Graduate Algebra - Noncommutative View
This is the second volume (parts IV-VI) of a comprehensive treatment of graduate algebra of which the first volume (parts I-III) dealt with commutative algebra and appeared as Vol. 73 in the same series in 2006. This volume presents a number of classic results from non-commutative algebra but also some of the highlights of much later achievements (e.g. Kemer's solution of Specht's conjecture, Tits' alternative for algebraic groups and Zelmanov's solution to the restricted Burnside problem). Similarly, as polynomials in finitely many variables over a field F form the basic example of a commutative F-algebra, the full matrix algebra Mn(F) is the key example for the non-commutative setting. The author therefore dedicates most of part IV to the classic Wedderburn-Artin structure theory of semisimple rings and modules, the Jacobson density theorem and the Goldie theorems (in chapters 14-16). In chapter 17, he introduces a new point of view: presentations of algebras and groups in terms of generators and relations, the Gelfand-Kirillov dimension and decision problems. Chapter 18 brings more powerful machinery to the scene: that of tensor products of modules and algebras.
Part V is dedicated to representation theory. First, classic results on group algebras and group representations are presented in chapter 19, including a section on representations of the symmetric group and an appendix on the Tits' alternative. Chapter 20 treats classical character theory. Elements of Lie theory (including Cartan's classification) are presented in chapter 21; this chapter ends with a brief overview of alternative and Jordan algebras. Part V ends with an exposition of the theory of Dynkin graphs and Coxeter groups and their applications. Part VI starts with a remarkable chapter on PI-algebras, presenting the main points of the theory up to a rather detailed sketch of Kemer's and Zelmanov's results mentioned above. Chapter 24 deals with central simple algebras and the Brauer group. Chapter 25 presents elements of homological algebra and Morita theory for module categories and the first steps of representation theory of finite dimensional algebras (Gabriel's theorem). The final chapter is an introduction to Hopf algebras, a structure that comprises group algebras, enveloping algebras of Lie algebras and algebraic groups but also has applications in quantum theory.
Each of the three parts of this volume ends with more than 30 pages of exercises, many of which introduce the reader to further research and applications. The volume ends with a 27 page list of major results, a bibliography, a list of names and an index. Thanks to his deep insight in the subject, the author has succeeded in presenting most of classic non-commutative algebra in a unified way together with some of the more recent highlights of the theory. This is a remarkable achievement making the book of interest not just for graduate students in algebra but to graduate students and researchers in a wide spectrum of areas of mathematics and physics.