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Progress in Mathematics #9: Linear Algebraic Groups

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Progress in Mathematics #9: Linear Algebraic Groups Cover

 

Synopses & Reviews

Publisher 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 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 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 self-contained 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 F-groups, F-tori, solvable F-groups, F- reductive groups, reductive F-groups, and classification of simple groups over a ground field.
Annotation c. Book News, Inc., Portland, OR (booknews.com)

Synopsis:

This concise and essentially self-contained 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 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.

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 11-17 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 self-contained. 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 Contents

Preface
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 Semi-simple 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. Semi-simple groups of rank one
7.3. Reductive groups of semi-simple 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. Pseudo-parabolic 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 An-1
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

Product Details

ISBN:
9780817640217
Author:
Springer, Tony A.
Author:
Springer, T. A.
Author:
Springer
Publisher:
Birkhauser
Subject:
Algebra - Linear
Subject:
Algebras, linear
Subject:
Geometry - Algebraic
Subject:
Linear algebraic groups
Subject:
Algebraic Geometry
Subject:
General Mathematics
Subject:
Lie theory
Subject:
Mathematics-Linear Algebra
Edition Number:
2
Edition Description:
2nd ed.
Series:
Progress in Mathematics
Series Volume:
9
Publication Date:
19981031
Binding:
Hardcover
Language:
English
Illustrations:
Y
Pages:
334
Dimensions:
9.60x6.44x.89 in. 1.47 lbs.

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Science and Mathematics » Mathematics » Algebra » General
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Progress in Mathematics #9: Linear Algebraic Groups New Hardcover
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Product details 334 pages Birkhauser Boston - English 9780817640217 Reviews:
"Synopsis" by , This concise and essentially self-contained 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" by , 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.
"Synopsis" by , "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 11-17 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 self-contained. 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 "
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