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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.
Cover
Modern Birkhauuser Classics
Linear Algebraic Groups (Reprint of the 1998 Second Edition)
Copyright
9780817648398
Linear Algebraic Groups, Second Edition
Copyright
3764340215
Contents
Preface to the Second Edition
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 Groups, 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 ofmorphisms
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. Maximal 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 a reductive group
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 n_a
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 ofthe 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
Notes
14. Solvable F -groups
14.1. Generalities
14.2. Action of G_a 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 U_{(a)}
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 A_{n-1}
17.2. Types B_n and C_n
17.3. Type D_n
17.4. Exceptional groups, type G_2
17.5. Indices for types F_4 and E_8
17.6. Descriptions for type F_4
17.7. Type E_6
17.8. Type E_7
17.9. Trialitarian type D_4
17.10. Special fields
Notes
Table of Indices
Bibliography
Index
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