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Journal of Lie Theory 24 (2014), No. 4, 1067--1113
Copyright Heldermann Verlag 2014

Jordan Geometries -- an Approach Via Inversions

Wolfgang Bertram
Université de Lorraine, Institut Elie Cartan, B.P. 70239, 54506 Vandoeuvre-lès-Nancy, France

Jordan geometries are defined as spaces X equipped with point reflections Jaxz depending on triples of points (x,a,z), exchanging x and z and fixing a. In a similar way, symmetric spaces have been defined by O. Loos as spaces equipped with point reflections Sx fixing x, and therefore the theories of Jordan geometries and of symmetric spaces are closely related to each other. In order to describe this link, the notion of inversive action of torsors and of symmetric spaces is introduced. Jordan geometries give rise both to inversive actions of certain abelian torsors and of certain symmetric spaces, which in a sense are dual to each other. By using an algebraic differential calculus generalizing the classical Weil functors, we attach a tangent object to such geometries, namely a Jordan pair, respectively, a Jordan algebra. The present approach works equally well over base rings in which 2 is not invertible (and in particular over Z), and hence can be seen as a globalization of quadratic Jordan pairs; it also has a very transparent relation with the theory of associative geometries as developed by M. Kinyon and the author.

Keywords: Inversion, torsor, symmetric space, inversive action, generalized projective geometry, Jordan algebra, Jordan pair, associative algebra, Lie algebra, modular group.

MSC: 17C37, 16W10, 32M15, 51C05, 53C35

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