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An [[Incidence system|incidence system]] constructed by means of an [[Algebra|algebra]]. Originally (1842), Ch. von Staudt defined, on the projective line/plane over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101601.png" />, a chain ( "Kette" ) in a synthetic way as a subline/plane over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101602.png" />. Much later, in 1973, W. Benz [[#References|[a1]]] gave a common frame for rather different phenomena (the geometries of Möbius, Laguerre and Lie, and Minkowsky; cf. [[Benz plane|Benz plane]]) using the concept of a geometry over an algebra. A recent survey of the development of this theory is [[#References|[a3]]].
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An [[Incidence system|incidence system]] constructed by means of an [[Algebra|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 [[#References|[a1]]] gave a common frame for rather different phenomena (the geometries of Möbius, Laguerre and Lie, and Minkowsky; cf. [[Benz plane|Benz plane]]) using the concept of a geometry over an algebra. A recent survey of the development of this theory is [[#References|[a3]]].
  
 
==Fundamental concepts.==
 
==Fundamental concepts.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101603.png" /> be a [[Ring|ring]] (associative with one) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101604.png" /> be its group of units. To define the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101605.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101606.png" />, one introduces an [[Equivalence|equivalence]] relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101607.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101608.png" />, as for the [[Projective straight line|projective straight line]] over a skew-field:
+
Let $  A $
 +
be a [[Ring|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|equivalence]] relation $  \equiv $
 +
on $  A \times A $,  
 +
as for the [[Projective straight line|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 (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c1101609.png" /></td> </tr></table>
+
\begin{array}{cc}
 +
a  & b  \\
 +
x  & y  \\
 +
\end{array}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016010.png" /> denote the equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016011.png" />. Then
+
\right ) \in { \mathop{\rm GL} } _ {2} ( A ) } \right \} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016012.png" /></td> </tr></table>
+
One says that  $  [ a,b ] $
 +
is distant to  $  [ c,d ] $
 +
if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016013.png" /></td> </tr></table>
+
$$
 +
\left (
  
One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016014.png" /> is distant to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016015.png" /> if
+
\begin{array}{cc}
 +
a  & b  \\
 +
c & d  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016016.png" /></td> </tr></table>
+
\right ) \in { \mathop{\rm GL} } _ {2} ( A ) .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016017.png" /> be a commutative ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016018.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016019.png" />-algebra, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016020.png" /> is imbedded in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016021.png" /> via the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016022.png" />. Regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016023.png" /> as a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016024.png" />, and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016025.png" />. Then the incidence system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016026.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016027.png" /> as incidence relation, is called a chain geometry. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016028.png" /> are called chains. Any three pairwise distant points belong to exactly one chain. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016029.png" /> is a group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016030.png" />; 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|cross ratio]] belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016031.png" />.
+
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|cross ratio]] belongs to $  K $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016032.png" /> is a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016033.png" /> is said to be a proper chain geometry. A proper chain geometry is a [[Chain space|chain space]].
+
If $  K $
 +
is a field, $  \Sigma ( K,A ) $
 +
is said to be a proper chain geometry. A proper chain geometry is a [[Chain space|chain space]].
  
Below, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016034.png" /> denotes a [[Field|field]].
+
Below, $  K $
 +
denotes a [[Field|field]].
  
 
==Affine case.==
 
==Affine case.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016035.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016036.png" /> is the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016037.png" /> distant to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016038.png" />. Consider the traces of the chains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016039.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016040.png" /></td> </tr></table>
+
$$
 +
{\widehat{\mathfrak C}  } _ {K} ( A ) = \left \{ {C \cap D } : {C \in \mathfrak C _ {K} ( A ) , \left | {C \cap D } \right | \geq  3 } \right \} .
 +
$$
  
There is a natural bijection from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016041.png" /> into the [[Affine space|affine space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016042.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016043.png" /> via the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016044.png" />. Under this mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016045.png" /> becomes the set
+
There is a natural bijection from $  D $
 +
into the [[Affine space|affine space]] $  A $
 +
over $  K $
 +
via the mapping $  [ 1,a ] \mapsto a $.  
 +
Under this mapping, $  {\widehat{\mathfrak C}  } _ {K} ( A ) $
 +
becomes the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016046.png" /></td> </tr></table>
+
$$
 +
{\mathcal K} _ {K} ( A ) = \left \{ {\mathbf K ( a,b,c ) } : {a \in A  ^ {*} , b,c \in A, \left | {\mathbf K ( a,b,c ) } \right | \geq  3 } \right \}
 +
$$
  
 
of affine chains, defined by
 
of affine chains, defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016047.png" /></td> </tr></table>
+
$$
 +
\mathbf K ( a,b,c ) = \left \{ {( a t + b ) ^ {- 1 } + c } : {t \in K, a t + b \in A  ^ {*} } \right \} \cup \{ c \} .
 +
$$
  
This trace geometry is called the affine chain geometry, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016048.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016049.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016050.png" /> is a bundle of parallel straight lines in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016051.png" />. For an algebraic (especially, finite-dimensional) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016052.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016053.png" /> (cf. also [[Algebraic algebra|Algebraic algebra]]), the affine chain geometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016054.png" /> is an affine chain space (cf. [[Chain space|Chain space]]).
+
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|Algebraic algebra]]), the affine chain geometry $  {\mathcal A} ( K,A ) $
 +
is an affine chain space (cf. [[Chain space|Chain space]]).
  
 
==Structure of morphisms.==
 
==Structure of morphisms.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016056.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016057.png" />-algebras. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016058.png" />-Jordan homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016059.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016060.png" />-semi-linear mapping satisfying: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016061.png" />; and ii) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016062.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016063.png" />. For a  "strong"  algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016064.png" /> (strongness guarantees a great richness in units for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016065.png" />), any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016066.png" /> can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016067.png" /> for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016068.png" />. Then a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016069.png" />-Jordan homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016070.png" /> induces a well-defined mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016072.png" />, which preserves pairs of distant points and maps chains to chains. Moreover, under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016073.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016076.png" />. Such a mapping is called a fundamental morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016077.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016078.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016079.png" /> having more than one chain in its image can be obtained in this manner (see [[#References|[a2]]] for a more general context).
+
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 [[#References|[a2]]] for a more general context).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016080.png" /> be the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016081.png" />-Jordan automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016082.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016083.png" /> be the group of fundamental automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016084.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016086.png" />.
+
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.==
 
==Rational representations.==
As in the affine case one tries to find kinds of representations for chain geometries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016087.png" /> on a part of a projective space where the chains become curves, at least when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016088.png" /> is finite dimensional. In this way one has discovered incidence systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016089.png" /> isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016091.png" /> is a projective variety (cf. also [[Projective scheme|Projective scheme]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016092.png" /> is a (Zariski-) closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016094.png" /> consists of rational curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016095.png" /> (cf. also [[Incidence system|Incidence system]]; [[Rational curve|Rational curve]]).
+
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|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|Incidence system]]; [[Rational curve|Rational curve]]).
  
For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016096.png" />-algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016097.png" />-dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016098.png" />, a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c11016099.png" /> on a part of the [[Grassmann manifold|Grassmann manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160100.png" /> is obtained as follows. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160101.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160102.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160103.png" />-dimensional subspace of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160104.png" />-vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160105.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160106.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160107.png" /> is mapped to a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160108.png" /> (see [[Exterior algebra|Exterior algebra]]). By this procedure, chains are mapped to normal rational curves of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160109.png" />, i.e., to images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160110.png" /> under the [[Veronese mapping|Veronese mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160111.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160112.png" /> is the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160113.png" /> with a linear subspace. Other examples can be obtained from this by suitable projection.
+
For a $  K $-
 +
algebra of $  K $-
 +
dimension $  n $,  
 +
a representation of $  \Sigma ( K,A ) $
 +
on a part of the [[Grassmann manifold|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|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160114.png" /> (i.e., any element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160115.png" /> has a quadratic minimal polynomial; cf. also [[Extension of a field|Extension of a field]]) has a representation as a quadric model (cf. [[Chain space|Chain space]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160116.png" /> is the [[Quadric|quadric]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160117.png" /> is its set of singular points; the chains are conics.
+
A quadratic algebra $  A $(
 +
i.e., any element of $  A \setminus  K $
 +
has a quadratic minimal polynomial; cf. also [[Extension of a field|Extension of a field]]) has a representation as a quadric model (cf. [[Chain space|Chain space]]). Here, $  V = Q $
 +
is the [[Quadric|quadric]] and $  W $
 +
is its set of singular points; the chains are conics.
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160118.png" />-chain geometries.==
+
== $  n $-chain geometries.==
These are generalizations of chain geometries (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160119.png" />) to higher dimensions. E.g., let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160120.png" /> be a quadratic field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160121.png" /> (cf. [[Extension of a field|Extension of a field]]). Then, in the projective plane over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160122.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160123.png" />-chains are the subplanes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160124.png" />; these are better known as Baer subplanes, especially in finite geometries (cf. also [[Geometry|Geometry]]).
+
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|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|Geometry]]).
  
A Burau geometry is a [[Projective space|projective space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160125.png" /> (again a quadratic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160126.png" />) together with all projective sublines over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160127.png" />. It can be characterized by the property that the incidence system consisting of a projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160128.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160129.png" /> (as a point set) and all sublines over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160130.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110160/c110160131.png" /> (considered as blocks) for a [[Möbius plane|Möbius plane]]. A more general concept can be found in [[#References|[a4]]].
+
A Burau geometry is a [[Projective space|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|Möbius plane]]. A more general concept can be found in [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Benz,  "Vorlesungen über Geometrie der Algebren" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Blunk,  "Chain spaces over Jordan systems"  ''Abh. Math. Sem. Hamburg'' , '''64'''  (1994)  pp. 33–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Herzer,  "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Pieconkowski,  "Projektive Räume über Schiefkörperpaaren" , W und T, Wiss.- und Technik-Verl.  (1994)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Benz,  "Vorlesungen über Geometrie der Algebren" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Blunk,  "Chain spaces over Jordan systems"  ''Abh. Math. Sem. Hamburg'' , '''64'''  (1994)  pp. 33–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Herzer,  "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Pieconkowski,  "Projektive Räume über Schiefkörperpaaren" , W und T, Wiss.- und Technik-Verl.  (1994)</TD></TR></table>

Revision as of 16:43, 4 June 2020


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)
How to Cite This Entry:
Chain geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain_geometry&oldid=18056
This article was adapted from an original article by A. Herzer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article