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

Abstract

Abstract. 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.Contents.

Divisible designs, Laguerre geometry, and beyond

“…alternative algebras or Jordan systems) in order to obtain a kind of "chain geometry". We refer to [11], [12], [18], [19], [20], [21], [28], [40,Chapter 3], [41], and [68].…”

On the Hermitian Projective Line as a Home for the Geometry of Quantum Theory

“…Then the involution * lifts to an involution of the projective line AP 1 whose fixed point set is called the Hermitian projective line, see [BeNe05]. Let us give here a slightly modified version of the construction given in loc.…”

On Complex Infinite-Dimensional Grassmann Manifolds

“…Moreover, it is also shown that G A /G A (p) is a complexification of U A /U A (p), and then these results are translated in terms of homogeneous vector bundles, in the spirit of Theorem 5.8 in [5]. (A complementary perspective on these manifolds can be found in [6].) Section 4 is devoted to exhibit how the above results look like in the (on the other hand well known) case of the algebra A = B(H) and corresponding universal, tautological vector bundles.…”