Chain geometry

An incidence system constructed by means of an algebra. Originally (1842), Ch. von Staudt defined, on the projective line/plane over , a chain ( "Kette" ) in a synthetic way as a subline/plane over . Much later, in 1973, W. Benz [a1] gave a common frame for rather different phenomena (the geometries of Möbius, Laguerre and Lie, and Minkowsky; cf. Benz plane) using the concept of a geometry over an algebra. A recent survey of the development of this theory is [a3].

Fundamental concepts.

Let be a ring (associative with one) and let be its group of units. To define the projective line over , one introduces an equivalence relation on , as for the projective straight line over a skew-field:

Let denote the equivalence class of . Then

One says that is distant to if

Let be a commutative ring and a -algebra, where is imbedded in the centre of via the mapping . Regard as a subset of , and define . Then the incidence system , with as incidence relation, is called a chain geometry. The elements of are called chains. Any three pairwise distant points belong to exactly one chain. is a group of automorphisms of ; it is transitive on the set of triples of pairwise distant points, and hence transitive on the set of chains. Four pairwise distant points belong to a common chain, provided their cross ratio belongs to .

If is a field, is said to be a proper chain geometry. A proper chain geometry is a chain space.

Below, denotes a field.

Affine case.

Let . Then is the set of all points of distant to . Consider the traces of the chains in :

There is a natural bijection from into the affine space over via the mapping . Under this mapping, becomes the set

of affine chains, defined by

This trace geometry is called the affine chain geometry, denoted by . For , the set is a bundle of parallel straight lines in the affine space . For an algebraic (especially, finite-dimensional) -algebra (cf. also Algebraic algebra), the affine chain geometry is an affine chain space (cf. Chain space).

Structure of morphisms.

Let , be -algebras. A -Jordan homomorphism is a -semi-linear mapping satisfying: i) ; and ii) for all one has . For a "strong" algebra (strongness guarantees a great richness in units for ), any point of can be written as for suitable . Then a -Jordan homomorphism induces a well-defined mapping , , which preserves pairs of distant points and maps chains to chains. Moreover, under : , , . Such a mapping is called a fundamental morphism from to .

Conversely, any fundamental morphism having more than one chain in its image can be obtained in this manner (see [a2] for a more general context).

Let be the group of -Jordan automorphisms of , and let be the group of fundamental automorphisms of . Then and .

Rational representations.

As in the affine case one tries to find kinds of representations for chain geometries on a part of a projective space where the chains become curves, at least when is finite dimensional. In this way one has discovered incidence systems isomorphic to , where is a projective variety (cf. also Projective scheme), is a (Zariski-) closed subset of and consists of rational curves on (cf. also Incidence system; Rational curve).

For a -algebra of -dimension , a representation of on a part of the Grassmann manifold is obtained as follows. For , the set is an -dimensional subspace of the -vector space of dimension . Then is mapped to a point of (see Exterior algebra). By this procedure, chains are mapped to normal rational curves of order , i.e., to images of under the Veronese mapping , and is the intersection of with a linear subspace. Other examples can be obtained from this by suitable projection.

A quadratic algebra (i.e., any element of has a quadratic minimal polynomial; cf. also Extension of a field) has a representation as a quadric model (cf. Chain space). Here, is the quadric and is its set of singular points; the chains are conics.

-chain geometries.

These are generalizations of chain geometries () to higher dimensions. E.g., let be a quadratic field extension of (cf. Extension of a field). Then, in the projective plane over the -chains are the subplanes over ; these are better known as Baer subplanes, especially in finite geometries (cf. also Geometry).

A Burau geometry is a projective space over (again a quadratic extension of ) together with all projective sublines over . It can be characterized by the property that the incidence system consisting of a projective line over (as a point set) and all sublines over contained in (considered as blocks) for a Möbius plane. A more general concept can be found in [a4].

References

[a1]	W. Benz, "Vorlesungen über Geometrie der Algebren" , Springer (1973)
[a2]	A. Blunk, "Chain spaces over Jordan systems" Abh. Math. Sem. Hamburg , 64 (1994) pp. 33–49
[a3]	A. Herzer, "Chain geometries" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)
[a4]	K. Pieconkowski, "Projektive Räume über Schiefkörperpaaren" , W und T, Wiss.- und Technik-Verl. (1994)