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) |
Chain geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_geometry&oldid=46301