Octonions, Jordan Algebras, and Exceptional GroupsSpringer Science & Business Media, 16 mai 2000 - 208 pages The 1963 Göttingen notes of T. A. Springer are well-known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. In the group-theoretical part use is made of some results from the theory of linear algebraic groups. The book will be useful to mathematicians interested in octonion algebras and Albert algebras, or in exceptional groups. It is suitable for use in a graduate course in algebra. |
Table des matières
1 Composition Algebras | 1 |
12 Composition Algebras The Minimum Equation | 4 |
13 Conjugation Inverses | 7 |
14 Moufang Identities Alternative Laws | 9 |
15 Subalgebras Doubling | 11 |
16 Structure and Dimension of a Composition Algebra | 14 |
17 A Composition Algebra is Determined by its Norm | 16 |
18 Split Composition Algebras | 18 |
48 A Criterion for Reduced Twisted Octonion Algebras Applications | 105 |
49 More on Isotropic Normal Twisted Octonion Algebras | 108 |
410 Nonnormal Twisted Octonion Algebras with Isotropic Norm | 110 |
411 Twisted Composition Algebras with Anisotropic Norm | 112 |
412 Historical Notes | 115 |
51 Jalgebras Definition and Basic Properties | 117 |
52 Cross Product Idempotents | 122 |
53 Reduced Jalgebras and Their Decomposition | 125 |
19 Center and Associating Elements | 20 |
110 Classification over Special Fields | 21 |
111 Historical Notes | 23 |
2 The Automorphism Group of an Octonion Algebra | 25 |
22 Connectedness and Dimension of the Automorphism Group | 26 |
23 The Automorphism Group is of Type G2 | 30 |
24 Derivations and the Lie Algebra of the Automorphism Group | 33 |
25 Historical Notes | 35 |
3 Triality | 37 |
32 The Principle of Triality | 42 |
33 Outer Automorphisms Defined by Triality | 45 |
34 Automorphism Group and Rotation Group of an Octonion Algebra | 48 |
35 Local Triality | 50 |
36 The Spin Group of an Octonion Algebra | 58 |
37 Fields of Definition | 65 |
38 Historical Notes | 66 |
4 Twisted Composition Algebras | 69 |
41 Normal Twisted Composition Algebras | 70 |
42 Nonnormal Twisted Composition Algebras | 79 |
43 Twisted Composition Algebras over Split Cubic Extensions | 89 |
44 Automorphism Groups of Twisted Octonion Algebras | 92 |
45 Normal Twisted Octonion Algebras with Isotropic Norm | 94 |
46 A Construction of Isotropic Normal Twisted Octonion Algebras | 99 |
47 A Related Central Simple Associative Algebra | 102 |
54 Classification of Reduced Jalgebras | 133 |
55 Further Properties of Reduced Jalgebras | 141 |
56 Uniqueness of the Composition Algebra | 145 |
57 Norm Class of a Primitive Idempotent | 149 |
58 Isomorphism Criterion Classification over Some Fields | 152 |
59 Isotopes Orbits of the Invariance Group of the Determinant | 154 |
510 Historical Notes | 159 |
6 Proper Jalgebras and Twisted Composition Algebras | 161 |
62 From Jalgebras to Twisted Composition Algebras | 163 |
63 From Twisted Composition Algebras to Jalgebras | 167 |
64 Historical Notes | 171 |
7 Exceptional Groups | 173 |
72 The Automorphism Group of an Albert Algebra | 178 |
73 The Invariance Group of the Determinant in an Albert Algebra | 180 |
74 Historical Notes | 182 |
8 Cohomological Invariants | 185 |
82 An Invariant of Composition Algebras | 189 |
83 An Invariant of Twisted Octonion Algebras | 191 |
84 An Invariant of Albert Algebras | 195 |
85 The FreudenthalTits Construction | 199 |
86 Historical Notes | 200 |
| 201 | |
| 205 | |
Autres éditions - Tout afficher
Octonions, Jordan Algebras and Exceptional Groups Tonny A. Springer,Ferdinand D. Veldkamp Aucun aperçu disponible - 2010 |
Octonions, Jordan Algebras and Exceptional Groups Tonny A. Springer,Ferdinand D. Veldkamp Aucun aperçu disponible - 2014 |
Expressions et termes fréquents
Albert algebra algebra F algebraic group algebraically closed associated assume automorphism group bijective bilinear form char(k characteristic computation Corollary crossed product cubic cyclic cubic extension cubic form D₁ defined denote det(x dimension division algebra E₁ F is reduced finite follows formula Galois GO(N group of type Hence homomorphism identity element implies invariant irreducible isomorphism isotropic Jordan algebras L₁ Lemma Lie algebra linear transformation multiplication nondegenerate nontrivially nonzero norm class norm N normal twisted composition normal twisted octonion notation orthogonal polynomial primitive idempotent Proof Prop proper reduced J-algebra Proposition prove quadratic form quaternion subalgebra reduced norm related triple restriction of Q rotation RT(C SGO(N similar SO(N SO(N₁ spin group Spin(N split subgroup subspace t₁ t₂ Theorem triality twisted composition algebra twisted octonion algebra unique V₁ vector space zero