Percorrer por autor "Marrani, Alessio"
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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 ) .
- A minimal and non-alternative realisation of the Cayley planePublication . Marrani, Alessio; Zucconi, Francesco; Corradetti, DanieleThe compact 16-dimensional Moufang plane, also known as the Cayley plane, has traditionally been defined through the lens of octonionic geometry. In this study, we present a novel approach, demonstrating that the Cayley plane can be defined in an equally clean, straightforward and more economic way using two different division and composition algebras: the paraoctonions and the Okubo algebra. The result is quite surprising since paraoctonions and Okubo algebra possess a weaker algebraic structure than the octonions, since they are non-alternative and do not satisfy the Moufang identities. Intriguingly, the real Okubo algebra has SU (3) as automorphism group, which is a classical Lie group, while octonions and paraoctonions have an exceptional Lie group of type G2. This is remarkable, given that the projective plane defined over the real Okubo algebra is nevertheless isomorphic and isometric to the octonionic projective plane which is at the very heart of the geometric realisations of all types of exceptional Lie groups. Despite its historical ties with octonionic geometry, our research underscores the real Okubo algebra as the weakest algebraic structure allowing the definition of the compact 16-dimensional Moufang plane.
- Physics with non-unital algebras? an invitation to the okubo algebraPublication . Marrani, Alessio; Corradetti, Daniele; Zucconi, FrancescoThis paper presents some preliminary discussion on the possible relevance of the Okubonions, i.e. the real Okubo algebra O, in quantum chromodynamics (QCD). The Okubo algebra lacks a unit element and sits in the adjoint representation of its automorphism group SUO, thus being fundamentally different from the better-known octonions O. While these latter may represent quarks (and color singlets), the Okubonions are conjectured to represent the gluons, i.e. the gauge bosons of the QCD SU(3) color symmetry. However, it is shown that the SU(3) groups pertaining to Okubonions and octonions are distinct and inequivalent subgroups of Spin(8) that share no common SU(2) subgroup. The unusual properties of Okubonions may be related to peculiar QCD phenomena like asymptotic freedom and color confinement, though the actual mechanisms remain to be investigated.