A quantum octonion algebra
Using the natural irreducible 8-dimensional representation and the two
spin representations of the quantum group Uq(D4)
of D4, 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 contruction, the quantum octonion algebra is a nonassociative
algebra with a Yang-Baxter operator action coming from the R-matrix of
Uq(D4). The product in the quantum octonions is a
Uq(D4) module homomorphism. Using that, we prove
identities for the quantum octonions, and 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.
(This paper has appeared in Trans. Amer. Math. Soc. 352 (2000), 935-968)
G. Benkart <benkart@math.wisc.edu>
J. M. Perez-Izquierdo <>