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  • © 1998

Linear Algebraic Groups

Birkhäuser

Authors:

  • An affordable softcover edition of a classic text

  • Introduces the theory of algebraic groups over an algebraically closed field

  • Second Edition thoroughly revised and expanded, extending the theory over arbitrary fields which are not necessarily algebraically closed

  • Excellent exercises, bibliography and index

  • May be used as a textbook in graduate and advanced undergraduate courses on Algebra, Modern Algebra, and Linear Algebra

  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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eBook USD 69.99
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  • ISBN: 978-0-8176-4840-4
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  • Own it forever
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  • Tax calculation will be finalised during checkout
Softcover Book USD 89.99
Price excludes VAT (USA)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xiii
  2. Some Algebraic Geometry

    • T. A. Springer
    Pages 1-20
  3. Linear Algebraic Groups, First Properties

    • T. A. Springer
    Pages 21-41
  4. Commutative Algebraic Groups

    • T. A. Springer
    Pages 42-56
  5. Derivations, Differentials, Lie Algebras

    • T. A. Springer
    Pages 57-77
  6. Weyl Group, Roots, Root Datum

    • T. A. Springer
    Pages 114-131
  7. Reductive Groups

    • T. A. Springer
    Pages 132-153
  8. The Isomorphism Theorem

    • T. A. Springer
    Pages 154-174
  9. The Existence Theorem

    • T. A. Springer
    Pages 175-184
  10. More Algebraic Geometry

    • T. A. Springer
    Pages 185-207
  11. F-groups: General Results

    • T. A. Springer
    Pages 208-222
  12. F-tori

    • T. A. Springer
    Pages 223-236
  13. Solvable F-groups

    • T. A. Springer
    Pages 238-251
  14. Freductive Groups

    • T. A. Springer
    Pages 252-268
  15. Reductive F-Groups

    • T. A. Springer
    Pages 269-284
  16. Classification

    • T. A. Springer
    Pages 285-319
  17. Back Matter

    Pages 320-334

About this book

"[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)

Keywords

  • Algebraic Geometry
  • Borel Subgroups
  • Commutative Algebraic Groups
  • Derivation
  • Lie
  • Lie Theory
  • Morphism
  • Parabolic Subgroups
  • The Isomorphism Theorem
  • Weyl Groups
  • algebra
  • algebraic varieties
  • geometry
  • linear algebra
  • theorem
  • matrix theory

Reviews

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)

Authors and Affiliations

  • Mathematisch Instituut, Rijksuniversiteit Utrecht, Utrecht, The Netherlands

    T. A. Springer

Bibliographic Information

Buying options

eBook USD 69.99
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4840-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 89.99
Price excludes VAT (USA)