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对合之书（影印版）

样章
作者：
Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pier
定价：
269.00元
ISBN：
978-7-04-053493-1
版面字数：
998.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2020-04-22
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数学
暂无
  • 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

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