Hyperplane section ${\mathbb {O}\mathbb {P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm {F_4/P_4}$

Karel Pazourek; Vít Tuček; Peter Franek

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 $3 \times 3$ octonionic matrices, generalizing previous results in the $2 \times 2$ 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 $n$ ....

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

A symplectic representation of E 7

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

Spin representations and binary numbers

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 $E_{6}$

A symplectic representation of $E_{7}$

A $ℤ_{4}^{3}$ -grading on a $56$ -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$