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A quantum octonion algebra

Authors:	Georgia Benkart José M. Pé rez-Izquierdo

Affiliation:	Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 ; Departamento de Matematicas, Universidad de la Rioja, 26004 Logroño, Spain

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 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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