Jordan Geometries - an Approach by Inversions

TL;DR

This paper introduces Jordan geometries defined via point reflections depending on triples of points, explores their relation to symmetric spaces, and connects them to algebraic structures like Jordan pairs and algebras, over general base rings.

Contribution

It develops a new approach to Jordan geometries using inversions, linking them to symmetric spaces and algebraic structures, applicable over arbitrary base rings including integers.

Findings

Jordan geometries are linked to symmetric spaces through symmetry actions.

A tangent object called a Jordan pair or algebra is attached to these geometries.

The approach generalizes classical differential calculus to base rings where 2 is not invertible.

Abstract

Jordan geometries are defined as spaces equipped with point reflections depending on triples of points, exchanging two of the points and fixing the third. In a similar way, symmetric spaces have been defined by Loos (Symmetric Spaces I, 1969) as spaces equipped with point reflections depending on a point and fixing this point; 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 symmetry actions of torsors and of symmetric spaces is introduced. Jordan geometries give rise both to symmetry 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 (see arxiv:1402.2619), 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 the integers), 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 developped by M. Kinyon and the author.