Comptes Rendus
no. 1-2
Lie Algebras
Generalized flag geometries and manifolds associated to short Z-graded Lie algebras in arbitrary dimension
Presented by: Michèle Vergne
Julien Chenal1
1 Institut Elie-Cartan Nancy (IECN), Nancy-Université, CNRS, INRIA, boulevard des Aiguillettes, B.P. 239, 54506 Vandoeuvre-lès-Nancy, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 21-25

The object of this Note is to define the generalized flag geometry of a graded Lie algebra which corresponds to the generalized projective geometry in the case of 3-gradings. Then we construct a structure of manifold on this generalized flag geometry. This result generalizes a result known for 3-graded Lie algebras to the more general case of (2k+1)-graded Lie algebras.

L'objet de cette Note est de définir la géométrie de drapeaux généralisée d'une algèbre de Lie graduée, qui correspond à la géométrie projective généralisée dans le cas des 3-graduations, puis de construire une structure de variété différentielle sur cette géométrie. Ce résultat généralise au cas des (2k+1)-graduations un résultat déjà connu pour les 3-graduations.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.12.001

Julien Chenal 1

1 Institut Elie-Cartan Nancy (IECN), Nancy-Université, CNRS, INRIA, boulevard des Aiguillettes, B.P. 239, 54506 Vandoeuvre-lès-Nancy, France 
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Julien Chenal. Generalized flag geometries and manifolds associated to short $ \mathbb{Z}$-graded Lie algebras in arbitrary dimension. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 21-25. doi: 10.1016/j.crma.2008.12.001

[1] W. Bertram Differential geometry, Lie groups and symmetric spaces over general base fields and rings, Mem. Amer. Math. Soc., Volume 192 (2008) no. 900, p. ix+202

[2] W. Bertram; K.-H. Neeb Projective completions of Jordan pairs. Part I: The generalized projective geometry of a Lie algebra, J. Algebra, Volume 277 (2004), pp. 474-519

[3] W. Bertram; K.-H. Neeb Projective completions of Jordan pairs. Part II: Manifold structure and symmetric spaces, Geom. Dedicata, Volume 112 (2005), pp. 73-113

[4] S. Kaneyuki; H. Asano Graded Lie algebras and generalized Jordan triple systems, Nagoya Math. J., Volume 112 (1988), pp. 81-115

[5] O. Loos Jordan Pairs, Lecture Notes in Math., vol. 460, Springer, Berlin, 1975

[6] O. Loos Elementary groups and stability for Jordan pairs, K-Theory, Volume 9 (1995), pp. 77-116

[7] H. Upmeier Symmetric Banach Manifolds and Jordan C*-algebras, North-Holland Math. Stud., vol. 104, 1985

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