Structures algébriques sur les espaces symétriques

Abstract : ln a first part, we describe a method for associating to a Lie algebra over a ring a polynomial group. ln the case where this Lie algebra comes from a Lie group, the polynomial group corresponds to the n-jet of the Lie group. Then we extend this construction to the case of Lie triple systems and associate to each Lie triple system over a ring a polynomial symmetric space. The second part of this thesis is devoted to the study of a geometric interpretation of the notion of Lie triple system representation, in the sense of the module concept, introduced by S. Eilenberg [10]. We establish that the geometric object naturally associated to a representation of a Lie triple system coming from a symmetric space is a vector bundle over this symmetric space, carrying a symmetric space structure, compatible with the one of the base. Such an object is called a symmetric bundle. A similar correspondence between Lie triple system representations and polynomial symmetric bundles is also pointed out in the general case.
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Manon Didry. Structures algébriques sur les espaces symétriques. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2006. Français. ⟨NNT : 2006NAN10055⟩. ⟨tel-01754319⟩

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