theory for Lie algebras and superalgebras |

17B30 | Solvable, nilpotent (super)algebras |

17B35 | Universal enveloping (super)algebras |

20C30 | Representations of finite symmetric groups |