A quantum octonion algebra

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 quant um analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality an d has a nondegenerate bilinear form which satisfies a q-version of the comp osition property. By its construction, the quantum octonion algebra is a no nassociative algebra with a Yang-Baxter operator action coming from the R-m atrix of U-q(D-4). The product in the quantum octonions is a U-q(D-4)-modul e homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at q = 1 new "representation theory" proofs fo r very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-ana logue of the 8-dimensional para-Hurwitz algebra.