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- Conjugation matters. Bioctonionic veronese vectors and cayley-rosenfeld planesPublication . Corradetti, Daniele; Marrani, Alessio; Chester, David; Aschheim, RayMotivated 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 OP2C, we present a novel, explicit construction of the bioctonionic Cayley–Rosenfeld plane (C⊗O)P2, 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 F4 and E6, and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.
- Dixon-Rosenfeld lines and the standard modelPublication . Chester, David; Marrani, Alessio; Corradetti, Daniele; Aschheim, Raymond; Irwin, KleeWe present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra C⊗H⊗O. Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal–Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within C ⊗ H ⊗ J2(O), O ⊗ J2(C ⊗ H), and C ⊗ O ⊗ J2(H). $$\mathbb {C}\otimes \mathbb {H}\otimes \mathbb {O}$$ C ⊗ H ⊗ O . Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal–Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within $$\mathbb {C}\otimes \mathbb {H}\otimes J_{2}(\mathbb {O})$$ C ⊗ H ⊗ J 2 ( O ) , $$\mathbb {O}\otimes J_{2}(\mathbb {C}\otimes \mathbb {H})$$ O ⊗ J 2 ( C ⊗ H ) , and $$\mathbb {C}\otimes \mathbb {O}\otimes J_{2}(\mathbb {H})$$ C ⊗ O ⊗ J 2 ( H ) .