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Preface Introduction Conventions and Notations Chapter I. Involutions and Hermitian Forms 1. Central Simple Algebras 1.A. Fundamental theorems 1.B. One-sided ideals in central simple algebras 1.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. Adjointinvolutions 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 11.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. B1 = C1 15.B. A2 1 = D2 15.C. B2 = C2 15.D. A3 = D3 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. SK1 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 Cohomology 28. Cohomology of Profinite Groups 28.A. Cohomology sets 28.B. Cohomology sequences 28.C. Twisting 28.D. Torsors 29. Galois Cohomology 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 31.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 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 2D4 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