Composition Algebras

Elduque, Alberto

doi:10.1002/9781119818175.ch2

Cited by 9 publications

(6 citation statements)

References 29 publications

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“…Nevertheless, the idempotent e gives rise to an automorphism of order two, i.e. an involution, x −→ x, e e − x and an automorphism of order three x −→ τ (x) = x, e e − x * e, whose geometrical meaning was partially investigated in [El18].…”

Section: Okubo Algebras and Octonionsmentioning

confidence: 99%

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A Geometrical Interpretation of Okubo Spin Group

Corradetti,

Zucconi

2022

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In 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.

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Section: Okubo Algebras and Octonionsmentioning

confidence: 99%

“…On the other side, in [El18] Elduque gave a description and a definition of the Spin group over Okubo algebras Spin (O) and their triality algebras tri (O). Here we will give a geometric interpretation of Spin (O), starting from the definition of the Okubic affine plane.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

A Geometrical Interpretation of Okubo Spin Group

Corradetti,

Zucconi

2022

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“…In particular their dimension is restricted to 1, 2, 4 or 8. The reader may consult [ZSSS82, Chapter 2], [KMRT98, Chapter VIII], or the survey paper [Eld21].…”

Section: From Octonions To Composition Superalgebrasmentioning

confidence: 99%

From octonions to composition superalgebras via tensor categories

Daza-Garcia¹,

Elduque²,

Sayın³

2022

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The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too.In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

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“…all Hurwitz algebras such as R, C, H,O and all their split companions, allow the construction of projective lines and planes which found interesting and useful applications in alternative formulations of Quantum Mechanics [JNW]. Inspired by an algebrical definition of the Spin group, given by Elduque in [El18], in a previous work two of the authors defined an affine and projective plane over the Okubo algebra [CZ], noting that, with little modifications, Veronese coordinates could still be used even though Okubo algebra is neither alternative nor unital (a review of this construction is done in sec. 4).…”

Section: Introductionmentioning

confidence: 99%

“…In section 2 we present symmetric composition algebras and Okubo algebras, mainly following [El08,El18,El15] but introducing also some new elements. In section 3 we present for the first time the Okubic projective line showing that is the one point compactification of the Okubo algebra O.…”

Section: Introductionmentioning

confidence: 99%