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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

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 $E_{8}$ .

A combinatorial construction of $G_{2}$ .

Wildberger, N.J. (2003)

Journal of Lie Theory

A symplectic representation of $E_{7}$

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

Commentationes Mathematicae Universitatis Carolinae

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.

An analogue of the Weil representation for G2.

Gordan Savin (1993)

Journal für die reine und angewandte Mathematik

Compression of root systems and the $E$ -sequence.

Purbhoo, Kevin (2008)

The Electronic Journal of Combinatorics [electronic only]

Conjugacy classes in the weyl group

R. W. Carter (1972)

Compositio Mathematica

Corrections for “The closure diagram for nilpotent orbits of the split real form of E 8”

Dragomir Đoković (2005)

Open Mathematics

Éléments de la géométrie des octaves de Cayley

Claude Brada (1986)

Publications du Département de mathématiques (Lyon)

Exceptional and orthosymplectic graded Lie algebras

L. Trlifaj (1978)

Annales de l'I.H.P. Physique théorique

Homomorphisms of semiholonomic Verma modules: An exceptional case.

Sawon, J. (1999)

Acta Mathematica Universitatis Comenianae. New Series

How to find the top rootform of a complex simple Lie algebra.

C.J. Henrich (1974)

Journal für die reine und angewandte Mathematik

Nonsplit extensions of modular Lie algebras of rank 2.

Dzhumadil'daev, A.S., Ibraev, S.S. (2002)

Homology, Homotopy and Applications

Propriétés de dualité dans les représentations coinduites de superalgèbres de Lie

Sophie Chemla (1994)

Annales de l'institut Fourier

Nous généralisons un résultat de dualité dans les représentations coinduites établi par M. Duflo (dans [Du]) dans le cas des algèbres de Lie de dimension finie. La démonstration que nous en proposons utilise la superalgèbre des opérateurs différentiels sur le module coinduit ainsi que la correspondance, mise en évidence par J. Bernstein, entre $D$ -modules à droite et $D$ -modules à gauche. Elle n’est valable qu’en caractéristique zéro. Nous donnons aussi une interprétation de ce théorème en termes de...

Quantification pour les paires symétriques et diagrammes de Kontsevich

Alberto S. Cattaneo, Charles Torossian (2008)

Annales scientifiques de l'École Normale Supérieure

In this article we use the expansion for biquantization described in [7] for the case of symmetric spaces. We introduce a function of two variables $E (X, Y)$ for any symmetric pairs. This function has an expansion in terms of Kontsevich’s diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this $E$ function. We prove that Cattaneo and Felder’s star product coincides with Rouvière’s for any symmetric pairs. We generalize some of Lichnerowicz’s...

$R$ -matrice universelle pour $U_{h} (D (2, 1, x))$ et invariant d’entrelacs associé

Henrik Thys (2002)

Bulletin de la Société Mathématique de France

En utilisant la méthode du double quantique, nous construisons une $R$ -matrice universelle pour la quantification de la superalgèbre de Lie $D (2, 1, x)$ . Nous utilisons ce résultat pour construire un invariant d’entrelacs et nous montrons qu’il est égal à une spécialisation du polynôme de Dubrovnik introduit par Kauffman.

Rotations in the space of Split octonions.

Gogberashvili, Merab (2009)

Advances in Mathematical Physics

S4-symmetry on the Tits construction of exceptional Lie algebras and superalgebras

A. Elduque, S. Okubo (2008)

Publicacions Matemàtiques

Semiholonomic jets and induced modules in Cartan geometry calculus

Jan Slovák, Vladimír Souček (2024)

Archivum Mathematicum

The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein’s models. The aim of this short survey is to explain carefully the basic...

Series of nilpotent orbits.

Landsberg, J.M., Manivel, Laurent, Westbury, Bruce W. (2004)

Experimental Mathematics

The closure diagram for nilpotent orbits of the split real form of E8

Dragomir Đoković (2003)

Open Mathematics

Let $𝒪_{1}$ and $𝒪_{2}$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $𝒪_{1}$ ≥ $𝒪_{2}$ if $𝒪_{2}$ is contained in the closure of $𝒪_{1}$ . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible...

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