Lectures on Chevalley Groups

This is a great unsurpassed introduction to the subject of Chevalley groups that influenced generations of mathematicians. I would recommend it to anybody whose interests include group theory." — Efin Zelmanov, University of California, Los Angeles"Robert Steinberg's lectures on Chevalley groups were given at Yale University in 1967. The notes for the lectures contain a wonderful exposition of the work of Chevalley, as well as important additions to that work due to Steinberg himself. The theory of Chevalley groups is of central importance not only for group theory, but also for number theory and theoretical physics, and is as relevant today as it was in 1967. The publication of these lecture notes in book form is a very welcome addition to the literature." — George Lusztig, Massachusetts Institute of Technology"Robert Steinberg gave a course at Yale University in 1967 and the mimeographed notes of that course have been read by essentially anyone interested in Chevalley groups. In this course, Steinberg presents the basic constructions of the Chevalley groups over arbitrary fields. He also presents fundamental material about generators and relations for these groups and automorphism groups. Twisted variations on the Chevalley groups are also introduced. There are several chapters on the representation theory of the Chevalley groups (over an arbitrary field) and for many of the finite twisted groups. Even 50 years later, this book is still one of the best introductions to the theory of Chevalley groups and should be read by anyone interested in the field." — Robert Guralnick, University of Southern California"A Russian translation of this lecture course by Robert Steinberg was published in Russia more than 40 years ago, but for some mysterious reason has never been published in the original language. This book is very dear to me. It is not only an important advance in the theory of algebraic groups, but it has also played a key role in more recent developments of the theory of Kac-Moody groups. The very different approaches, one by Tits and another by Peterson and myself, borrowed heavily from this remarkable book." — Victor Kac, Massachusetts Institute of Technology"Before this publication, to get your hands on this masterly exposition, you had to hope that your local library had a copy or a copy of a copy of it…I must emphasize that, as anyone involved in these subject knows, that the exposition in these lectures includes many original contributions of the author that have now become the standard ones which one finds in the literature…The AMS edition of these notes has been reset in LaTeX and includes some corrections and a few additions by Steinberg himself." — Filipe Zaldivar, MAA Reviews

• * Introduction* A basis for $\mathcal{L}$* A basis for $\mathcal{U}$* The Chevalley groups* Simplicity of $G$* Chevalley groups and algebraic groups* Generators and relations* Central extensions* Variants of the Bruhat lemma* The orders of the finite Chevalley groups* Isomorphisms and automorphisms* Some twisted groups* Representations* Representations continued* Representations completed* Appendix on finite reflection groups* Bibliography* Index