Isotopes of octonion algebras, G2-torsors and triality

Hopf Algebras with Triality

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and ...

Octonion X,Y-Product G2 Variants

The automorphism group G2 of the octonions changes when octonion X,Y-product variants are used. I present here a general solution for how to go from G2 to its X,Y-product variant. * Happy Birthday to Larry Horwitz. If there’s a next time you must come.

Lie Algebras with Triality

Some Properties of Octonion and Quaternion Algebras

In 1988, J.R. Faulkner has given a procedure to construct an octonion algebra on a finite dimensional unitary alternative algebra of degree three over a field K. Here we use a similar procedure to get a quaternion algebra. Then we obtain some conditions for these octonion and quaternion algebras to be split or division algebras. Then we consider the implications of the found conditions to the u...

Sums of Squares in Certain Quaternion and Octonion Algebras

Formulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented. Let R be a not necessarily associative ring with unity. The level and sublevel of R, respectively denoted by s(R) and s(R), are defined as follows: s(R) = in...