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- 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.
- 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 ) .