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Progress in Mathematics #9: Linear Algebraic Groupsby Tony A. Springer
Synopses & ReviewsPublisher Comments: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 selfcontained 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 Tannakaa (TM)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 textbooksa e.g., Rosenlichta (TM)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. Book News Annotation:This second edition, thoroughly revised and expanded, extends the theory of linear algebra groups over arbitrary fields, which are not necessarily algebraically closed. The book includes a selfcontained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results from reductive groups. Chapters one through ten, essentially unchanged in this edition, can serve as a text for an introductory graduate course. Chapters 11 through 17 are new and include more algebraic geometry, such as Fgroups, Ftori, solvable Fgroups, F reductive groups, reductive Fgroups, and classification of simple groups over a ground field.
Annotation c. Book News, Inc., Portland, OR (booknews.com) Synopsis:This concise and essentially selfcontained book should appeal to a wide audience of graduate students and researchers in the field. The first part of "Linear Algebraic Groups" is quite suitable as a textbook for a course on the theory. The second part can be used in an advanced course or a seminar. 12 illus.
Synopsis:The work is concise and selfcontained, 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.
Synopsis:"The structure and classification of reductive groups over arbitrary fields has become a standard part of mathematics, with many connections to other areas: Lie groups (classical and finite), number theory (Langlands program, arithmetic groups), invariant theory. It is an active field, to which the author T. A. Springer has made invaluable contributions. Linear Algebraic Groups is rather different from the existing texts on the subject. The first ten chapters cover the theory of linear algebraic groups over algebraically closed fields, including the required results from algebraic geometry. These chapters culminate in the theory of reductive groups. The uniqueness and existence theorems are also included. Chapters 1117 cover the theory of linear algebraic groups over fields which are not algebraically closed. Much of this material cannot be found in existing texts. The last chapters deal with the Tits classification of simple groups. The book is concise and essentially selfcontained. It should appeal to a wide audience of graduate students and researchers in the field. The first part of Linear Algebraic Groups is quite suitable as a textbook for a course on the theory. The second part can be used in an advanced course or a seminar. There are many exercises, an excellent bibliography, and subject index. Series: Progress In Mathematics, Volume 9 "
Table of ContentsPreface
1. Some algebraic geometry 1.1. The Zariski topology 1.2. Irreducibility of topological spaces 1.3 Affine algebras 1.4 Regular functions, ringed spaces 1.5 Products 1.6 Prevarieties and varieties 1.7 Projective varieties 1.8 Dimension 1.9 Some results on morphisms Notes 2. Linear algebraic grops, first properties 2.1 Algebraic groups 2.2 Some basic results 2.3 G spaces 2.4 Jordan decomposition 2.5 Recovering a group from its representations Notes 3. Commutative algebraic groups 3.1 Structure of commutative algebraic groups 3.2 Diagonalizable groups and tori 3.3 Additive functions 3.4 Elementary unipotent groups Notes 4. Derivations, differentials, Lie algebras 4.1 Derivations and tangent spaces 4.2 Differentials, separability 4.3 Simple points 4.4 The Lie algebra of a linear algebraic group Notes 5. Topological properties of morphisms, applications 5.1 Topological properties of morphisms 5.2 Finite morphisms, normality 5.3 Homogeneous spaces 5.4 Semisimple automorphisms 5.5 Quotients Notes 6. Parabolic subgroups, Borel subgroups, Solvable groups 6.1 Complete varieties 6.2 Parabolic subgroups and Borel subgroups 6.3 Connected solvable groups 6.4 Maximalt tori, further properties of Borel groups Notes 7. Weyl group, roots, root datum 7.1. The Weyl group 7.2. Semisimple groups of rank one 7.3. Reductive groups of semisimple rank one 7.4. Root data 7.5. Two roots 7.6. The unipotent radical Notes 8. Reductive groups 8.1. Structural properties of reductive groups 8.2. Borel subgroups and systems of positive roots 8.3. The Bruhat decomposition 8.4. Parabolic subgroups 8.5. Geometric questions related to the Bruhat decomposition Notes 9. The isomorphism theorem 9.1. Two dimensional root systems 9.2. The structure constants 9.3. The elements nalpha 9.4. A presentation of G 9.5. Uniqueness of structure constants 9.6. The isomorphism theorem Notes 10. The existence theorem 10.1. Statement of the theorem, reduction 10.2. Simply laced root systems 10.3. Automorphisms, end of the proof of 10.1.1 Notes 11. More algebraic geometry 11.1. F structures on vector spaces 11.2. F varieties, density, criteria for ground fields 11.3. Forms 11.4. Restriction of the ground field Notes 12. F groups, general results 12.1. Field of definition of subgroups 12.2. Complements on quotients 12.3. Galois cohomology 12.4.Restriction of the ground field Notes 13. F tori 13.1. Diagonalizable groups over F 13.2. F tori 13.3. Tori in F groups 13.4. The groups P (lambda) Notes 14. Solvable F groups 14.1. Generalities 14.2. Action of Galpha on an affine variety, applications 14.3. F split solvable groups 14.4. Structural properties of solvable groups Notes 15. F reductive groups 15.1. Pseudoparabolic F subgroups 15.2. A fixed point theorem 15.3. The root datum of an F reductive group 15.4. The groups Ualpha 15.5. The index Notes 16. Reductive F groups 16.1. Parabolic subgroups 16.2. Indexed root data 16.3. F split groups 16.4. The isomorphism theorem 16.5. Existence Notes 17. Classification 17.1. Type An1 17.2. Types Bn and C n 17.3. Type Dn 17.4. Exceptional groups, type G2 17.5. Indices for the types F4 and E8 17.6. Descriptions for type F4 17.7.E6 17.8. Type E7 17.9. Trialtarian type D4 17.10. Special fields Notes Table of indices References Subject Index What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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