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 .

Contact and conformal maps on Iwasawa N groups

Michael Cowling, Filippo De Mari, Adam Korányi, Hans Martin Reimann (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The action of the conformal group O ⁢ 1 , n + 1 on R n ∪ ∞ may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a C 4 map between domains U and V in R n whose differential is a (variable) multiple of a (variable) isometry at each point of U is the restriction to U of a transformation x → g ⋅ x , for some g in O ⁢ 1 , n + 1 . In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G / P , where P is a parabolic subgroup....

Equivalence bimodule between non-commutative tori

Sei-Qwon Oh, Chun-Gil Park (2003)

Czechoslovak Mathematical Journal

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The non-commutative torus C * ( ℤ n , ω ) is realized as the C * -algebra of sections of a locally trivial C * -algebra bundle over S ω ^ with fibres isomorphic to C * ( ℤ n / S ω , ω 1 ) for a totally skew multiplier ω 1 on ℤ n / S ω . D. Poguntke [9] proved that A ω is stably isomorphic to C ( S ω ^ ) ⊗ C * ( ℤ n / S ω , ω 1 ) ≅ C ( S ω ^ ) ⊗ A ϕ ⊗ M k l ( ℂ ) for a simple non-commutative torus A ϕ and an integer k l . It is well-known that a stable isomorphism of two separable C * -algebras is equivalent to the existence of equivalence bimodule between them. We construct an A ω - C ( S ω ^ ) ⊗ A ϕ -equivalence bimodule.