Conjugation Matters. Bioctonionic Veronese Vectors and Cayley-Rosenfeld Planes

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 $\left(\mathbb{C}\otimes \mathbb{O}\right)P^2$, 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.

arXiv comment1+18 pages, 3 figures, 2 tables

References in corpus (14)

• Locally supersymmetric D=3 non-linear sigma models
• On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds
• A magic pyramid of supergravities
• Graviweak Unification
• Gravitational origin of the weak interaction's chirality
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• SO(3) family symmetry and axions
• The standard model, the Pati-Salam model, and "Jordan geometry"
• Subgroups of Clifford algebras
• SO(9) characterisation of the Standard Model gauge group
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• M-theory Cosmology, Octonions, Error Correcting Codes
• Graviweak Unification, Invisible Universe and Dark Energy
• A Kind of Magic

Cited by in corpus (1)

• A Geometrical Interpretation of Okubo Spin Group