A comprehensive study of the main research done in polynomial identities over the last 25 years, including Kemer's solution to the Specht problem in characteristic O and examples in the characteristic p situation. The authors also cover codimension theory, starting with Regev's theorem and continuing through the Giambruno-Zaicev exponential rank. The "best" proofs of classical results, such as the existence of central polynomials, the tensor product theorem, the nilpotence of the radical of an affine PI-algebra, Shirshov's theorem, and characterization of group algebras with PI, are presented.
" ""The present book is written by two of the leading experts in the thoery of PI-algebras and fills a serious gap… It not only contains a comprehensive study of the main research done in polynomial identitites over the last 25 years. Its purpose is also to make more transparent important and difficult topics. 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."" -Vesselin Drensky, Mathematiacl Reviews 2006b, February 2006
""As a summary, the book contains a wealth of contemporary material about polynomial identities in associative algebras. It can be recommended to an advanced reader with substantial experience in the theory of PI-algebras."" -CMS Reviews, November 2006
""All topics of the monograph are well-arranged and developed in a clear way… suitable not only as a useful reference for researchers but also as part of a course on PI-algebras for graduate students."" -EMS, December 2006
""This book, written by masters in this area, beautifully describe [approaches to polynomial identities], partly following the historic development of some famous problems."" -Internationale Mathematische Nachrichten, August 2008"
1. Basic Results 2. Affine Pl-algebras 3. T-ldeals and Relatively Free Algebras 4. Specht's Problem in the Affine Case 5. Representations of Sn and Their Applications 6. Superidentities and Kemer's Main Theorem 7. Pi-Algebras in Characteristic p 8. Recent Structural Results 9. Poincare-Hilbert Series and Gelfand-Kirillov Dimension 10. More Representation Theory 11. Unified Theory of Identities 12. Trace Identities 13. Exercises 14. Lists of Theorems and Examples 15. Some Open Questions