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

Author(s): Hubert Rubenthaler
Journal: Trans. Amer. Math. Soc. 360 (2008), 347-367.
MSC (2000): Primary 17A75; Secondary 17B25, 11S90
Posted: August 14, 2007
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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$.

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Additional Information:

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

DOI: 10.1090/S0002-9947-07-04269-9
PII: S 0002-9947(07)04269-9
Received by editor(s): October 4, 2004
Received by editor(s) in revised form: February 13, 2006
Posted: August 14, 2007
Dedicated: A la mémoire de Maurice Drexler
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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