Chain geometry

An incidence system constructed by means of an algebra. Originally (1842), Ch. von Staudt defined, on the projective line/plane over $ \mathbf C $, a chain ( "Kette" ) in a synthetic way as a subline/plane over $ \mathbf R $. 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 $ A $ be a ring (associative with one) and let $ A ^ {*} $ be its group of units. To define the projective line $ \mathbf P ( A ) $ over $ A $, one introduces an equivalence relation $ \equiv $ on $ A \times A $, as for the projective straight line over a skew-field:

$$ ( a,b ) \equiv ( a ^ \prime ,b ^ \prime ) \iff \exists u \in A ^ {*} : a ^ \prime = ua \& b ^ \prime = ub. $$

Let $ [ a,b ] $ denote the equivalence class of $ ( a,b ) $. Then

$$ \mathbf P ( A ) = $$

$$ = \left \{ {[ a,b ] } : {a,b \in A, \exists x,y \in A: \left ( \begin{array}{cc} a & b \\ x & y \\ \end{array} \right ) \in { \mathop{\rm GL} } _ {2} ( A ) } \right \} . $$

One says that $ [ a,b ] $ is distant to $ [ c,d ] $ if

$$ \left ( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ) \in { \mathop{\rm GL} } _ {2} ( A ) . $$

Let $ K $ be a commutative ring and $ A $ a $ K $- algebra, where $ K $ is imbedded in the centre of $ A $ via the mapping $ k \mapsto k \cdot 1 $. Regard $ \mathbf P ( K ) $ as a subset of $ \mathbf P ( A ) $, and define $ \mathfrak C _ {K} ( A ) = \{ {\mathbf P ( K ) ^ \gamma } : {\gamma \in { \mathop{\rm GL} } _ {2} ( A ) } \} $. Then the incidence system $ \Sigma ( K,A ) = ( \mathbf P ( A ) , \mathfrak C _ {K} ( A ) ) $, with $ \in $ as incidence relation, is called a chain geometry. The elements of $ \mathfrak C _ {K} ( A ) $ are called chains. Any three pairwise distant points belong to exactly one chain. $ { \mathop{\rm PGL} } _ {2} ( A ) $ is a group of automorphisms of $ \Sigma ( K,A ) $; 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 $ K $.

If $ K $ is a field, $ \Sigma ( K,A ) $ is said to be a proper chain geometry. A proper chain geometry is a chain space.

Below, $ K $ denotes a field.

Affine case.

Let $ D = \{ {[ 1,x ] } : {x \in A } \} $. Then $ D $ is the set of all points of $ \mathbf P ( A ) $ distant to $ [ 0,1 ] $. Consider the traces of the chains in $ D $:

$$ {\widehat{\mathfrak C} } _ {K} ( A ) = \left \{ {C \cap D } : {C \in \mathfrak C _ {K} ( A ) , \left | {C \cap D } \right | \geq 3 } \right \} . $$

There is a natural bijection from $ D $ into the affine space $ A $ over $ K $ via the mapping $ [ 1,a ] \mapsto a $. Under this mapping, $ {\widehat{\mathfrak C} } _ {K} ( A ) $ becomes the set

$$ {\mathcal K} _ {K} ( A ) = \left \{ {\mathbf K ( a,b,c ) } : {a \in A ^ {*} , b,c \in A, \left | {\mathbf K ( a,b,c ) } \right | \geq 3 } \right \} $$

of affine chains, defined by

$$ \mathbf K ( a,b,c ) = \left \{ {( a t + b ) ^ {- 1 } + c } : {t \in K, a t + b \in A ^ {*} } \right \} \cup \{ c \} . $$

This trace geometry is called the affine chain geometry, denoted by $ {\mathcal A} ( K,A ) = ( A, {\mathcal K} _ {K} ( A ) ) $. For $ a \in A $, the set $ \{ {\mathbf K ( a,0,c ) } : {c \in A } \} $ is a bundle of parallel straight lines in the affine space $ A $. For an algebraic (especially, finite-dimensional) $ K $- algebra $ A $( cf. also Algebraic algebra), the affine chain geometry $ {\mathcal A} ( K,A ) $ is an affine chain space (cf. Chain space).

Structure of morphisms.

Let $ A $, $ A ^ \prime $ be $ K $- algebras. A $ K $- Jordan homomorphism $ \alpha : A \rightarrow {A ^ \prime } $ is a $ K $- semi-linear mapping satisfying: i) $ 1 ^ \alpha = 1 $; and ii) for all $ a,b \in A $ one has $ ( aba ) ^ \alpha = a ^ \alpha b ^ \alpha a ^ \alpha $. For a "strong" algebra $ A $( strongness guarantees a great richness in units for $ A $), any point of $ \mathbf P ( A ) $ can be written as $ [ 1 + ab,a ] $ for suitable $ a,b \in A $. Then a $ K $- Jordan homomorphism $ \alpha : A \rightarrow {A ^ \prime } $ induces a well-defined mapping $ \sigma : {\mathbf P ( A ) } \rightarrow {\mathbf P ( A ^ \prime ) } $, $ [ 1 + ab,a ] \mapsto [ 1 + a ^ \alpha b ^ \alpha ,a ^ \alpha ] $, which preserves pairs of distant points and maps chains to chains. Moreover, under $ \sigma $: $ [ 1,0 ] \mapsto [ 1,0 ] $, $ [ 0,1 ] \mapsto [ 0,1 ] $, $ [ 1,1 ] \mapsto [ 1,1 ] $. Such a mapping is called a fundamental morphism from $ \Sigma ( K,A ) $ to $ \Sigma ( K,A ^ \prime ) $.

Conversely, any fundamental morphism $ \Sigma ( K,A ) \rightarrow \Sigma ( K,A ^ \prime ) $ having more than one chain in its image can be obtained in this manner (see [a2] for a more general context).

Let $ { \mathop{\rm Aut} } _ {K} ( A ) $ be the group of $ K $- Jordan automorphisms of $ A $, and let $ F ( K,A ) $ be the group of fundamental automorphisms of $ \Sigma ( K,A ) $. Then $ { \mathop{\rm Aut} } _ {K} ( A ) \simeq F ( K,A ) $ and $ { \mathop{\rm Aut} } _ {K} \Sigma ( K,A ) \simeq F ( K,A ) \cdot { \mathop{\rm PGL} } _ {2} ( A ) $.

Rational representations.

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

For a $ K $- algebra of $ K $- dimension $ n $, a representation of $ \Sigma ( K,A ) $ on a part of the Grassmann manifold $ G _ {2n,n } ( K ) $ is obtained as follows. For $ [ a,b ] \in \mathbf P ( A ) $, the set $ A ( a,b ) = \{ {( xa,xb ) } : {x \in A } \} $ is an $ n $- dimensional subspace of the $ K $- vector space $ A \times A $ of dimension $ 2n $. Then $ A ( a,b ) $ is mapped to a point of $ G _ {2n,n } ( K ) $( see Exterior algebra). By this procedure, chains are mapped to normal rational curves of order $ n $, i.e., to images of $ \mathbf P ( K ) $ under the Veronese mapping $ v _ {n} $, and $ V $ is the intersection of $ G _ {2n,n } ( K ) $ with a linear subspace. Other examples can be obtained from this by suitable projection.

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

$ n $-chain geometries.

These are generalizations of chain geometries ($n = 1$) to higher dimensions. E.g., let $A$ be a quadratic field extension of $K$ (cf. Extension of a field). Then, in the projective plane over $A$ the $2$-chains are the subplanes over $K$; these are better known as Baer subplanes, especially in finite geometries (cf. also Geometry).

A Burau geometry is a projective space over $A$ (again a quadratic extension of $K$) together with all projective sublines over $K$. It can be characterized by the property that the incidence system consisting of a projective line $L$ over $A$ (as a point set) and all sublines over $K$ contained in $L$ (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)