Combinatorial aspects in polynomial identities

Nowadays many of the main results on PI-algebras have simple and elegant proofs which can be found, for example, in the recent books on PI-algebras. Nevertheless, there are several important results which still wait for their easy proofs. The present book is written by two of the leading experts in the theory of PI-algebras and fills a serious gap in this direction. It not only contains a comprehensive study of the main research done in polynomial identities over the last 25 years. Its purpose is also to make more transparent important and difficult topics. The first one is the Shirshov height theorem. It implies directly the positive solution for PI-algebras of the Kurosh problem for the finite dimensionality of finitely generated algebraic algebras (established by Levitzki and Kaplansky using deep structure theory of PI-algebras) and the theorem of Berele that the finitely generated PI-algebras have finite Gelfand-Kirillov dimension. The second example is the Razmyslov-Kemer-Braun theorem for the nilpotency of the Jacobson radical of finitely generated PI-algebras. The third important topic in the book is the structure theory of T-ideals developed by Kemer, in the spirit of ideal theory of polynomial algebras. It led him to the positive solution of the Specht problem in characteristic 0 which asks whether the polynomial identities of an arbitrary PI-algebra follow from a finite collection of its identities. Other important applications of the theory of Kemer are the Belov theorem for the rationality of Hilbert (or Poincaré) series of a finitely generated relatively free algebra over an infinite field and the recent results of Giambruno and Zaicev on the codimension growth of PI-algebras. The theory of Kemer demonstrates the importance of the applications of the theory of superalgebras to the PI-theory. The main achievement of the book is the improved and simplified exposition of the results of Kemer. Also, the book contains different counterexamples to the Specht problem in positive characteristic. Besides the authors cover other classical results, such as the codimension theorem of Regev and his tensor product theorem, the existence of central polynomials, and the characterization of group algebras with polynomial identities. The approach of the authors is mainly combinatorial. They also emphasize the computational aspects of the PI-theory. They study numerical parameters of PI-algebras, as the Shirshov height, the index of nilpotency of the Jacobson radical, the Kemer index of a T-ideal, etc. A recurring theme is the Grassmann algebra, not only as the easiest nontrivial example of a PI-algebra, but also as a link between algebras and superalgebras, and finally, as a test algebra for the counterexamples to the Specht problem in characteristic p. I believe that the algebraic community will find the book interesting and useful. The text is suitable both for beginners and experts. The book (or parts of it) may serve as a graduate course on PI-algebras and on combinatorial ring theory. It can be used as a good source of references. In more detail, Chapter 1 contains basic results on PI-algebras. Chapter 2 deals with finitely generated (or affine) PI-algebras, the Shirshov theorem, the trace ring, the Razmyslov-Kemer-Braun theorem for the nilpotency of the radical. Chapter 3 introduces T-ideals and relatively free algebras, and poses the Specht problem. Chapter 4 considers the Specht problem in the finitely generated case. Chapter 5 deals with applications to PI-algebras of representation theory of the symmetric group and the Regev theorem for the exponential bound of the codimensions of PI-algebras. Chapter 6 introduces superidentities and their application to the main theorem in the theory of Kemer. Chapter 7 considers PI-algebras in positive characteristic and the negative solution of the Specht problem in this case. Chapter 8 contains some recent structural results. Chapter 9 is devoted to the Hilbert series of relatively free algebras and Gelfand-Kirillov dimension of finitely generated PI-algebras. Chapter 10 includes more representation theory, cocharacters and applications of representation theory of the general linear group. Chapter 11 goes to the general theory of identities and varieties in universal algebra. Chapter 12 returns to trace identities for their deeper study. Finally, Chapters 13, 14, and 15 contain, respectively, a big collection of exercises, lists of main theorems and major examples and counterexamples, and some open questions.