Geodesic
Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2
Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 776-806

Voir la notice de l'article provenant de la source Cambridge University Press

It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ(W) of “maximal” forms. If ψ ∈ Ψ(W) then the subgroup of AUTC(W) consisting of the invertible complex-linear transformations S such that ψ(S•, S•, S•) = ψ(•, •, •) is denoted G(ψ), and, in Proposition 3.6. we prove where G 1(ψ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G 2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking. 
Herz, Carl. Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2. Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 776-806. doi: 10.4153/CJM-1983-045-2
@article{10_4153_CJM_1983_045_2,
     author = {Herz, Carl},
     title = {Alternating {3-Forms} and {Exceptional} {Simple} {Lie} {Groups} of {Type} {G2}},
     journal = {Canadian journal of mathematics},
     pages = {776--806},
     year = {1983},
     volume = {35},
     number = {5},
     doi = {10.4153/CJM-1983-045-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-045-2/}
}
TY  - JOUR
AU  - Herz, Carl
TI  - Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2
JO  - Canadian journal of mathematics
PY  - 1983
SP  - 776
EP  - 806
VL  - 35
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-045-2/
DO  - 10.4153/CJM-1983-045-2
ID  - 10_4153_CJM_1983_045_2
ER  - 
%0 Journal Article
%A Herz, Carl
%T Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2
%J Canadian journal of mathematics
%D 1983
%P 776-806
%V 35
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-045-2/
%R 10.4153/CJM-1983-045-2
%F 10_4153_CJM_1983_045_2

[1] 1. E., Cartan, Sur la structure des groupes simples finis et continus, C. R. Acad. Sci. Paris 116 (1893), p. 784–786 = Oeuvres complètes II (Gauthier-Villars, Paris 1952), 99-101. Google Scholar

[2] 2. E., Cartan, Sur la structure des groupes de transformations finis et continus, Thesis ( 1894) = Oeuvres complètes II, 137-287. Google Scholar

[3] 3. E., Cartan, Les groupes réels simples, finis et continus, Ann. Ecole Norm. Sup. Paris 31 (1914), 263–355 = Oeuvres complètes II, 399-491. Google Scholar

[4] 4. E., Cartan, Sur certaines formes riemanniennes remarquables des geometries à groupe fondamental simple, Ann. Ecole Norm. Sup. Paris 44 (1927), 345–467 = Oeuvres complètes 12, 867-988. Google Scholar

[5] 5. F., Engel, Sur un groupe simple à quatorze paramètres, C. R. Acad. Sci. Paris 116 (1893), 786–788. Google Scholar

[6] 6. S., Helgason, Differential geometry, Lie groups, and symmetric spaces (Academic Press, New York, 1978). Google Scholar

[7] 7. C., Herz, Symmetric spaces, (to appear). Google Scholar

[8] 8. R. D., Schafer, An introduction to nonassociative algebras (Academic Press, New York, 1966). Google Scholar

[9] 9. M., Zorn, The automorphisms of Cayley's non-associative algebra, Proc. Nat. Acad. Sci. U.S.A. 21 (1935), 355–358. Google Scholar

Cité par Sources :