The Book of Involutions

Max-Albert Knus

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The Book of Involutions

Max-Albert Knus

Alexander Merkurjev

Markus Rost

Jean-Pierre Tignol

Author address:

Dept. Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland

E-mail address: knus@math.ethz.ch
URL: http://www.math.ethz.ch/~knus/

Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA

E-mail address: merkurev@math.ucla.edu
URL: http://www.math.ucla.edu/~merkurev/

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516/

Département de mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@agel.ucl.ac.be
URL: http://www.math.ucl.ac.be/tignol/

Contents

Préface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . xv

Chapter I. Involutions and Hermitian Forms . . . . . . . . . . . . . 1

§1. Central Simple Algebras . . . . . . . . . . . . . . . . . . . 3 1.A. Fundamental theorems . . . . . . . . . . . . . . . . . 3 1.B. One-sided ideals in central simple algebras . . . . . . . . . 5 1.C. Severi-Brauer varieties . . . . . . . . . . . . . . . . . 9

§2. Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.A. Involutions of the first kind . . . . . . . . . . . . . . . 13 2.B. Involutions of the second kind . . . . . . . . . . . . . . 20 2.C. Examples . . . . . . . . . . . . . . . . . . . . . . . 23 2.D. Lie and Jordan structures . . . . . . . . . . . . . . . . 27

§3. Existence of Involutions . . . . . . . . . . . . . . . . . . . 31 3.A. Existence of involutions of the first kind . . . . . . . . . . 32 3.B. Existence of involutions of the second kind . . . . . . . . 36

§4. Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . 41 4.A. Adjoint involutions . . . . . . . . . . . . . . . . . . . 42 4.B. Extension of involutions and transfer . . . . . . . . . . . 45

§5. Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . 53 5.A. Standard identifications . . . . . . . . . . . . . . . . . 53 5.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 55

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter II. Invariants of Involutions . . . . . . . . . . . . . . . . . 71

§6. The Index . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.A. Isotropic ideals . . . . . . . . . . . . . . . . . . . . . 72 6.B. Hyperbolic involutions . . . . . . . . . . . . . . . . . 74 6.C. Odd-degree extensions . . . . . . . . . . . . . . . . . 79

§7. The Discriminant . . . . . . . . . . . . . . . . . . . . . . 80 7.A. The discriminant of orthogonal involutions . . . . . . . . 80 7.B. The discriminant of quadratic pairs . . . . . . . . . . . . 83

§8. The Clifford Algebra . . . . . . . . . . . . . . . . . . . . . 87 8.A. The split case . . . . . . . . . . . . . . . . . . . . . 87 8.B. Definition of the Clifford algebra . . . . . . . . . . . . . 91 8.C. Lie algebra structures . . . . . . . . . . . . . . . . . . 95 8.D. The center of the Clifford algebra . . . . . . . . . . . . 99 8.E. The Clifford algebra of a hyperbolic quadratic pair . . . . . 106

§9. The Clifford Bimodule . . . . . . . . . . . . . . . . . . . . 107 9.A. The split case . . . . . . . . . . . . . . . . . . . . . 107 9.B. Definition of the Clifford bimodule . . . . . . . . . . . . 108 9.C. The fundamental relations . . . . . . . . . . . . . . . . 113

§10. The Discriminant Algebra . . . . . . . . . . . . . . . . . . 114 10.A. The λ-powers of a central simple algebra . . . . . . . . . 115 10.B. The canonical involution . . . . . . . . . . . . . . . . 117 10.C. The canonical quadratic pair . . . . . . . . . . . . . . . 119 10.D. Induced involutions on λ-powers . . . . . . . . . . . . . 123 10.E. Definition of the discriminant algebra . . . . . . . . . . . 127 10.F. The Brauer class of the discriminant algebra . . . . . . . . 130

§11. Trace Form Invariants . . . . . . . . . . . . . . . . . . . . 132 11.A. Involutions of the first kind . . . . . . . . . . . . . . . 133 11.B. Involutions of the second kind . . . . . . . . . . . . . . 138

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter III. Similitudes . . . . . . . . . . . . . . . . . . . . . . 153

§12. General Properties . . . . . . . . . . . . . . . . . . . . . . 153 12.A. The split case . . . . . . . . . . . . . . . . . . . . . 153 12.B. Similitudes of algebras with involution . . . . . . . . . . 158 12.C. Proper similitudes . . . . . . . . . . . . . . . . . . . 163 12.D. Functorial properties . . . . . . . . . . . . . . . . . . 168

§13. Quadratic Pairs . . . . . . . . . . . . . . . . . . . . . . . 172 13.A. Relation with the Clifford structures . . . . . . . . . . . 172 13.B. Clifford groups . . . . . . . . . . . . . . . . . . . . . 177 13.C. Multipliers of similitudes . . . . . . . . . . . . . . . . 190

§14. Unitary Involutions . . . . . . . . . . . . . . . . . . . . . 193 14.A. Odd degree . . . . . . . . . . . . . . . . . . . . . . 193 14.B. Even degree . . . . . . . . . . . . . . . . . . . . . . 194 14.C. Relation with the discriminant algebra . . . . . . . . . . 195

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter IV. Algebras of Degree Four . . . . . . . . . . . . . . . . 205

§15. Exceptional Isomorphisms . . . . . . . . . . . . . . . . . . 205 15.A. B_{1} ≡ C_{1} . . . . . . . . . . . . . . . . . . . . . . . . 207 15.B. A_{1}^{2} ≡ D_{2} . . . . . . . . . . . . . . . . . . . . . . . . 210 15.C. B_{2} ≡ C_{2} . . . . . . . . . . . . . . . . . . . . . . . . 216 15.D. A_{3} ≡ D_{3} . . . . . . . . . . . . . . . . . . . . . . . . 220

§16. Biquaternion Algebras . . . . . . . . . . . . . . . . . . . . 233 16.A. Albert forms . . . . . . . . . . . . . . . . . . . . . . 235 16.B. Albert forms and symplectic involutions . . . . . . . . . . 237 16.C. Albert forms and orthogonal involutions . . . . . . . . . . 245

§17. Whitehead Groups . . . . . . . . . . . . . . . . . . . . . . 253 17.A. SK_{1} of biquaternion algebras . . . . . . . . . . . . . . . 253 17.B. Algebras with involution . . . . . . . . . . . . . . . . 266 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Chapter V. Algebras of Degree Three . . . . . . . . . . . . . . . . 279

§18. Étale and Galois Algebras . . . . . . . . . . . . . . . . . . 279 18.A. Étale algebras . . . . . . . . . . . . . . . . . . . . . 280 18.B. Galois algebras . . . . . . . . . . . . . . . . . . . . . 287 18.C. Cubic étale algebras . . . . . . . . . . . . . . . . . . 296

§19. Central Simple Algebras of Degree Three . . . . . . . . . . . . 302 19.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 302 19.B. Classification of involutions of the second kind . . . . . . . 304 19.C. Étale subalgebras . . . . . . . . . . . . . . . . . . . . 307

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Chapter VI. Algebraic Groups . . . . . . . . . . . . . . . . . . . 323

§20. Hopf Algebras and Group Schemes . . . . . . . . . . . . . . 324

20.A. Group schemes . . . . . . . . . . . . . . . . . . . . . 325 §21. The Lie Algebra and Smoothness . . . . . . . . . . . . . . . 334

21.A. The Lie algebra of a group scheme . . . . . . . . . . . . 334 §22. Factor Groups . . . . . . . . . . . . . . . . . . . . . . . 339

22.A. Group scheme homomorphisms . . . . . . . . . . . . . . 339 §23. Automorphism Groups of Algebras . . . . . . . . . . . . . . 344

23.A. Involutions . . . . . . . . . . . . . . . . . . . . . . 345 23.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 350

§24. Root Systems . . . . . . . . . . . . . . . . . . . . . . . . 352

24.A. Classification of irreducible root systems . . . . . . . . . 354 §25. Split Semisimple Groups . . . . . . . . . . . . . . . . . . . 355

25.A. Simple split groups of type A, B, C, D, F, and G . . . . . 357 25.B. Automorphisms of split semisimple groups . . . . . . . . . 360

§26. Semisimple Groups over an Arbitrary Field . . . . . . . . . . . 360 26.A. Basic classification results . . . . . . . . . . . . . . . . 364 26.B. Algebraic groups of small dimension . . . . . . . . . . . 373

§27. Tits Algebras of Semisimple Groups . . . . . . . . . . . . . . 375 27.A. Definition of the Tits algebras . . . . . . . . . . . . . . 376 27.B. Simply connected classical groups . . . . . . . . . . . . 378 27.C. Quasisplit groups . . . . . . . . . . . . . . . . . . . . 379

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Chapter VII. Galois Cohomology . . . . . . . . . . . . . . . . . . 383

§28. Cohomology of Profinite Groups . . . . . . . . . . . . . . . . 383 28.A. Cohomology sets . . . . . . . . . . . . . . . . . . . . 383 28.B. Cohomology sequences . . . . . . . . . . . . . . . . . 385 28.C. Twisting . . . . . . . . . . . . . . . . . . . . . . . 387 28.D. Torsors . . . . . . . . . . . . . . . . . . . . . . . . 388

§29. Galois Cohomology of Algebraic Groups . . . . . . . . . . . . 391 29.A. Hilbert’s Theorem 90 and Shapiro’s lemma . . . . . . . . 392 29.B. Classification of algebras . . . . . . . . . . . . . . . . 395 29.C. Algebras with a distinguished subalgebra . . . . . . . . . 398 29.D. Algebras with involution . . . . . . . . . . . . . . . . 399 29.E. Quadratic spaces . . . . . . . . . . . . . . . . . . . . 406 29.F. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 408

§30. Galois Cohomology of Roots of Unity . . . . . . . . . . . . . 413 30.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 414 30.B. Twisted coefficients . . . . . . . . . . . . . . . . . . . 416 30.C. Cohomological invariants of algebras of degree three …. 420

§31. Cohomological Invariants . . . . . . . . . . . . . . . . . . . 423 31.A. Connecting homomorphisms . . . . . . . . . . . . . . . 423 31.B. Cohomological invariants of algebraic groups . . . . . . . . 429

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

Chapter VIII. Composition and Triality . . . . . . . . . . . . . . . 451 §32. Nonassociative Algebras . . . . . . . . . . . . . . . . . . . 451 §33. Composition Algebras . . . . . . . . . . . . . . . . . . . . 454

33.A. Multiplicative quadratic forms . . . . . . . . . . . . . . 454 33.B. Unital composition algebras . . . . . . . . . . . . . . . 456 33.C. Hurwitz algebras . . . . . . . . . . . . . . . . . . . . 458 33.D. Composition algebras without identity . . . . . . . . . . 462

§34. Symmetric Compositions . . . . . . . . . . . . . . . . . . . 463 34.A. Para-Hurwitz algebras . . . . . . . . . . . . . . . . . 464 34.B. Petersson algebras . . . . . . . . . . . . . . . . . . . 466 34.C. Cubic separable alternative algebras . . . . . . . . . . . 469 34.D. Alternative algebras with unitary involutions . . . . . . . 477 34.E. Cohomological invariants of symmetric compositions …. 480

§35. Clifford Algebras and Triality . . . . . . . . . . . . . . . . . 481 35.A. The Clifford algebra . . . . . . . . . . . . . . . . . . 481 35.B. Similitudes and triality . . . . . . . . . . . . . . . . . 483 35.C. The group Spin and triality . . . . . . . . . . . . . . . 485

§36. Twisted Compositions . . . . . . . . . . . . . . . . . . . . 489 36.A. Multipliers of similitudes of twisted compositions . . . . . . 493 36.B. Cyclic compositions . . . . . . . . . . . . . . . . . . . 495 36.C. Twisted Hurwitz compositions . . . . . . . . . . . . . . 499 36.D. Twisted compositions of type A′_{2} . . . . . . . . . . . . . 504 36.E. The dimension 2 case . . . . . . . . . . . . . . . . . . 506

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Chapter IX. Cubic Jordan Algebras . . . . . . . . . . . . . . . . . 513

§37. Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . . 513 37.A. Jordan algebras of quadratic forms . . . . . . . . . . . . 514 37.B. Jordan algebras of classical type . . . . . . . . . . . . . 515 37.C. Freudenthal algebras . . . . . . . . . . . . . . . . . . 516

§38. Cubic Jordan Algebras . . . . . . . . . . . . . . . . . . . . 519

38.A. The Springer decomposition . . . . . . . . . . . . . . . 522 §39. The Tits Construction . . . . . . . . . . . . . . . . . . . . 524

39.A. Symmetric compositions and Tits constructions . . . . . . 528

39.B. Automorphisms of Tits constructions . . . . . . . . . . . 529 §40. Cohomological Invariants . . . . . . . . . . . . . . . . . . . 533

40.A. Invariants of twisted compositions . . . . . . . . . . . . 539 §41. Exceptional Simple Lie Algebras . . . . . . . . . . . . . . . . 539 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Chapter X. Trialitarian Central Simple Algebras . . . . . . . . . . . 547

§42. Algebras of Degree 8 . . . . . . . . . . . . . . . . . . . . . 547 42.A. Trialitarian triples . . . . . . . . . . . . . . . . . . . 547 42.B. Decomposable involutions . . . . . . . . . . . . . . . . 550

§43. Trialitarian Algebras . . . . . . . . . . . . . . . . . . . . . 552 43.A. A definition and some properties . . . . . . . . . . . . . 552 43.B. Quaternionic trialitarian algebras . . . . . . . . . . . . . 555 43.C. Trialitarian algebras of type ^{2}D_{4} . . . . . . . . . . . . . 558

§44. Classification of Algebras and Groups of Type D_{4} . . . . . . . . 560 44.A. Groups of trialitarian type D_{4} . . . . . . . . . . . . . . 561 44.B. The Clifford invariant . . . . . . . . . . . . . . . . . . 563

§45. Lie Algebras and Triality . . . . . . . . . . . . . . . . . . . 565 45.A. Local triality . . . . . . . . . . . . . . . . . . . . . 567 45.B. Derivations of twisted compositions . . . . . . . . . . . . 569 45.C. Lie algebras and trialitarian algebras . . . . . . . . . . . 569

Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

xx CONTENTS

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Index

absolute Galois group, 279 absolutely simple group, 364 additive group, 326 adjoint anti-automorphism, 1 adjoint group, 364 adjoint involution, 2, 24, 42 adjoint representation, 343 adjoint semisimple group, 356 admissible pair, 525 affine group scheme, 325 Albert algebra, 517 Albert form, 235 Albert quadratic space, 235 Albert’s theorem, 236, 308 algebraic group, 338 algebraic group scheme, 326 algebraic torus, 333 Allen invariant, 570 alternating bilinear form, xvi alternating hermitian form, 42 alternative algebra, 451 annihilator of an ideal, 8 Arason invariant, 436 Arf invariant, xix Artin’s theorem, 454 Artin-Schreier sequence, 414 associative bilinear form, 453 associator, 454 augmentation Hopf ideal, 327 automorphism of a quadratic pair, 159

base of a root system, 353 bilinear trace form, 132, 454 biquaternion algebra, 233 Brauer group, 4 Brauer-equivalent algebras, 4

canonical involution, 89, 94, 119 canonical quadratic pair on λ^{m}A, 123 canonical quadratic pair on tensor products,

63 Cartan subspace, 146 Cartan-Dieudonné theorem, 178 Cartier duality, 333 Cayley algebra, 459 Cayley parametrization, 201 Cayley-Dickson algebra, 458 Cayley-Dickson process, 459 central algebra, 451 central isogeny, 342 central simple L-algebra, 552 central simple algebra, 3 centralizer, 5 centroid, 451 character, 327 characteristic polynomial, xv Clifford algebra, 87, 92 Clifford bimodule, 110 Clifford group of a quadratic pair, 178, 351 Clifford invariant, 481, 563 closed embedding, 327 closed subgroup, 327 cohomological invariant of algebraic group,

429 cohomologous cocycles, 384 commutative group scheme, 325 comodule, 343 comorphism, 325 composition algebra, 456 composition algebra of type ^{1}A_{2}, 475 composition algebra of type G_{2}, 466 composition algebra of type A_{1}, 466 composition of quadratic spaces, 488 cone of dominant weights, 354 connected component, 331 connected group scheme, 330 connected groupoid, 360 connecting map, 385 constant group scheme, 332 continuous Γ-action, 279 coordinate algebra, 543 corestriction, 329 coroot, 353 crossed product construction, 397 cubic cyclic algebra, 302 cubic norm, 489 cubic norm structure, 520 cyclic algebra of degree n, 414 cyclic composition, 495

degree, 452 degree of a central simple algebra, 4 degree of an algebra with involution, 21 Δ-associate space, 499 derivation, 334 derivation of a twisted composition, 569 descent condition, 361 determinant of a form, xvi, 80 determinant of a hermitian form, 114 determinant of a quadratic form, xvii determinant of an element, xv determinant of an involution, 81 diagonalizable group scheme, 332 Dickson invariant, 157, 167 Dieudonné’s theorem, 453 differential, 335 dimension of an algebraic group, 337 direct sum of root systems, 353 discriminant, 291 discriminant algebra, 128 discriminant class of a quadratic pair, 409 discriminant of a form, xvi, 80 discriminant of a hermitian form, 114 discriminant of a quadratic form, xvii discriminant of an involution, 81 discriminant quadratic extension, 409 distinguished involution, 306 division algebra, 3 division cubic norm structure, 521 double centralizer theorem, 5 dual numbers, 334 dual root system, 353 Dynkin diagram, 353

equivalent cocycles, 384 equivalent quadratic forms, xix étale algebra, 281 étale group scheme, 332 even Clifford algebra, 87 even Clifford group, 349 even hermitian form, 42 exact sequence of pointed sets, 385 exceptional Jordan algebra, 513 exchange involution, 21 extended Clifford group, 181, 352

factor group scheme, 340 fiber functor, 342 fiber product, 328 field of definition, 378 finite group scheme, 333 first Tits construction, 526 first Tits process, 526 flag, 280 formally real Jordan Algebra, 542 Freudenthal “×”-product, 519 Freudenthal adjoint, 519 Freudenthal algebra, 517 Freudenthal’s “magic square”, 540

G-invariant vector, 343 Galois G-algebra, 288 Galois S_{n}-closure, 291 Galois descent, 279 Γ-embedding, 360 Γ-extension, 361 Γ-group, 383 Γ-module, 383 general linear group, 326 generalized cyclic algebra, 191 generalized pfaffian, 103 generic element, 451 generic minimal polynomial, 452 generic norm, 452 generic trace, 452 Goldman element, 32 Grassmannian, 9 group of R-points, 326 group of orthogonal similitudes, 350 group of symplectic similitudes, 347 group of unitary similitudes, 346 group scheme homomorphism, 325 group scheme of isometries, 345 group scheme of similitudes, 345 group scheme over an étale algebra, 330 group-like elements, 327 groupoid, 170

hermitian form, 23, 42 highest weight, 375 Hilbert’s Theorem 90, 392 Hopf algebra, 324 Hopf algebra homomorphism, 324 Hopf ideal, 325 Hurwitz algebra, 460 hyperbolic algebra, 75 hyperbolic form, 74 hyperbolic involution, 75 hyperbolic quadratic pair, 77 hyperbolic quadratic space, 77 hyperplane reflection, 157

idealizer, 9 idempotent, 466 image, 327 improper similitude, 157, 164 index of a central simple algebra, 4 index of an algebra with involution, 73 induced Γ-group, 390 induced torsor, 390 injective homomorphism, 339 inner automorphism, 2 Inner form of an algebraic group, 425 intersection of algebraic groups, 328 invariant bilinear form, 453 inverse image, 328 involution, 13 involution of orthogonal type, 16 involution of symplectic type, 16 involution of unitary type, 21 involution trace, 45 involutions of the first kind, 13 involutions of the second kind, 13 irreducible representation, 343 irreducible root system, 353 isogeny, 342 isometry, 153, 159 isometry of a space, 155 isomorphism of root systems, 353 isotropic algebra, 74 isotropic algebra with involution, 73 isotropic ideal, 72, 73 isotropic involution, 73

Jordan algebra, 513 Jordan algebra of a quadratic form, 514 Jordan product, 27

kernel, 328, 385 Kummer sequence, 413

left-invariant derivation, 334 Lie algebra of a group scheme, 334 Lie algebra of Jordan type, 569 Lie algebra of type D_{4}, 569 Lie invariant function, 452 Lie product, 27 local triality, 567 loop in algebraic group, 432

maximal subtorus, 356 minimal dominant weight, 354 minimal Tits algebra, 377 Moore determinant, 239 multiplication algebra, 451 multiplicative group, 326 multiplicative quadratic form, 454 multiplicative type, 332 multiplicity of a weight, 344 multiplier, 488, 490 multiplier map, 184 multiplier of a similitude, 153, 158

nondegenerate bilinear form, xvi nondegenerate quadratic space, xvii nonsingular bilinear form, xvi, 1 nonsingular cubic form, 474 nonsingular hermitian form, 23, 42 nonsingular quadratic space, xvii norm of a vector space, 37 norm of an algebra, 36, 554 normal subgroup, 327 normalizer, 327

Okubo composition algebra, 475, 480 1-cocycle, 361, 384 opposite algebra, 3 order of a finite group scheme, 333 orientation, 207 orientation of a quadratic space, 171 oriented quadratic space, 171 orthogonal group, 348, 350 orthogonal ideal, 72 orthogonal idempotents, 63 orthogonal involution, 16

para-Cayley algebra, 464 para-Hurwitz algebra, 464 para-quadratic algebra, 464 para-quaternion algebra, 464 para-unit, 466 Peirce decomposition, 516 Petersson algebra, 466 pfaffian, 20 pfaffian adjoint, 275 pfaffian characteristic polynomial, 19 pfaffian norm, 19 pfaffian trace, 19 Pfister bilinear form, xvi Pfister form of an involution, 304 Pfister quadratic form, xix pointed quadratic space, 520 polar of a cubic form, 474 polar of a quadratic form, xvii principal homogeneous set, 288, 388 product algebra, 416 product of group schemes, 328 profinite group, 383 projective general linear group, 345 projective orthogonal group, 350 projective similitude, 159 projective symplectic group, 347 projective unitary group, 346 proper isometry, 154, 164 proper similitude, 154, 157, 164, 167 pseudodiscriminant, xix

quadratic map, 489 quadratic norm, 489 quadratic pair, 55 quadratic trace, 517 quasisplit group, 379 quaternion algebra, 25 quaternion basis, 25

radical of a quadratic form, 348 reduced characteristic polynomial, 5 reduced dimension, 6 reduced Freudenthal algebra, 517, 520 reduced norm, 5 reduced root system, 352 reduced trace, 5 reduced unitary Whitehead group, 267 reduced Whitehead group, 253 reflection, 352 regular bilinear form, xvi regular form, 348 regular local ring, 337 regular quadratic space, xvii regular space, 348 representation, 342 restriction map, 385 restriction scheme, 327 root, 353, 356 root lattice, 353 root system, 356, 364 roots of unity, 328 Rost invariant, 435

sandwich map, 32 Schur index, 4 second involution, 550 second Tits construction, 525 second Tits process, 526 semilinear action, 279 semisimple algebraic group, 355 separability idempotent, 285 separable algebra, 344, 453 separable homomorphism, 341 Severi-Brauer variety, 10 Shapiro’s Lemma, 394 signature of a form, 136 signature of an involution, 137 similar spaces, 168 similitude, 488, 490 similitude of a bilinear space, 153 similitude of a quadratic pair, 159 similitude of an algebra with involution, 158 simple group, 357 simply connected group, 364 simply connected semisimple group, 356 simply transitive action, 388 skew field, 3 skew-hermitian form, 42 skew-symmetric bilinear form, xvi Skolem-Noether theorem, 5 smooth algebraic group, 338 smooth group scheme, 338 SN-condition, 344 solvable algebraic group, 355 special Clifford group, 177 special Jordan algebra, 513 special linear group, 328 special unitary group, 346 spin group, 187 spinor group, 349, 351 spinor norm, 351 spinor norms, 187 split algebra, 4 split Cayley algebra, 462 split Freudenthal algebra, 517 split semisimple group, 356 split torus, 333 split trialitarian algebra, 553 split twisted composition, 490 splitting field, 4 Springer construction, 523 Springer decomposition, 523 stabilizer, 327 standard identification, 55 ∗-action, 376 strictly power-associative algebra, 451 sub-Grassmannian variety, 12 surjective homomorphism, 327 symbol algebra, 27 symmetric bilinear form, xvi symmetric composition algebra, 464 symmetric composition of type ^{i}A′_{2}, 480 symplectic basis, 16 symplectic group, 347 symplectic involution, 16 system of simple roots, 353

τ′ (second involution), 550

Tits algebra, 377

Tits class of a simply connected semisimple

group, 426 Tits class of an algebra with unitary involu-

tion, 427 Tits construction, 525 Tits process, 525 torsor, 288, 388 trace of an element, xv transfer of a bilinear form, 524 transpose of a linear map, 1 trialitarian algebra, 552, 553 trialitarian algebra of type G_{2}, 553 trialitarian algebra of type ^{1}D_{4}, 558 trialitarian algebra of type ^{6}D_{4}, 558 trialitarian algebra of type ^{3}D_{4}, 558 trialitarian algebra of type ^{2}D_{4}, 558 trialitarian isomorphism, 553 trialitarian triple, 548 triality, 451, 547 trivial group, 326 twisted ρ-form, 392 twisted Cayley composition, 490 twisted composition of type G_{2}, 499 twisted composition of type ^{1}A′_{2}, 505 twisted composition of type ^{2}A_{2}, 506 twisted composition over L, 489 twisted form, 361 twisted Hurwitz composition, 499 twisting, 387 2-cocycle, 384 type of a root system, 356

unital algebra, 451 unitary group, 346 unitary involution, 21 unitary Whitehead group, 267

vector representation, 177

Wedderburn’s theorem, 3, 303

weight, 344

weight lattice, 353 Weyl group, 353

Zorn matrices, 507

Notation

()^{#} (Freudenthal adjoint), 470, 519 × (Freudenthal product), 470, 519 (Jordan product), 513 ⊥, xvii ⊗ = ⊗_{F} (tensor product), xix (C, n)_{Δ}, 500 (L, a) (cyclic algebra), 302, 414 (U, q)_{K}, 499 (V, Φ) (root system), 353 (V, q) (quadratic space), 74 (V, q, ζ) (oriented quadratic space), 171 (Z, α)_{F} (quaternion algebra), 190 (σ, f) (quadratic pair), 55 (a, b)_{F} (quaternion algebra), 25 (a, b)_{ζ,K} (symbol algebra), 27 ⟨⟨α_{1}, …, α_{n}⟩⟩ (Pfister bilinear form), xvi ⟨⟨α_{1}, …, α_{n}]], xix ⟨α_{1}, …, α_{n}⟩, xix ⟨f, ξ⟩, 282 [α_{1}], xix [α_{1}, α_{2}], xix [a, b)_{F} (quaternion algebra in char. 2), 25 {a, b}_{i}, 452

A(ρ, w) (twisted ρ-forms of w), 362, 392 Ã(ρ, w), 362 A^{+} (Jordan algebra of A), 27, 513 A^{0} (trace zero elements in A), 28, 472 1A, 472 1A_{2}, 475 1A_{3}, 227 1A′_{n} (central simple algebras), 471 2A, 478 2A_{2}, 480 A_{1}, 207 \overline{A_{1}}, 464 A_{1}^{2}, 207 A_{3} (alternating group), 421 A_{n} (root system), 354 A_{n}, 207 A_{n}^{+}, 515 A^{n}, 366 \overline{A}^{n}, 366 A_{α} (twisted action), 387 A_{χ} (minimal Tits algebra), 426 a^{χ}, 431 Ad (adjoint representation), 343 ad (inner derivation), 344 A^{Γ} (Γ-invariant elements), 383 A_{λ} (Tits algebra), 377 Alg_{F}, 280 α_{A}^{ϕ}, 549 α_{S}, 481 α_{σ,τ}, 386 Alt(A, σ) (alternating elements in A), 14 A^{op} (opposite algebra of A), 3 A_{red}, 331 (A^{×} × A^{×})^{Det}, 529 A–Tors_{Γ} (A-torsors), 388 Aut(A, σ), 345 Aut(J, A), 529 Aut(J/A), 529 AutÃ, 362 Aut_{alg}(A), 328 Aut_{F} (A, σ, f), 159 Aut_{F} (J, A), 529 Aut_{F} (J/A), 529 Aut_{G}(v), 328 Aut_{K}(A, σ), 345 Aut_{R}(A_{R}), 328

B(A, σ, f) (Clifford bimodule of A), 110 B(θ) (Clifford bimodule functor), 175 B_{n} (root system), 354 B_{n}, 207 B′_{n}, 170, 369 B_{n}^{+}, 515 B_{n}, 172 B^{n}, 368 \overline{B}^{n}, 368 B/A (left cosets), 385 β(v, w), 522 β_{ℭ}, 490 β_{S}, 490 b̂, xvi b_{K/F}, 534 b_{q} (polar form of q), xvii Br(F) (Brauer group of F), 4 b ∗ (ℓ_{1} ⊗ ℓ_{2}), 311

2C, 478

C(−) (Clifford algebra functor), 173 C(A, σ), 93, 175 C(A, σ, f) (even Clifford algebra of A), 92 C(G) (center of G), 357 c(T) (Clifford invariant), 563 C(V, q) (Clifford algebra of (V, q)), 87 C_{0}(V, q) (even Clifford algebra of (V, q)), 87 1C, 472 C_{3} (cyclic group), 302 C_{n} (root system), 354 C_{n}, 207 C_{n}^{+}, 515 C^{n}, 369 \overline{C}^{n}, 369 C^{+}(V, q), 359 C^{−}(V, q), 359 C_{A}B (centralizer of B in A), 5 ℭ_{s} (split Cayley algebra), 462 ℭ̃, 490 CD(B, λ) (Cayley-Dickson process), 458 Comp_{m} (composition algebras), 458 Comp_{m}^{+} (unital composition algebras), 458 C^{⊥}, 454 C_{φ} (Petersson algebra), 466 Cubjord (cubic norm structures), 520

D(B, τ) (discriminant algebra of (B, τ)),

128 1D_{3}, 227 1D_{4}, 558 2D_{4}, 558 3D_{4}, 558 6D_{4}, 558 D_{4}, 562 D^{4}, 562 \overline{D}^{4}, 562 D_{n} (root system), 355 D_{n}, 207 D_{n}^{+}, 515 D^{n}, 370 \overline{D}^{n}, 370 deg A (degree of A), 4 Δ (Dickson invariant), 167 Δ(L), 291 Δ(X), 291 δ^{0} (connecting map), 385 δ^{1} (connecting map), 386 Δ(g) (Dickson invariant of g), 157 Der(A, M) (derivation module), 334 Der(Γ), 569 Der(J), 569 Der(J/L), 569 Der(V, L, Q, β) , 569 det(m) (determinant of m), xv det b (determinant of b), xvi det q, xvii det σ (determinant of σ), 81 df (differential of F), 335 dim G (dimension of G), 337 disc b (discriminant of b), xvi disc q, xvii disc σ (discriminant of σ), 81 Dyn(Φ) (Dynkin diagram), 353

E(a), 557

E(𝔏) (Allen invariant), 570

E_{6} (root system), 355 E_{7} (root system), 355 E_{8} (root system), 355 (e_{ij})_{1≤i,j≤n} (standard basis of M_{n}), 30 e_{n}(π) (cohom. inv. of the Pfister form π),

413 End, 171 End(ℭ̃) (split trialitarian algebra), 553 End(Γ), 555 Et_{F} (étale algebras), 281

f_{1}(B, τ) (cohomology class), 422 f_{1}(L) (cohomology class), 421 F_{4} (root system), 355 F_{4}, 372 F^{4}, 372 f_{3}(B, τ) (cohomology class), 423 f_{i}(J) (cohomology class), 534 f_{n}(q_{n}) (cohomological invariant), 462 F [V], 326 F [G], 325 F ⟨H⟩, 332 F_{alg} (algebraic closure), 323 f_{alg}, 339 F_{χ} (field of definition of χ), 426 F_{λ}, 378 φ_{q} (standard identification for b = b_{q}), 58 f_{q} (quadratic pair of q), 92 F_{sep} (separable closure), 280 f^{∗} (comorphism), 327 f^{t} (transpose of f), 1 F_{v} (completion of F at v), 449

G(A, σ) (multipliers), 162 G(R) (group of R-points), 326 G^{0} (connected component of G), 331 g_{1}(L) (cohomology class), 421 G_{1} × G_{2}, 328 G_{2}, 373 g_{2}(B, τ) (cohomology class), 420 \overline{G_{2}}, 464 G^{2}, 373 G_{2} (root system), 355 g_{3}(J), 537 G_{L} (restriction scheme), 326 G^{A}(R), 325 G_{a} (additive group functor), 326 Γ (absolute Galois group), 279 Γ^{+}(V, q), 349 Γ(A, σ, f) (Clifford group), 351 Γ(A, σ, f) (Clifford group), 178 Γ(C, L), 499 Γ^{+}(V, q) (special Clifford group), 177 Γ_{λ}, 490 γξ, 281 G^{D} (Cartier dual), 333 G–Gal_{F} (Galois algebras), 290 GL(V) (general linear group), 326 GL_{1}(A), 326 GL_{n}, 326 G_{m} (multiplicative group functor), 326 G_{m,K}^{1} (norm one elements), 402 𝔤(A, σ), 28 GO(A, σ, f), 350 GO(A, σ), 159 GO(A, σ), 347 GO(A, σ, f), 159 GO(V, b), 154 GO(V, q), 155 GO^{+}(V, q), 157 GO^{+}(A, σ), 164 GO^{+}(A, σ, f), 167 GO^{+}(V, b), 154 GO^{−}(A, σ, f), 202 G^{q} (quasisplit inner form of G), 426 Gr(rn, A) (Grassmannian), 9 Groups, 325 GSp(A, σ), 159 GSp(V, b), 154 GSp(A, σ), 347 G^{∗}, 327 G–Tors_{Γ} (G-torsors), 290 GU(A, σ), 159 GU(B, τ), 193 GU(B, τ), 346 GU(V, h), 155

H(A, σ) (Jordan algebra of Sym(A, σ)), 28 H_{n}(C, α), 516 H_{3}(C, α)^{0}, 517 H^{0}(Γ, A), 383 H^{1}(Γ, A) (cohomology set), 384 H_{1} ∩ H_{2}, 328 H^{2}(Γ, A) (cohomology group), 384 H^{i}(F, A) (cohomology groups), 391 H_{const} (constant group scheme), 332 H_{diag} (diagonalizable group scheme), 332 H_{et}, 332 nH (n-torsion of H), 413 Hom_{F} (G, H) (group scheme homom.), 325 Hurw_{m} (Hurwitz algebras), 460 \overline{Hurw}(para-Hurwitz algebras), 464

i(G) (Rost invariant), 435 I^{0} (annihilator of I), 8 IF (fundamental ideal of W F), 141 I^{n}F (powers of the fundamental ideal), 141 im(f) (image of F), 327 Ind_{Γ_{0}}^{Γ} A_{0} (induced Γ-group), 390 ind A (Schur index of A), 4 Ind_{H}^{G} M, 288 Int(v) (inner automorphism), 2 I^{⊥} (ideal orthogonal to I), 72 Isom(X ), 170 Iso(A, σ), 159 Iso(V, b) (isometries of (V, b)), 153 Iso(A, σ) (isometries of (A, σ)), 345

J(A, λ) (first Tits construction), 526

J(B, τ, u, ν) (second Tits construction), 524 J(L, V) (Springer construction), 523 J(V, q) (Jordan algebra of a quadratic

form), 514 J^{0}, 514

K ∗ K′, 416

ker(f) (kernel of f), 328

L^{0}, 284

Λ (root lattice), 353

Λ_{+} (cone of dominant weights), 353

Λ_{r}, 353 λ^{k}a, 195 L^{G} (invariant elements), 287 𝔏(A) (Lie algebra of A), 27 Lie(G) (Lie algebra of G), 334 𝔏(T), 569 L_{K}, 280 ℓ_{x} (left multiplication with x), 455

M(A) (multiplication algebra), 451

M_{[K]} (twisted module), 416 M_{n}(C) (matrix algebra), 516 M_{r,s}(D) (r × s-matrices), 7 μ(t) (multiplier of t), 483 μ_{i}(a, b), 452 μ_{n} (roots of unity), 328 μ_{n[K]} (twisted group), 418 μ_{n}^{⊗i}(F), 431 \overline{μ} (multiplier map), 182 \underline{μ} (multiplier map), 184

N(ℓ) (norm of ℓ), xvi

N(x, y, z) (cubic form), 474 N_{A} (generic norm), 452 N_{C} (norm of x), 457 nil(A) (nilpotent elements), 331 N_{J} (norm of J), 516 N_{K/F} (A) (norm (corestriction) of A), 37 N(K/F) (group of norms), 114 N_{L/F} (ℓ) (norm of ℓ), xvi Nrd_{A}(a) (reduced norm of a), 5 N_{U}, 327

O(A, σ), 347 O(A, σ, f), 160 O(A, σ, f), 350 O(V, q), 155 O(V, q), 348 O(A, σ), 159 O^{+}(A, σ), 164 O^{+}(A, σ, f), 351 O^{+}(V, b), 154 O^{+}(V, q), 157 O^{+}(V, q), 348 O(V, b), 154 Oku, 480 1Oku, 475 2Oku, 480 Ω(A, σ), 181 Ω(A, σ, f) (extended Clifford group), 181 Ω(A, σ, f) (extended Clifford group), 352

P (R) (fiber functor), 342 ℘ (Artin-Schreier map), xviii, 414 P_{A,x}(X) (generic polynomial), 452 PGL(V), 345 PGL_{1}(A), 344 PGL_{n} (projective linear group), 345 PGO(A, σ), 159 PGO(A, σ, f), 159 PGO(A, σ), 347 PGO(A, σ, f), 350 PGO^{+}(A, σ, f), 351 PGO(V, b), 154 PGO(V, q), 350 PGO^{+}(A, σ, f), 167 PGO^{+}(V, b), 154 PGSp(A, σ), 159 PGSp(A, σ), 347 PGSp(V, b), 154 PGU(A, σ), 159 PGU(B, τ), 346 PGU(V, h), 155 π(τ) (Pfister form of an involution), 304 π_{0}(A), 330 Pic(G) (Picard group), 431 P_{J,a}(X) (generic minimal polynomial), 519 P_{L,ℓ}(X) (characteristic polynomial), xvi P_{m}(X) (characteristic polynomial), xv PGO(V, q), 155 PQ_{m} (Pfister forms), 460 Prd_{A,a}(X) (reduced characteristic

polynomial of a), 5 Prp_{σ,s}(X) (pfaffian characteristic

polynomial of s), 19 p_{σ} (pfaffian adjoint), 245 PSim(A, σ), 159 PSim(V, b) (projective similitudes of (V, b)),

153 p_{θ}, 246

Q^{+}(A, σ, ℓ), 411 Q(A, σ), 411 Q_{0}(A, σ, ℓ), 411 q_{b}, xvii Q_{n}, 171 Q_{n}^{1}, 171 Q_{τ}^{0}, 306 Quot(S) (field of fractions of S), 341 ℚ/ℤ(i)(F), 431 R(C), 455 rdim M (reduced dimension of M), 6 res (restriction map), 385 ρ_{v} (hyperplane reflection), 157 ρ(E ⊗ Δ, σ ⊗ 1), 552 ρ^{−1}α_{^{ρ}E}, 554 ρ_{R}, 325 R_{L/F} (G) (corestriction functor), 329 R_{L/F} (G) (corestriction of G), 329

S, 187 S(ℓ), xvi S(V^{∗}) (symmetric algebra), 326 S^{0}, 284 \overline{S_{2}}, 464 S_{L/F}, 283 S_{A}, 469 Sand(x) (sandwich map), 32 SB_{r}(A) (Severi-Brauer variety of A), 10 Scomp_{m} (symmetric composition algebras),

464 2Sepalt_{m}(3) (cubic separable algebras), 478 Sepjord_{n}(m) (separable Jordan algebras),

514 Sets_{Γ} (Γ-sets), 281 sgn_{P} σ (signature of σ), 138 𝔰(B, τ), 198 SGU(B, τ), 194 s_{i}(m) (coefficients of P_{m}(X)), xv Σ(L), 291 Σ(X), 290 σ_{b} (involution adjoint to b), 1 σ_{g} (involution), 13 σ_{q} (adjoint involution of q), 58 Σ_{σ}(A), 266 σ_{∗}, 18 \underline{σ} (induced involution on C(A, σ, f)), 94 Sim(A, σ) (similitudes of (A, σ)), 158 Sim(A, σ) (similitudes of (A, σ)), 345 Sim(V, b) (similitudes of (V, b)), 153 S_{J} (a), 516, 519 SK_{1} U(B) (reduced unitary Whitehead

group), 267 SK_{1}(A) (reduced Whitehead group), 253 Skew(A, σ) (skew-symmetric elements

in A), 14 Skew(A, σ)^{0} (trace zero elements), 28 SL_{1}(A), 253 SL(E), 194 S_{L/F} (ℓ), xvi SL(V), 328 SL_{1}(A), 328 Sn (spinor norm), 186 Sn(A, σ, f), 187 Sp(A, σ), 159 Sp(A, σ), 347 Sp(V, b), 154 Sp(V, h), 347 Spin(A, σ, f), 187 Spin(A, σ, f), 351 Spin(V, q), 187 Spin(V, q), 349 Spin_{8}, 486 Spin^{±}(V, q), 359 Srd_{A}(a), 5 SSym(A, σ)^{×}, 406 SSym(B, τ)^{×}, 531 SSym(B, τ)^{×}, 401 SU(B, τ), 194 SU(B, τ), 346 SU(V, h), 155 S_{v}, 327 Sym(A, σ) (symmetric elements in A), 14 Sym(A, σ)′, 400 Sym(B, τ)^{0}, 306 Symd(A, σ) (symmetrized elements in A),

14

T (ℓ) (trace of ℓ), xvi T (A, σ) (trialitarian algebra), 558 t(B, τ) (Tits class), 427 T (𝔏) (trialitarian algebra of a Lie algebra),

570 T (x, y) (bilinear trace form), 281 t^{+}, 483 t^{−}, 483 T_{A} (generic trace), 452 τ′, 550 T_{C}(x) (trace of x), 457 θ^{+}, 485 θ^{−}, 485 θ_{α}, 387 T_{J} (a), 516, 519 T_{L/F} (ℓ) (trace of ℓ), xvi (T_{L/F})_{∗}(b), 524 tr(m) (trace of m), xv Trd_{A}(a) (reduced trace of a), 5 T_{τ}, 534 T_{τ,τ′}(x, y), 534

U(B, τ), 346 UK_{1}(B) (unitary Whitehead group), 267 U(A, σ), 159 U(B, τ), 193 U(V, h), 155

V^{⊥}, xvii V^{∗} (dual space of V), xvi V^{G} (G-invariant subspace), 343

W (Φ) (Weyl group), 353 W F (Witt ring of F), 141

x^{#}, 470, 519 X(f), 281 X(L), 280 x ⋆ y, 472 X_{c}, 444 x^{f} (left module homomorphism), xv χ(λ), 354 χ_{i}, 357 x ∗_{σ} (a ⊗ b) (right module structure), 32

Z(A) (centroid), 451

Z(A, σ, f), 187

Z^{1}(Γ, A) (1-cocycles), 384 Z^{2}(Γ, A) (2-cocycles), 384