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arXiv · 2311.11907

Collineation groups of octonionic and split-octonionic planes

Daniele Corradetti·Alessio Marrani·Francesco Zucconi
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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 $G_{2},F_{4}$ and $E_{6}$.

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Daniele Corradetti, Alessio Marrani, Francesco Zucconi. 2023-11-20. Collineation groups of octonionic and split-octonionic planes. https://doi.org/10.1142/s0129055x24500272

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