Conjugation Matters. Bioctonionic Veronese Vectors and Cayley-Rosenfeld Planes
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Publication:6390117
DOI10.1142/S0219887822501420arXiv2202.02050WikidataQ114072311 ScholiaQ114072311MaRDI QIDQ6390117
Alessio Marrani, Daniele Corradetti, David Chester, Ray Aschheim
Publication date: 4 February 2022
Abstract: Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane $mathbb{O}P_{mathbb{C}}^{2}$, we present a novel, explicit construction of the bioctonionic Cayley-Rosenfeld plane $left( mathbb{C}otimes mathbb{O}
ight) P^{2}$, again by exploiting Veronese coordinates. We discuss the isometry groups of all generalized bioctonionic planes, recovering all complex and real forms of the exceptional groups $F_{4}$ and $E_{6}$, and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.
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