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Projective Completions of Jordan Pairs, Part II: Manifold Structures and Symmetric Spaces

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Abstract

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields \(\mathbb{K}\), and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.

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Authors and Affiliations

  1. Institut Elie Cartan, Faculté des Sciences, Université Nancy I, BP 239, F-54506, Vandœ uvre-lès-Nancy Cedex, France

    Wolfgang Bertram

  2. Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289, Darmstadt, Germany

    Karl-Hermann Neeb

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  1. Wolfgang Bertram
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  2. Karl-Hermann Neeb
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Correspondence to Wolfgang Bertram.

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Mathematics Subject Classiffications (2000). primary: 17C36, 46H70, 17C65; secondary: 17C30, 17C90

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Bertram, W., Neeb, KH. Projective Completions of Jordan Pairs, Part II: Manifold Structures and Symmetric Spaces. Geom Dedicata 112, 73–113 (2005). https://doi.org/10.1007/s10711-004-4197-6

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