Linear algebraic groups

Author(s)

• Springer, Tonny Albert

Bibliographic Information

Linear algebraic groups

T.A. Springer

（Progress in mathematics, v. 9）

Birkhäuser, c1998

2nd ed

• : us
• : sz

Note

Includes bibliographical references (p. [323]-329) and index

Description and Table of Contents

Volume

: us ISBN 9780817640217

Description

'[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).

Table of Contents

Preface to the Second Edition.- 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.- F-reductive Groups.- Reductive F-groups.- Classification.- Table of Indices.- Bibliography.- Index.

Volume

: sz ISBN 9783764340216

Description

The structure and classification of reductive groups over arbitrary fields has become a standard part of mathematics, with broad connections to many aspects of group theory (Lie groups), number theory (Langlands program, arithmetic groups), algebraic geometry and invariant theory. The first ten chapters of this text cover the theory of linear algebraic groups over algebraically closed fields, culminating in the theory of reductive groups, and includes the uniqueness and existence theorems. Chapters 11-17 cover the theory of linear algebraic groups which are not algebraically closed. The last seven chapters deal with the Tits classification of simple groups. The work is concise and self-contained, and should appeal to a broad audience of graduate students and researchers in the field. It is suitable for use as a textbook for a course on the theory, and contains exercises.

Table of Contents

• Some algebraic geometry
• linear algebraic groups, first properties
• communtative algebraic groups
• derivations, differntials, 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
• F-reductive groups
• reductive F-groups
• classification.

by "Nielsen BookData"

Details

• NCID: BA38018862
• ISBN:
  • 0817640215
  • 3764340215
• LCCN: 98009333
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Boston
• Pages/Volumes: x, 334 p.
• Size: 25 cm
• Classification:
  • LCC : QA179
  • DC21 : 512/.55
  • NDC8 : 411.6
• Subject Headings:
  • LCSH : Linear algebraic groups
• Parent Bibliography ID:
  • BA00018010