# 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].

How to Cite This Entry:
Chain geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_geometry&oldid=46301
This article was adapted from an original article by A. Herzer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article