THE BOOK OF INVOLUTIONS
CONTENTS 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 ?RST 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 ?RST 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 IDENTI?CATIONS 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 CLI?ORD ALGEBRA 8 A THE SPLIT CASE 8 B DE?NITION OF THE CLI?ORD ALGEBRA 8 C LIE ALGEBRA STRUCTURES 8 D THE CENTER OF THE CLI?ORD ALGEBRA 8 E THE CLI?ORD ALGEBRA OF A HYPERBOLIC QUADRATIC PAIR §9 THE CLI?ORD BIMODULE 9 A THE SPLIT CASE 9 B DE?NITION OF THE CLI?ORD BIMODULE 9 C THE FUNDAMENTAL RELATIONS §10 THE DISCRIMINANT ALGEBRA 10 A THE ?-POWERS OF A CENTRAL SIMPLE ALGEBRA 10 B THE CANONICAL INVOLUTION 10 C THE CANONICAL QUADRATIC PAIR 10 D INDUCED INVOLUTIONS ON ?-POWERS 10 E DE?NITION OF THE DISCRIMINANT ALGEBRA 10 F THE BRAUER CLASS OF THE DISCRIMINANT ALGEBRA §11 TRACE FORM INVARIANTS 11 A INVOLUTIONS OF THE ?RST KIND 11 B INVOLUTIONS OF THE SECOND KIND 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 CLI?ORD STRUCTURES 13 B CLI?ORD GROUPS 13 C MULTIPLIERS OF SIMILITUDES §14 UNITARY INVOLUTIONS 14 A ODD DEGREE 14 B EVEN DEGREE 14 C RELATION WITH THE DISCRIMINANT ALGEBRA CHAPTER IV ALGEBRAS OF DEGREE FOUR §15 EXCEPTIONAL ISOMORPHISMS 15 A B1 ?C1 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 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 CLASSI?CATION OF INVOLUTIONS OF THE SECOND KIND 19 C ´ETALE SUBALGEBRAS 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 §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 CLASSI?CATION RESULTS 26 B ALGEBRAIC GROUPS OF SMALL DIMENSION §27 TITS ALGEBRAS OF SEMISIMPLE GROUPS 27 A DE?NITION OF THE TITS ALGEBRAS 27 B SIMPLY CONNECTED CLASSICAL GROUPS 27 C QUASISPLIT GROUPS CHAPTER VII GALOIS COHOMOLOGY §28 COHOMOLOGY OF PRO?NITE 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 CLASSI?CATION 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 COE?CIENTS 30 C COHOMOLOGICAL INVARIANTS OF ALGEBRAS OF DEGREE THREE §31 COHOMOLOGICAL INVARIANTS 31 A CONNECTING HOMOMORPHISMS 31 B COHOMOLOGICAL INVARIANTS OF ALGEBRAIC GROUPS CHAPTER VIII COMPOSITION AND TRIALITY §32 NONASSOCIATIVE ALGEBRAS §33 COMPOSITION ALGEBRAS 33 A MULTIPLICATIVE QUADRATIC FORMS 33 B UNITAL COMPOSITION ALGEBRAS