Displaying similar documents to “A symplectic representation of E 7 ”

Weyl algebra and a realization of the unitary symmetry

Strasburger, Aleksander

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In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the d -dimensional quantum...

Relation of the spectra of symplectic Rarita-Schwinger and Dirac operators on flat symplectic manifolds

Svatopluk Krýsl (2007)

Archivum Mathematicum

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Consider a flat symplectic manifold ( M 2 l , ω ) , l ≥ 2 , admitting a metaplectic structure. We prove that the symplectic twistor operator maps the eigenvectors of the symplectic Dirac operator, that are not symplectic Killing spinors, to the eigenvectors of the symplectic Rarita-Schwinger operator. If λ is an eigenvalue of the symplectic Dirac operator such that - ı l λ is not a symplectic Killing number, then l - 1 l λ is an eigenvalue of the symplectic Rarita-Schwinger operator.

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...

On compact homogeneous symplectic manifolds

P. B. Zwart, William M. Boothby (1980)

Annales de l'institut Fourier

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In this paper the authors study compact homogeneous spaces G / K (of a Lie group G ) on which there if defined a G -invariant symplectic form Ω . It is an important feature of the paper that very little is assumed concerning G and K . The essential assumptions are: (1) G is connected and (2) K is uniform (i.e., G / K is compact). Further, for convenience only and with no loss of generality, it is supposed that G is simply connected and K contains no connected normal subgroup of G , i.e., that G acts...

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 .