Browsing by Author "Chester, David"
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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 ) .
- Three Fibonacci-Chain aperiodic algebrasPublication . Corradetti, Daniele; Chester, David; Aschheim, Raymond; Irwin, KleeAperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera and Twarock we present three aperiodic algebras based on Fibonacci-chain quasicrystals: a quasicrystal Lie algebra, an aperiodic Witt algebra and, finally, an aperiodic Jordan algebra. While a quasicrystal Lie algebra was already constructed from a modification of the Fibonacci chain, we here present an aperiodic algebra that matches exactly the original quasicrystal. Moreover, this is the first time to our knowledge, that an aperiodic Jordan algebra is presented leaving room for both theoretical and applicative developments.