A QUANTUM OCTONION ALGEBRA

GEORGIA BENKART JOSÉ M. PÉREZ-IZQUIERDO

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.