Displaying similar documents to “Hyperplane section 𝕆 ℙ 0 2 of the complex Cayley plane as the homogeneous space F 4 / P 4 ”

Octonionic Cayley spinors and E 6

Tevian Dray, Corinne A. Manogue (2010)

Commentationes Mathematicae Universitatis Carolinae

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Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group E 6 , and of its subgroups. We are therefore led to a description of E 6 in terms of 3 × 3 octonionic matrices, generalizing previous results in the 2 × 2 case. Our treatment naturally includes a description of several important subgroups of E 6 , notably G 2 , F 4 , and (the double cover of) S O ( 9 , 1 ) . An interpretation of...

A symplectic representation of E 7

Tevian Dray, Corinne A. Manogue, Robert A. Wilson (2014)

Commentationes Mathematicae Universitatis Carolinae

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We explicitly construct a particular real form of the Lie algebra 𝔢 7 in terms of symplectic matrices over the octonions, thus justifying the identifications 𝔢 7 ≅ 𝔰𝔭 ( 6 , 𝕆 ) and, at the group level, E 7 ≅ Sp ( 6 , 𝕆 ) . Along the way, we provide a geometric description of the minimal representation of 𝔢 7 in terms of rank 3 objects called cubies.

A ℤ 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna, Alberto Elduque, Mikhail Kochetov (2014)

Commentationes Mathematicae Universitatis Carolinae

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We describe two constructions of a certain ℤ 4 3 -grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1 ) over an algebraically closed field of characteristic different from 2 . The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E 6 , E 7 and E 8 .

Spin representations and binary numbers

Henrik Winther (2024)

Archivum Mathematicum

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We consider a construction of the fundamental spin representations of the simple Lie algebras 𝔰𝔬 ( n ) in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a ℤ -graded associative algebra (rather than the usual ℕ -filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some n ....