Jordan Geometries by Inversions
The present work is closely related to Preprint 90 on this server,
where I defined a class of geometries corresponding to Jordan structures
(pairs, triple systems and algebras). The new approach given here is simpler
and has the advantage to work also over base rings in which the scalar 2 is not
invertible, and to have a very transparent relation with geometries related to
associative structures (pairs and algebras, see preprints 271 and 276 on this
server). To be more specific, 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 the preprint "Weil spaces and
Weil-Lie groups")
we attach a tangent object to such
geometries, namely a Jordan pair, respectively a Jordan algebra. Conversely,
every Jordan pair or algebra gives rise to a Jordan geometry; thus we have a
Jordan analog of the Lie functor from Lie theory. However, in contrast to Lie
theory, the Jordan functor is defined in a purely algebraic way, and therefore
it exists in the very general context of Jordan structures over rings.
(Revised version 17 Feb 2014, to appear in J. Lie Theory under the title
"Jordan Geometries - an Approach by Inversions")
Wolfgang Bertram < wolfgang.bertram@univ-lorraine.fr >