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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A quantum octonion algebra
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by Georgia Benkart and José M. Pérez-Izquierdo PDF
Trans. Amer. Math. Soc. 352 (2000), 935-968 Request permission

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.
References
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Additional Information
  • Georgia Benkart
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 34650
  • 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
  • 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. 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. Both authors acknowledge with gratitude the support and hospitality of the Mathematical Sciences Research Institute, Berkeley.

  • Dedicated: To the memory of Alberto Izquierdo
  • © Copyright 1999 American Mathematical Society
  • 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
  • MathSciNet review: 1637137