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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
版面字数：
998千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2020年
ISBN：
978-7-04-053493-1
物料号：
53493-00
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
  • 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

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