Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces

Details Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces

2004-01-19

arxiv.org/abs/math/0401236v1

Mathematics

Group Theory

Scientific paper

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, 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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