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The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim.

As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.

The material of the first ten chapters of the new edition is roughly the same as that of the first edition. The arrangement of the material is somewhat different, and a few things have been added, such as the basic facts about algebraic varieties and algebraic groups over a ground field, and an elementary testament of Tannaka’s theorem in Chapter2.

The last seven chapters are new and contain the extension of the theory to algebraic groups over arbitrary fields. Some of the material has not been dealt with before in textbooks—e.g., Rosenlicht’s results about solvable groups in chapter 14, the theory of Borel and Tits on the conjugacy over the ground field of a maximal split torus in an arbitrary linear algebraic group in Chapter 15, and the Tits classification of simple groups over a ground field in Chapter 17.

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It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades..
Price: $105.01 [Notify me when price goes down.]

From the reviews of Vols. I-III:

"Since their publication in 1986 J-P. Serre's Collected Papers have already become one of the classical references in mathematical research. This is on the one hand due to the completeness of the collection (132 items) and on the other, of course, due to the beautiful and clear expositions of Serre's papers and their influence on mathematics.

As listed in the preface, the three volumes cover almost all articles published in mathematical journals between 1949 and 1984, the summaries of the author's courses at the Collège de France since 1956, some of his Séminaire notes, and some items not previously published. [...]

The author's notes at the end of each volume giving corrections and important recent progress as well as improvements of the main results represent a highlight of this collection. The mathematical community definitely looks forward to further volume(s) of Serre's outstanding work."

Zentralblatt MATH

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Price: $59.46 [Notify me when price goes down.]

This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections.
Price: $141.52 [Notify me when price goes down.]

The first part of this book on Discrete Subgroups of Lie Groups is written by E.B. Vinberg, V.V. Gorbatsevich, and O.V. Shvartsman Various types of discrete subgroups of Lie groups arise in the theory of functions of complex variables, arithmetic, geometry, and crystallography. Since the foundation of their general theory in the 50-60s of this century, considerable and in many respects exhaustive results were obtained. This development is reflected in this survey. Both semisimple and general Lie groups are considered. Part II on Cohomologies of Lie Groups and Algebras is written by B.L. Feigin and D. B.Fuchs. It contains different definitions of cohomologies of Lie groups and (both finite-dimensional and some infinite-dimensional) Lie algebras, the main methods of their calculation, and the results of these calculations. The book can be useful as a reference and research guide to graduate students and researchers in different areas of mathematics and theoretical physics..
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