目录
 前辅文
 Preface
  Introduction
  Conventions and Notations
 Chapter I. Involutions and Hermitian Forms
  § 1. Central Simple Algebras
   l.A. Fundamental theorems
   l.B. One-sided ideals in central simple algebras
   l.C. Severi-Brauer varieties
  §2. Involutions
   2.A. Involutions of the first kind
   2.B. Involutions of the second kind
   2.C. Examples
   2.D. Lie and Jordan structures
  §3. Existence of Involutions
   3.A. Existence of involutions of the first kind
   3.B. Existence of involutions of the second kind
  §4. Hermitian Forms
   4.A. Adjoint involutions
   4.B. Extension of involutions and transfer
  §5. Quadratic Forms
   5.A. Standard identifications
   5.B. Quadratic pairs
  Exercises
  Notes
 Chapter II. Invariants of Involutions
  §6. The Index
   6.A. Isotropic ideals
   6.B. Hyperbolic involutions
   6.C. Odd-degree extensions
  §7. The Discriminant
   7. A. The discriminant of orthogonal involutions
   7.B. The discriminant of quadratic pairs
  §8. The Clifford Algebra
   8.A. The split case
   8.B. Definition of the Clifford algebra
   8.C. Lie algebra structures
   8.D. The center of the Clifford algebra
   8.E. The Clifford algebra of a hyperbolic quadratic pair
  §9. The Clifford Bimodule
   9.A. The split case
   9.B. Definition of the Clifford bimodule
   9.C. The fundamental relations
  §10. The Discriminant Algebra
   10.A. The A-powers of a central simple algebra
   10.B. The canonical involution
   10.C. The canonical quadratic pair
   10.D. Induced involutions on A-powers
   10.E. Definition of the discriminant algebra
   10.F. The Brauer class of the discriminant algebra
  §11. Trace Form Invariants
   11.A. Involutions of the first kind
   l l . B . Involutions of the second kind
  Exercises
  Notes
 Chapter III. Similitudes
  §12. General Properties
   12.A. The split case
   12.B. Similitudes of algebras with involution
   12.C. Proper similitudes
   12.D. Functorial properties
  §13. Quadratic Pairs
   13.A. Relation with the Clifford structures
   13.B. Clifford groups
   13.C. Multipliers of similitudes
  § 14. Unitary Involutions
   14.A. Odd degree
   14.B. Even degree
   14.C. Relation with the discriminant algebra
  Exercises
  Notes
 Chapter IV. Algebras of Degree Four
  §15. Exceptional Isomorphisms
   15.A. Bi = d
   15.B. A\ = D2
   15.C. B2 = C2
   15.D. A3 = Ds
  §16. Biquaternion Algebras
   16.A. Albert forms
   16.B. Albert forms and symplectic involutions
   16.C. Albert forms and orthogonal involutions
  §17. Whitehead Groups
   17.A. SKi of biquaternion algebras
   17.B. Algebras with involution
  Exercises
  Notes
 Chapter V. Algebras of Degree Three
  §18. Etale and Galois Algebras
   18.A. Etale algebras
   18.B. Galois algebras
   18.C. Cubic etale algebras
  §19. Central Simple Algebras of Degree Three
   19.A. Cyclic algebras
   19.B. Classification of involutions of the second kind
   19.C. Etale subalgebras
  Exercises
  Notes
 Chapter VI. Algebraic Groups
  §20. Hopf Algebras and Group Schemes
   20.A. Group schemes
  §21. The Lie Algebra and Smoothness
   21.A. The Lie algebra of a group scheme
  §22. Factor Groups
   22.A. Group scheme homomorphisms
  §23. Automorphism Groups of Algebras
   23.A. Involutions
   23.B. Quadratic pairs
  §24. Root Systems
   24.A. Classification of irreducible root systems
  §25. Split Semisimple Groups
   25.A. Simple split groups of type A, B, C, D, F , and G
   25.B. Automorphisms of split semisimple groups
  §26. Semisimple Groups over an Arbitrary Field
   26.A. Basic classification results
   26.B. Algebraic groups of small dimension
  § 27. Tits Algebras of Semisimple Groups
   27.A. Definition of the Tits algebras
   27.B. Simply connected classical groups
   27.C. Quasisplit groups
  Exercises
  Notes
 Chapter VII. Galois Cohomoiogy
  §28. Cohomoiogy of Profinite Groups
   28.A. Cohomoiogy sets
   28.B. Cohomoiogy sequences
   28.C. Twisting
   28.D. Torsors
  §29. Galois Cohomoiogy of Algebraic Groups
   29.A. Hilbert's Theorem 90 and Shapiro's lemma
   29.B. Classification of algebras
   29.C. Algebras with a distinguished subalgebra
   29.D. Algebras with involution
   29.E. Quadratic spaces
   29.F. Quadratic pairs
  §30. Galois Cohomology of Roots of Unity
   30.A. Cyclic algebras
   30.B. Twisted coefficients
   30.C. Cohomological invariants of algebras of degree three . . . .
  §31. Cohomological Invariants
   31.A. Connecting homomorphisms
   3l.B. Cohomological invariants of algebraic groups
  Exercises
  Notes
 Chapter VIII. Composition and Triality
  §32. Nonassociative Algebras
  §33. Composition Algebras
   33.A. Multiplicative quadratic forms
   33.B. Unital composition algebras
   33.C. Hurwitz algebras
   33.D. Composition algebras without identity
  §34. Symmetric Compositions
   34.A. Para-Hurwitz algebras
   34.B. Petersson algebras
   34.C. Cubic separable alternative algebras
   34.D. Alternative algebras with unitary involutions
   34.E. Cohomological invariants of symmetric compositions . . . .
  §35. Clifford Algebras and Triality
   35.A. The Clifford algebra
   35.B. Similitudes and triality
   35.C. The group Spin and triality
  §36. Twisted Compositions
   36.A. Multipliers of similitudes of twisted compositions
   36.B. Cyclic compositions
   36.C. Twisted Hurwitz compositions
   36.D. Twisted compositions of type A'2
   36.E. The dimension 2 case
  Exercises
  Notes
 Chapter IX. Cubic Jordan Algebras
  §37. Jordan Algebras
   37.A. Jordan algebras of quadratic forms
   37.B. Jordan algebras of classical type
   37.C. Freudenthal algebras
  §38. Cubic Jordan Algebras
   38.A. The Springer decomposition
  §39. The Tits Construction
   39.A. Symmetric compositions and Tits constructions
   39.B. Automorphisms of Tits constructions
  §40. Cohomological Invariants
   40.A. Invariants of twisted compositions
  §41. Exceptional Simple Lie Algebras
  Exercises
  Notes
 Chapter X. Trialitarian Central Simple Algebras
  §42. Algebras of Degree 8
   42.A. Trialitarian triples
   42.B. Decomposable involutions
  §43. Trialitarian Algebras
   43.A. A definition and some properties
   43.B. Quaternionic trialitarian algebras
   43.C. Trialitarian algebras of type 2D^
  §44. Classification of Algebras and Groups of Type D4
   44.A. Groups of trialitarian type D4
   44.B. The Clifford invariant
  §45. Lie Algebras and Triality
   45.A. Local triality
   45.B. Derivations of twisted compositions
   45.C. Lie algebras and trialitarian algebras
  Exercise
  Notes
 Bibliography
 Index
 Notation
 前辅文
 Preface
  Introduction
  Conventions and Notations
 Chapter I. Involutions and Hermitian Forms
  § 1. Central Simple Algebras
   l.A. Fundamental theorems
   l.B. One-sided ideals in central simple algebras
   l.C. Severi-Brauer varieties
  §2. Involutions
   2.A. Involutions of the first kind
   2.B. Involutions of the second kind
   2.C. Examples
   2.D. Lie and Jordan structures
  §3. Existence of Involutions
   3.A. Existence of involutions of the first kind
   3.B. Existence of involutions of the second kind
  §4. Hermitian Forms
   4.A. Adjoint involutions
   4.B. Extension of involutions and transfer
  §5. Quadratic Forms
   5.A. Standard identifications
   5.B. Quadratic pairs
  Exercises
  Notes
 Chapter II. Invariants of Involutions
  §6. The Index
   6.A. Isotropic ideals
   6.B. Hyperbolic involutions
   6.C. Odd-degree extensions
  §7. The Discriminant
   7. A. The discriminant of orthogonal involutions
   7.B. The discriminant of quadratic pairs
  §8. The Clifford Algebra
   8.A. The split case
   8.B. Definition of the Clifford algebra
   8.C. Lie algebra structures
   8.D. The center of the Clifford algebra
   8.E. The Clifford algebra of a hyperbolic quadratic pair
  §9. The Clifford Bimodule
   9.A. The split case
   9.B. Definition of the Clifford bimodule
   9.C. The fundamental relations
  §10. The Discriminant Algebra
   10.A. The A-powers of a central simple algebra
   10.B. The canonical involution
   10.C. The canonical quadratic pair
   10.D. Induced involutions on A-powers
   10.E. Definition of the discriminant algebra
   10.F. The Brauer class of the discriminant algebra
  §11. Trace Form Invariants
   11.A. Involutions of the first kind
   l l . B . Involutions of the second kind
  Exercises
  Notes
 Chapter III. Similitudes
  §12. General Properties
   12.A. The split case
   12.B. Similitudes of algebras with involution
   12.C. Proper similitudes
   12.D. Functorial properties
  §13. Quadratic Pairs
   13.A. Relation with the Clifford structures
   13.B. Clifford groups
   13.C. Multipliers of similitudes
  § 14. Unitary Involutions
   14.A. Odd degree
   14.B. Even degree
   14.C. Relation with the discriminant algebra
  Exercises
  Notes
 Chapter IV. Algebras of Degree Four
  §15. Exceptional Isomorphisms
   15.A. Bi = d
   15.B. A\ = D2
   15.C. B2 = C2
   15.D. A3 = Ds
  §16. Biquaternion Algebras
   16.A. Albert forms
   16.B. Albert forms and symplectic involutions
   16.C. Albert forms and orthogonal involutions
  §17. Whitehead Groups
   17.A. SKi of biquaternion algebras
   17.B. Algebras with involution
  Exercises
  Notes
 Chapter V. Algebras of Degree Three
  §18. Etale and Galois Algebras
   18.A. Etale algebras
   18.B. Galois algebras
   18.C. Cubic etale algebras
  §19. Central Simple Algebras of Degree Three
   19.A. Cyclic algebras
   19.B. Classification of involutions of the second kind
   19.C. Etale subalgebras
  Exercises
  Notes
 Chapter VI. Algebraic Groups
  §20. Hopf Algebras and Group Schemes
   20.A. Group schemes
  §21. The Lie Algebra and Smoothness
   21.A. The Lie algebra of a group scheme
  §22. Factor Groups
   22.A. Group scheme homomorphisms
  §23. Automorphism Groups of Algebras
   23.A. Involutions
   23.B. Quadratic pairs
  §24. Root Systems
   24.A. Classification of irreducible root systems
  §25. Split Semisimple Groups
   25.A. Simple split groups of type A, B, C, D, F , and G
   25.B. Automorphisms of split semisimple groups
  §26. Semisimple Groups over an Arbitrary Field
   26.A. Basic classification results
   26.B. Algebraic groups of small dimension
  § 27. Tits Algebras of Semisimple Groups
   27.A. Definition of the Tits algebras
   27.B. Simply connected classical groups
   27.C. Quasisplit groups
  Exercises
  Notes
 Chapter VII. Galois Cohomoiogy
  §28. Cohomoiogy of Profinite Groups
   28.A. Cohomoiogy sets
   28.B. Cohomoiogy sequences
   28.C. Twisting
   28.D. Torsors
  §29. Galois Cohomoiogy of Algebraic Groups
   29.A. Hilbert's Theorem 90 and Shapiro's lemma
   29.B. Classification of algebras
   29.C. Algebras with a distinguished subalgebra
   29.D. Algebras with involution
   29.E. Quadratic spaces
   29.F. Quadratic pairs
  §30. Galois Cohomology of Roots of Unity
   30.A. Cyclic algebras
   30.B. Twisted coefficients
   30.C. Cohomological invariants of algebras of degree three . . . .
  §31. Cohomological Invariants
   31.A. Connecting homomorphisms
   3l.B. Cohomological invariants of algebraic groups
  Exercises
  Notes
 Chapter VIII. Composition and Triality
  §32. Nonassociative Algebras
  §33. Composition Algebras
   33.A. Multiplicative quadratic forms
   33.B. Unital composition algebras
   33.C. Hurwitz algebras
   33.D. Composition algebras without identity
  §34. Symmetric Compositions
   34.A. Para-Hurwitz algebras
   34.B. Petersson algebras
   34.C. Cubic separable alternative algebras
   34.D. Alternative algebras with unitary involutions
   34.E. Cohomological invariants of symmetric compositions . . . .
  §35. Clifford Algebras and Triality
   35.A. The Clifford algebra
   35.B. Similitudes and triality
   35.C. The group Spin and triality
  §36. Twisted Compositions
   36.A. Multipliers of similitudes of twisted compositions
   36.B. Cyclic compositions
   36.C. Twisted Hurwitz compositions
   36.D. Twisted compositions of type A'2
   36.E. The dimension 2 case
  Exercises
  Notes
 Chapter IX. Cubic Jordan Algebras
  §37. Jordan Algebras
   37.A. Jordan algebras of quadratic forms
   37.B. Jordan algebras of classical type
   37.C. Freudenthal algebras
  §38. Cubic Jordan Algebras
   38.A. The Springer decomposition
  §39. The Tits Construction
   39.A. Symmetric compositions and Tits constructions
   39.B. Automorphisms of Tits constructions
  §40. Cohomological Invariants
   40.A. Invariants of twisted compositions
  §41. Exceptional Simple Lie Algebras
  Exercises
  Notes
 Chapter X. Trialitarian Central Simple Algebras
  §42. Algebras of Degree 8
   42.A. Trialitarian triples
   42.B. Decomposable involutions
  §43. Trialitarian Algebras
   43.A. A definition and some properties
   43.B. Quaternionic trialitarian algebras
   43.C. Trialitarian algebras of type 2D^
  §44. Classification of Algebras and Groups of Type D4
   44.A. Groups of trialitarian type D4
   44.B. The Clifford invariant
  §45. Lie Algebras and Triality
   45.A. Local triality
   45.B. Derivations of twisted compositions
   45.C. Lie algebras and trialitarian algebras
  Exercise
  Notes
 Bibliography
 Index
 Notation