##
**Free algebras and PI-algebras. Graduate course in algebra.**
*(English)*
Zbl 0936.16001

Singapore: Springer. xii, 271 p. (2000).

The author of the book under review is one of the leading specialists in the area of PI algebras; for the last 30 years he has made a large list of important and significant contributions to the theory of algebras satisfying polynomial identities. These contributions are concentrated mainly in the combinatorial study of the PI algebras.

The book is a (quite) extended version of a graduate course given by the author in 1996. It should be noted that the book is not only an introduction to the PI theory but it also contains various topics that make it look like a monograph. For the last decade we have seen several monographs devoted to polynomial identities; a brief list of them would consist of the following books: Yu. P. Razmyslov, Identities of algebras and their representations [Transl. Math. Monogr. 138, AMS, Providence RI (1994; Zbl 0827.17001)], A. Kemer, Ideals of identities of associative algebras [Transl. Math. Monogr. 87, AMS, Providence RI (1991; Zbl 0732.16001)], E. Formanek, The polynomial identities and invariants of \(n\times n\) matrices [Reg. Conf. Ser. Math. 78. AMS, Providence RI (1991; Zbl 0714.16001)]. While the first two of them are certainly not for beginners, and the third also requires some knowledge, the book under review can be used by readers having quite modest knowledge of algebra. This does not mean at all that the book is at an elementary level, just on the contrary. The author discusses the very basics of the PI theory and further topics that one could find in journal publications only. Thus the book will be useful for post-graduate students who aim at extending their algebraic “luggage”. It will be useful also for researchers as a rich source of facts, examples, and bibliographic references. The specialists in the area will also find the book extremely useful and interesting.

The first two chapters of the book are preliminary, and they contain the basic definitions and properties of free algebras, T-ideals, varieties. In Chapter 3, the author gives a brief account of results on the finite basis problem for various algebraic systems. In more detail is considered the existence of a variety of Lie algebras over a field of characteristic \(2\) that is not finitely based.

Chapter 4 introduces the Hilbert series of a variety of algebras and the very important notion of a commutator (or proper) identity. The next chapter shows how the methods work in two concrete algebras – the Grassmann algebra and the algebra of the upper triangular matrices. Chapter 6 considers the rationality of Hilbert series of algebras that satisfy some nonmatrix identity (i.e., an identity that does not hold in the algebra of \(2\times 2\) matrices).

Chapter 7 is devoted to the polynomial identities of matrix algebras. It starts with the Amitsur and Levitzki Theorem, and then gives a good account of the known “bits and pieces”. The important concepts of central polynomials and of generic matrices are introduced and illustrated with various examples. (It should be noted that as a rule these examples are not very simple, they have their origin in research papers by the author of the book and many other algebraists.)

In Chapter 8, multilinear identities are considered. Since in characteristic \(0\) they determine the respective T-ideals their importance is obvious. The author discusses the codimension sequences and their growth, and at the end of the chapter he gives an account on the structure theory of the T-ideals developed by A. Kemer (see the book by Kemer cited above).

Chapter 9 discusses another important combinatorial aspect of the PI theory. There the author considers finiteness properties namely Shirshov’s theorem, and Gelfand-Kirillov dimension of a PI algebra.

The next two chapters are devoted to the study of automorphisms of free and relatively free associative and Lie algebras.

In the last, eleventh chapter, the author discusses an extremely important and fruitful method in the theory of PI algebras. It is the representation theory of the symmetric and of the general linear groups. This method is shown “in action”, i.e., how it works in several important situations.

The book reflects the author’s tastes and preferences. It discusses topics where he has made significant contributions to the theory. The exposition is lively and “intriguing.” One big plus of the book is the abundance of exercises. In general they come with hints, or at least with bibliographic references. In addition there are lots of open problems in the book, generally with comments and remarks on them.

In my opinion the book is a must for all algebraists interested in combinatorial ring theory. It will be useful for research students, and it can serve as a base for graduate courses in Algebra.

The book is a (quite) extended version of a graduate course given by the author in 1996. It should be noted that the book is not only an introduction to the PI theory but it also contains various topics that make it look like a monograph. For the last decade we have seen several monographs devoted to polynomial identities; a brief list of them would consist of the following books: Yu. P. Razmyslov, Identities of algebras and their representations [Transl. Math. Monogr. 138, AMS, Providence RI (1994; Zbl 0827.17001)], A. Kemer, Ideals of identities of associative algebras [Transl. Math. Monogr. 87, AMS, Providence RI (1991; Zbl 0732.16001)], E. Formanek, The polynomial identities and invariants of \(n\times n\) matrices [Reg. Conf. Ser. Math. 78. AMS, Providence RI (1991; Zbl 0714.16001)]. While the first two of them are certainly not for beginners, and the third also requires some knowledge, the book under review can be used by readers having quite modest knowledge of algebra. This does not mean at all that the book is at an elementary level, just on the contrary. The author discusses the very basics of the PI theory and further topics that one could find in journal publications only. Thus the book will be useful for post-graduate students who aim at extending their algebraic “luggage”. It will be useful also for researchers as a rich source of facts, examples, and bibliographic references. The specialists in the area will also find the book extremely useful and interesting.

The first two chapters of the book are preliminary, and they contain the basic definitions and properties of free algebras, T-ideals, varieties. In Chapter 3, the author gives a brief account of results on the finite basis problem for various algebraic systems. In more detail is considered the existence of a variety of Lie algebras over a field of characteristic \(2\) that is not finitely based.

Chapter 4 introduces the Hilbert series of a variety of algebras and the very important notion of a commutator (or proper) identity. The next chapter shows how the methods work in two concrete algebras – the Grassmann algebra and the algebra of the upper triangular matrices. Chapter 6 considers the rationality of Hilbert series of algebras that satisfy some nonmatrix identity (i.e., an identity that does not hold in the algebra of \(2\times 2\) matrices).

Chapter 7 is devoted to the polynomial identities of matrix algebras. It starts with the Amitsur and Levitzki Theorem, and then gives a good account of the known “bits and pieces”. The important concepts of central polynomials and of generic matrices are introduced and illustrated with various examples. (It should be noted that as a rule these examples are not very simple, they have their origin in research papers by the author of the book and many other algebraists.)

In Chapter 8, multilinear identities are considered. Since in characteristic \(0\) they determine the respective T-ideals their importance is obvious. The author discusses the codimension sequences and their growth, and at the end of the chapter he gives an account on the structure theory of the T-ideals developed by A. Kemer (see the book by Kemer cited above).

Chapter 9 discusses another important combinatorial aspect of the PI theory. There the author considers finiteness properties namely Shirshov’s theorem, and Gelfand-Kirillov dimension of a PI algebra.

The next two chapters are devoted to the study of automorphisms of free and relatively free associative and Lie algebras.

In the last, eleventh chapter, the author discusses an extremely important and fruitful method in the theory of PI algebras. It is the representation theory of the symmetric and of the general linear groups. This method is shown “in action”, i.e., how it works in several important situations.

The book reflects the author’s tastes and preferences. It discusses topics where he has made significant contributions to the theory. The exposition is lively and “intriguing.” One big plus of the book is the abundance of exercises. In general they come with hints, or at least with bibliographic references. In addition there are lots of open problems in the book, generally with comments and remarks on them.

In my opinion the book is a must for all algebraists interested in combinatorial ring theory. It will be useful for research students, and it can serve as a base for graduate courses in Algebra.

Reviewer: Plamen Koshlukov (Campinas)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |

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

16R30 | Trace rings and invariant theory (associative rings and algebras) |

20C30 | Representations of finite symmetric groups |

20B27 | Infinite automorphism groups |

16W20 | Automorphisms and endomorphisms |