Clifford parallelisms defined by octonions

Andrea Blunck , Universität Hamburg

Clifford's classical parallelims in real projective 3-space can be described using the multiplicative group of Hamilton's quaternions. This can be generalized to real 7-space using the 8-dimensional alternative division algebra of octonions (Cayley numbers) instead, even though then the multiplicative structure is no longer a group but only a loop. We show that in this case the parallel classes are regular spreads that are indicated by totally isotropic 3-spaces on a hyperbolic quadric in complex 7-space.

Moreover, we make some remarks on how the situation changes over an arbitrary ground field.