Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $ ({\bf A_2,G_2})$ duality in $ {\bf E_6}$, octonions and the triality principle


Author: Hubert Rubenthaler
Journal: Trans. Amer. Math. Soc. 360 (2008), 347-367
MSC (2000): Primary 17A75; Secondary 17B25, 11S90
DOI: https://doi.org/10.1090/S0002-9947-07-04269-9
Published electronically: August 14, 2007
MathSciNet review: 2342006
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the existence of a dual pair of type $ (A_2, G_{2})$ in $ E_6$ leads to a definition of the product of octonions on a specific $ 8$-dimensional subspace of $ E_6$. This product is expressed only in terms of the Lie bracket of $ E_6$. The well known triality principle becomes an easy consequence of this definition, and $ G_2$ acting by the adjoint action is shown to be the algebra of derivations of the octonions. The real octonions are obtained from two specific real forms of $ E_6$.


References [Enhancements On Off] (What's this?)

  • [All-Fer-1] H.P ALLEN, J.C. FERRAR | Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 No. 1 (1977),1-35. MR 0444729 (56:3079)
  • [All-Fer-2] H.P ALLEN, J.C. FERRAR | Jordan algebras and exceptional subalgebras of the exceptional algebra $ E_{6}$, Pacific J. Math. 32 (1970), 283-297 . MR 0262308 (41:6917)
  • [Al-1] B.N. ALLISON | A construction of Lie algebras from J-ternary algebras, Americ. J. Math., 98 (1976), 285-294. MR 0430010 (55:3018)
  • [Al-2] B.N. ALLISON | A class of nonassociative algebras with involutions containing the class of Jordan algebras, Math. Ann., 237 (1978), no 2, 133-156. MR 507909 (81h:17003)
  • [Al-3] B.N. ALLISON | Models of isotropic simple Lie algebras, Comm. Algebra., 7 (1979), no 17, 1835-1875. MR 547712 (81d:17005)
  • [Ba] J. BAEZ | The Octonions, Bul. Amer. Mat. Soc.(NS), 39 (2002) Number 2, 145-205. MR 1886087 (2003f:17003)
  • [B-S] C.H. BARTON - A. SUDBERY | Magic squares of Lie Algebras, arXiv:math.RA/0001083.
  • [Bro-1] R. B. BROWN | A new type of associative algebras, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 947-949. MR 0158913 (28:2135)
  • [Bro-2] R. B. BROWN | Groups of Type $ E_{7}$, J. Reine Angew. Math. 236 (1969), 79-102. MR 0248185 (40:1439)
  • [Che] C. CHEVALLEY | The Algebraic Theory of Spinors, Columbia University Press, 1954 MR 0060497 (15:678d)
  • [Fau] J.R. FAULKNER | A construction of Lie algebras from a class of ternary algebras, Trans. Amer. Math. Soc., 155 (1971),397-408. MR 0294424 (45:3494)
  • [Fer] J.C. FERRAR | Lie algebras of type $ E_{6}$, J. Algebra 13 (1969), 57-72, | Lie algebras of type $ E_{6}$ II, J. Algebra 52 (1978) No. 1 , 201-209. MR 0263881 (41:8480)
  • [Freu] H. FREUDENTHAL | Lie groups and the foundations of geometry, Adv. Math. 1 (1964), 145-190. MR 0170974 (30:1208)
  • [Gar] S. GARIBALDI | Structurable algebras and groups of type $ E_{6}$ and $ E_{7}$ , J. Algebra 236, No.2 (2001), 651-691. MR 1813495 (2001m:20069)
  • [Hei] W. HEIN | A construction of Lie algebras by triple systems, Trans. Amer. Math. Soc., 205 (1975), 79-95. MR 0393153 (52:13963)
  • [Hu] J.E. HUMPHREYS | Linear Algebraic Groups, Graduate Texts in Math. 21. Springer, Berlin, New York, 1975. MR 0396773 (53:633)
  • [Ig] J-I. IGUSA | A Classification of Spinors up to dimension twelve, Amer. J. Math. 92 (1970) 997-1028. MR 0277558 (43:3291)
  • [Iw] N. IWAHORI | On real irreducible representation of Lie algebras, Nagoya. Math. J. vol. 14 (1959), 59-83. MR 0102534 (21:1325)
  • [Jac] N. JACOBSON | Exceptional Lie Algebras, Marcel Dekker, New York, 1971. MR 0284482 (44:1707)
  • [Ka] I.L. KANTOR | Models of exceptional Lie algebras, Soviet. Math. Doklady, 14 (1973),254-258.
  • [Ka-Sko] I.L. KANTOR, I. M. SKOPETS | Some Results on Freudenthal Triple Systems, Sel. Math. Sov. vol. 2. No. 3. (1982), 293-305 (originally published in Trudy Sem. Vektor. Tenzor. Anal 18 (1978) 250-263).
  • [Ki] T. KIMURA | Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs, 215. American Mathematical Society, Providence, RI (2003). MR 1944442 (2003k:11180)
  • [Ko] M. KOECHER | Imbedding of Jordan algebras into Lie algebras I, Americ. J. Math., 89 (1967),787-815. MR 0214631 (35:5480)
  • [L-M] J.M. LANDSBERG - L. MANIVEL | Triality, exceptional Lie algebras and Deligne dimension formulas , Adv. Math. 171 (2002), no. 1, 59-85. MR 1933384 (2003i:17012)
  • [Lo] O. LOOS | Jordan Pairs, Lecture Notes in Math., vol. 460, Springer-Verlag, New York (1975) . MR 0444721 (56:3071)
  • [M-R-S] I. MULLER - H. RUBENTHALER - G. SCHIFFMANN | Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées, Math. Ann. 274 (1986), 95-123. MR 834108 (88e:17025)
  • [Mey] K. MEYBERG | Zur Konstruktion von Lie-Algebren aus Jordan Triple-Systemen, Manuscripta math. 3, (1970), 115-132 . MR 0285576 (44:2794)
  • [Ru-1] H. RUBENTHALER | Construction de certaines sous-algèbres remarquables dans les algèbres de Lie semi-simples, J. Algebra 81 (1983) No. 1, 268-278. MR 696139 (84g:17010)
  • [Ru-2] H. RUBENTHALER | Algèbres de Lie et espaces préhomogènes, Travaux en Cours, Hermann, Paris (1992).
  • [Ru-3] H. RUBENTHALER | Les paires duales dans les algèbres de Lie réductives, Astérisque No. 219 (1994), 121 pp. MR 1264015 (95b:17010)
  • [Ru-4] H. RUBENTHALER | Non Parabolic Prehomogeneous Vector Spaces and Exceptional Lie Algebras, J. Algebra 281 (2004) 366-394. MR 2091977 (2005k:20110)
  • [Ru-5] H. RUBENTHALER | Formes réelles des espaces préhomogènes irréductibles réguliers, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 1, 1-38. MR 840712 (87k:17010)
  • [Sa-Ki] M. SATO - T. KIMURA | A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. MR 0430336 (55:3341)
  • [S-V] T. A. SPRINGER - F. D. VELDKAMP | Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer, Berlin (2000) MR 1763974 (2001f:17006)
  • [Ti-1] J. TITS | Une classe d'algèbres de Lie en relation avec les algèbres de Jordan, Indag. Math. 24 (1962), 530-535. MR 0146231 (26:3753)
  • [Ti-2] J. TITS | Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Indag. Math. 28 (1966), 223-237. MR 0219578 (36:2658)
  • [Wa] G. WARNER | Harmonic Analysis on Semi-Simple Lie Groups I, Die Grundlehren der mathematischen Wissenschaften, Band 188. Springer-Verlag, New York-Heidelberg, 1972. MR 0498999 (58:16979)
  • [Yam] K. YAMAGUTI | On algebras of totally geodesic spaces (Lie triple systems), J. Sci. Hiroshima Univ. Ser. A 21 (1957/58), 107-113 . MR 0100046 (20:6482)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17A75, 17B25, 11S90

Retrieve articles in all journals with MSC (2000): 17A75, 17B25, 11S90


Additional Information

Hubert Rubenthaler
Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France

DOI: https://doi.org/10.1090/S0002-9947-07-04269-9
Received by editor(s): October 4, 2004
Received by editor(s) in revised form: February 13, 2006
Published electronically: August 14, 2007
Dedicated: A la mémoire de Maurice Drexler
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society