alexa Symmetric bundles and representations of Lie triple sy
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Review Article

Symmetric bundles and representations of Lie triple systems

Wolfgang BERTRAM and Manon DIDRY

Institut Élie Cartan de Nancy, Universite Nancy, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, F-54506 Vanduvre-les-Nancy, France

*Corresponding Author:
Institut Élie Cartan de Nancy
UniversitÉe Nancy, CNRS, INRIA
Boulevard des Aiguillettes
B.P. 239, F-54506 Vandeauvre-lÉes-Nancy, France

Received Date: January 9, 2009; Revised Date: May 07, 2009

 

Abstract

We de ne symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying re ection space, and we investigate the corresponding forgetful functor both from the point of view of di erential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing \unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.

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