Generalized projective geometries. Part I: From linear algebra via affine algebra to projective algebra
We introduce an algebraic formalism, called "affine algebra",
which corresponds
to affine geometry over a field or ring K in a similar
way as linear algebra corresponds to affine geometry with
respect to a fixed base point. In a second step,
we describe projective geometry over K by a similar
formalism, called "projective algebra".
We observe that this formalism not only applies to ordinary
projective geometry, but also to several other geometries
such as, e.g., Grassmannian geometry, Lagrangian geometry
and conformal geometry. These are examples of ``generalized
projective geometries".
The axiomatic definition and
general theory of such geometries is given in Part II
of this work.
W. Bertram <bertram@iecn.u-nancy.fr>