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

DOI:
https://doi.org/10.1090/S0002-9947-99-02415-0

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 (D) of D, 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 (D). The product in the quantum octonions is a (D)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at 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

Email:
benkart@math.wisc.edu

**José M. Pérez-Izquierdo**

Affiliation:
Departamento de Matematicas, Universidad de la Rioja, 26004 Logroño, Spain

Email:
jm.perez@dmc.unirioja.es

DOI:
https://doi.org/10.1090/S0002-9947-99-02415-0

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