The $(\textbf {A_2,G_2})$ duality in $\textbf {E_6}$, octonions and the triality principle
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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 $E_6$.References
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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
- Received by editor(s): October 4, 2004
- Received by editor(s) in revised form: February 13, 2006
- Published electronically: August 14, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2342006
Dedicated: A la mémoire de Maurice Drexler