Percorrer por autor "Zucconi, Francesco"
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- A geometrical interpretation of Okubo spin groupPublication . Corradetti, Daniele; Zucconi, FrancescoIn this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin(O) as the group of collineations that preserve the axis of the plane.
- 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.