Displaying similar documents to “Octonionic Cayley spinors and E 6 ”

Hyperplane section 𝕆 ℙ 0 2 of the complex Cayley plane as the homogeneous space F 4 / P 4

Karel Pazourek, Vít Tuček, Peter Franek (2011)

Commentationes Mathematicae Universitatis Carolinae

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We prove that the exceptional complex Lie group F 4 has a transitive action on the hyperplane section of the complex Cayley plane 𝕆 ℙ 2 . Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of Spin ( 9 , ℂ ) ≤ F 4 . Moreover, we identify the stabilizer of the F 4 -action as a parabolic subgroup P 4 (with Levi factor B 3 T 1 ) of the complex Lie group F 4 . In the real case we obtain an analogous realization of F 4 ( - 20 ) / ℙ .

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.

Quadro-quadric Cremona transformations in low dimensions via the  J C -correspondence

Luc Pirio, Francesco Russo (2014)

Annales de l’institut Fourier

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It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

A matrix formalism for conjugacies of higher-dimensional shifts of finite type

Michael Schraudner (2008)

Colloquium Mathematicae

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We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ℤ⁺-matrices. Using the decomposition theorem every topological conjugacy between two ℤ d -shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matrices of the two subshifts. Our results may be used algorithmically in computer explorations on...