Collineation groups of octonionic and split-octonionic planes

Abstract

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley–Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type [Formula: see text] and [Formula: see text].