Projective completions of Jordan pairs. Part I: The generalized
projective geometry of a Lie algebra
This is the first part in a series of two papers in
which we construct the projective completion of certain Jordan pairs,
possibly infinite dimensional and defined over topological fields
or even over certain topological rings. This first part is purely
algebraic and applies to any Jordan pair: we describe a realization
of the projective completion of a Jordan pair as a space of
3-filtrations of the associated Kantor-Koecher-Tits algebra.
Our structure theorem (Theorem 1.6) asserts that the
remoteness relation on the projective completion corresponds
to the natural relation of transversality of flags and that
the set of flags transversal to a given one carries a natural
structure of an affine space. Based on this theorem, a
structure theory of the projective completion is developed:
we can define, in a purely algebraic way,
tangent bundles and vector fields, leading finally to
a description of the action of the automorphism group by
fractional quadratic maps (Theorem 2.8).
This will be used in the second part of this work in order to define
a manifold structure on the projective completion,
where we will use suitable concepts of differential calculus
and manifolds over general topological fields and rings
developed in joint work with H. Gloeckner
(W. Bertram, H. Gloeckner, K.-H. Neeb, "Differential
Calculus, Manifolds and Lie Groups over arbitrary
infinite Fields", see arXive: math.GM/0303300).
(This paper has appeared in J. Algebra 227 (2004), 474--519)
W. Bertram < bertram@iecn.u-nancy.fr >
K.-H. Neeb < neeb@mathematik.tu-darmstadt.de >