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 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 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 (see [Adv. Geom. 2 (2002) 329] for the axiomatic definition and general theory). The corresponding generalized polar geometries give rise to certain “symmetric spaces over ” generalizing the symmetric spaces known from the real case; we give here some important examples of this construction.