Transactions of the American Mathematical Society

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



A quantum octonion algebra

Authors: Georgia Benkart and José M. Pérez-Izquierdo
Journal: Trans. Amer. Math. Soc. 352 (2000), 935-968
MSC (1991): Primary 17A75, 17B37, 81R50
Published electronically: August 10, 1999
MathSciNet review: 1637137
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Abstract: Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_{q}$(D$_{4}$) of D$_{4}$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_{q}$(D$_{4}$). The product in the quantum octonions is a $U_{q}$(D$_{4}$)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation theory'' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.

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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

José M. Pérez-Izquierdo
Affiliation: Departamento de Matematicas, Universidad de la Rioja, 26004 Logroño, Spain

Received by editor(s): November 28, 1997
Published electronically: August 10, 1999
Additional Notes: The first author gratefully acknowledges support from National Science Foundation Grant #DMS–9622447.\endgraf The second author is grateful for support from the Programa de Formación del Personal Investigador en el Extranjero and from Pb 94-1311-C03-03, DGICYT. \endgraf Both authors acknowledge with gratitude the support and hospitality of the Mathematical Sciences Research Institute, Berkeley.
Dedicated: To the memory of Alberto Izquierdo
Article copyright: © Copyright 1999 American Mathematical Society