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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.
List of contents
Some Algebraic Geometry.- Linear Algebraic Groups, First Properties.- Commutative Algebraic Groups.- Derivations, Differentials, Lie Algebras.- Topological Properties of Morphisms, Applications.- Parabolic Subgroups, Borel Subgroups, Solvable Groups.- Weyl Group, Roots, Root Datum.- Reductive Groups.- The Isomorphism Theorem.- The Existence Theorem.- More Algebraic Geometry.- F-groups: General Results.- F-tori.- Solvable F-groups.- Freductive Groups.- Reductive F-Groups.- Classification.
Summary
The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. 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.
Report
From the reviews of the second edition:
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style... [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature...appear for the first time in a text." -Mathematical Reviews (Review of the Second Edition)
"This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of the first ten chapters covers the contents of the old book, but the arrangement is somewhat different and there are additions, such as the basic facts about algebraic varieties and algebraic groups over a ground field, as well as an elementary treatment of Tannaka's theorem. These chapters can serve as a text for an introductory course on linear algebraic groups. The last seven chapters are new. They deal with algebraic groups over arbitrary fields. Some of the material has not been dealt with before in other texts, such as Rosenlicht's results about solvable groups in Chapter 14, the theorem of Borel and Tits on the conjugacy over the ground field of maximal split tori in an arbitrary linear algebraic group in Chapter 15, and the Tits classification of simple groups over a ground field in Chapter 17. The book includes many exercises and a subject index." -Zentralblatt Math(Review of the Second Edition)
"In Linear Algebraic Groups Springer aims at a self-contained treatment of the subject in the title and he certainly succeeds ... . each chapter comes equipped with an endnote for a bit of history and context, as well as indications of where to go next. And all of it is done in a very clear style, making for a smooth and readable presentation. ... a superb choice for any one wishing to learn the subject and go deeply into it quickly and effectively." (Michael Berg, The Mathematical Association of America, March, 2009)
