Description
Book SynopsisHe has previously held posts at the University of Oregon and New York University. His main research interests include group theory and Lie algebras, and this graduate level text is an exceptionally well-written introduction to everything about linear algebraic groups.
Trade ReviewJ.E. Humphreys
Linear Algebraic Groups
"Exceptionally well-written and ideally suited either for independent reading or as a graduate level text for an introduction to everything about linear algebraic groups."—MATHEMATICAL REVIEWS
Table of ContentsI. Algebraic Geometry.- 0. Some Commutative Algebra.- 1. Affine and Projective Varieties.- 1.1 Ideals and Affine Varieties.- 1.2 Zariski Topology on Affine Space.- 1.3 Irreducible Components.- 1.4 Products of Affine Varieties.- 1.5 Affine Algebras and Morphisms.- 1.6 Projective Varieties.- 1.7 Products of Projective Varieties.- 1.8 Flag Varieties.- 2. Varieties.- 2.1 Local Rings.- 2.2 Prevarieties.- 2.3 Morphisms.- 2.4 Products.- 2.5 Hausdorff Axiom.- 3. Dimension.- 3.1 Dimension of a Variety.- 3.2 Dimension of a Subvariety.- 3.3 Dimension Theorem.- 3.4 Consequences.- 4. Morphisms.- 4.1 Fibres of a Morphism.- 4.2 Finite Morphisms.- 4.3 Image of a Morphism.- 4.4 Constructible Sets.- 4.5 Open Morphisms.- 4.6 Bijective Morphisms.- 4.7 Birational Morphisms.- 5. Tangent Spaces.- 5.1 Zariski Tangent Space.- 5.2 Existence of Simple Points.- 5.3 Local Ring of a Simple Point.- 5.4 Differential of a Morphism.- 5.5 Differential Criterion for Separability.- 6. Complete Varieties.- 6.1 Basic Properties.- 6.2 Completeness of Projective Varieties.- 6.3 Varieties Isomorphic to P1.- 6.4 Automorphisms of P1.- II. Affine Algebraic Groups.- 7. Basic Concepts and Examples.- 7.1 The Notion of Algebraic Group.- 7.2 Some Classical Groups.- 7.3 Identity Component.- 7.4 Subgroups and Homomorphisms.- 7.5 Generation by Irreducible Subsets.- 7.6 Hopf Algebras.- 8. Actions of Algebraic Groups on Varieties.- 8.1 Group Actions.- 8.2 Actions of Algebraic Groups.- 8.3 Closed Orbits.- 8.4 Semidirect Products.- 8.5 Translation of Functions.- 8.6 Linearization of Affine Groups.- III. Lie Algebras.- 9. Lie Algebra of an Algebraic Group.- 9.1 Lie Algebras and Tangent Spaces.- 9.2 Convolution.- 9.3 Examples.- 9.4 Subgroups and Lie Subalgebras.- 9.5 Dual Numbers.- 10. Differentiation.- 10.1 Some Elementary Formulas.- 10.2 Differential of Right Translation.- 10.3 The Adjoint Representation.- 10.4 Differential of Ad.- 10.5 Commutators.- 10.6 Centralizers.- 10.7 Automorphisms and Derivations.- IV. Homogeneous Spaces.- 11. Construction of Certain Representations.- 11.1 Action on Exterior Powers.- 11.2 A Theorem of Chevalley.- 11.3 Passage to Projective Space.- 11.4 Characters and Semi-Invariants.- 11.5 Normal Subgroups.- 12. Quotients.- 12.1 Universal Mapping Property.- 12.2 Topology of Y.- 12.3 Functions on Y.- 12.4 Complements.- 12.5 Characteristic 0.- V. Characteristic 0 Theory.- 13. Correspondence between Groups and Lie Algebras.- 13.1 The Lattice Correspondence.- 13.2 Invariants and Invariant Subspaces.- 13.3 Normal Subgroups and Ideals.- 13.4 Centers and Centralizers.- 13.5 Semisimple Groups and Lie Algebras.- 14. Semisimple Groups.- 14.1 The Adjoint Representation.- 14.2 Subgroups of a Semisimple Group.- 14.3 Complete Reducibility of Representations.- VI. Semisimple and Unipotent Elements.- 15. Jordan-Chevalley Decomposition.- 15.1 Decomposition of a Single Endomorphism.- 15.2 GL(n, K) and gl(n, K).- 15.3 Jordan Decomposition in Algebraic Groups.- 15.4 Commuting Sets of Endomorphisms.- 15.5 Structure of Commutative Algebraic Groups.- 16. Diagonalizable Groups.- 16.1 Characters and d-Groups.- 16.2 Tori.- 16.3 Rigidity of Diagonalizable Groups.- 16.4 Weights and Roots.- VII. Solvable Groups.- 17. Nilpotent and Solvable Groups.- 17.1 A Group-Theoretic Lemma.- 17.2 Commutator Groups.- 17.3 Solvable Groups.- 17.4 Nilpotent Groups.- 17.5 Unipotent Groups.- 17.6 Lie-Kolchin Theorem.- 18. Semisimple Elements.- 18.1 Global and Infinitesimal Centralizers.- 18.2 Closed Conjugacy Classes.- 18.3 Action of a Semisimple Element on a Unipotent Group.- 18.4 Action of a Diagonalizable Group.- 19. Connected Solvable Groups.- 19.1 An Exact Sequence.- 19.2 The Nilpotent Case.- 19.3 The General Case.- 19.4 Normalizer and Centralizer.- 19.5 Solvable and Unipotent Radicals.- 20. One Dimensional Groups.- 20.1 Commutativity of G.- 20.2 Vector Groups and e-Groups.- 20.3 Properties of p-Polynomials.- 20.4 Automorphisms of Vector Groups.- 20.5 The Main Theorem.- VIII. Borel Subgroups.- 21. Fixed Point and Conjugacy Theorems.- 21.1 Review of Complete Varieties.- 21.2 Fixed Point Theorem.- 21.3 Conjugacy of Borel Subgroups and Maximal Tori.- 21.4 Further Consequences.- 22. Density and Connectedness Theorems.- 22.1 The Main Lemma.- 22.2 Density Theorem.- 22.3 Connectedness Theorem.- 22.4 Borel Subgroups of CG(S).- 22.5 Cartan Subgroups: Summary.- 23. Normalizer Theorem.- 23.1 Statement of the Theorem.- 23.2 Proof of the Theorem.- 23.3 The variety G/B.- 23.4 Summary.- IX. Centralizers of Tori.- 24. Regular and Singular Tori.- 24.1 Weyl Groups.- 24.2 Regular Tori.- 24.3 Singular Tori and Roots.- 24.4 Regular 1-Parameter Subgroups.- 25. Action of a Maximal Torus on G/?.- 25.1 Action of a 1-Parameter Subgroup.- 25.2 Existence of Enough Fixed Points.- 25.3 Groups of Semisimple Rank 1.- 25.4 Weyl Chambers.- 26. The Unipotent Radical.- 26.1 Characterization of Ru(G).- 26.2 Some Consequences.- 26.3 The Groups U?.- X. Structure of Reductive Groups.- 27. The Root System.- 27.1 Abstract Root Systems.- 27.2 The Integrality Axiom.- 27.3 Simple Roots.- 27.4 The Automorphism Group of a Semisimple Group.- 27.5 Simple Components.- 28. Bruhat Decomposition.- 28.1 T-Stable Subgroups of Bu.- 28.2 Groups of Semisimple Rank 1.- 28.3 The Bruhat Decomposition.- 28.4 Normal Form in G.- 28.5 Complements.- 29. Tits Systems.- 29.1 Axioms.- 29.2 Bruhat Decomposition.- 29.3 Parabolic Subgroups.- 29.4 Generators and Relations for W.- 29.5 Normal Subgroups of G.- 30. Parabolic Subgroups.- 30.1 Standard Parabolic Subgroups.- 30.2 Levi Decompositions.- 30.3 Parabolic Subgroups Associated to Certain Unipotent Groups.- 30.4 Maximal Subgroups and Maximal Unipotent Subgroups.- XI. Representations and Classification of Semisimple Groups.- 31. Representations.- 31.1 Weights.- 31.2 Maximal Vectors.- 31.3 Irreducible Representations.- 31.4 Construction of Irreducible Representations.- 31.5 Multiplicities and Minimal Highest Weights.- 31.6 Contragredients and Invariant Bilinear Forms.- 32. Isomorphism Theorem.- 32.1 The Classification Problem.- 32.2 Extension of ?T to N(T).- 32.3 Extension of ?T to Z?.- 32.4 Extension of ?T to TU?.- 32.5 Extension of ?T to ?.- 32.6 Multiplicativity of ?.- 33. Root Systems of Rank 2.- 33.1 Reformulation of (?), (?), (?).- 33.2 Some Preliminaries.- 33.3 Type A2.- 33.4 Type B2.- 33.5 Type G2.- 33.6 The Existence Problem.- XII. Survey of Rationality Properties.- 34. Fields of Definition.- 34.1 Foundations.- 34.2 Review of Earlier Chapters.- 34.3 Tori.- 34.4 Some Basic Theorems.- 34.5 Borel-Tits Structure Theory.- 34.6 An Example: Orthogonal Groups.- 35. Special Cases.- 35.1 Split and Quasisplit Groups.- 35.2 Finite Fields.- 35.3 The Real Field.- 35.4 Local Fields.- 35.5 Classification.- Appendix. Root Systems.- Index of Terminology.- Index of Symbols.
