Quelle xmod.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>

<Chapter Label="chap-xmod">

<Heading>Crossed modules</Heading>

In this chapter we will present the notion of crossed modules of commutative
algebras and their implementation in this package.

<Section>
<Heading>Definition and Examples</Heading>

<Index>crossed module</Index>
<Index>precrossed module</Index>
<Index>2d-algebra</Index>

A <E>crossed module</E> is a <B>k</B>-algebra morphism
<M>\mathcal{X}:=(\partial:S\rightarrow R)</M>
with a left action of <M>R</M> on <M>S</M> satisfying
<Display>
{\bf XModAlg\ 1} ~:~  \partial(r \cdot s)
   = r(\partial s),
\qquad
{\bf XModAlg\ 2} ~:~  (\partial s) \cdot s^{\prime} = ss^{\prime},
</Display>
for all <M>s,s^{\prime }\in S, \ r\in R</M>.
The morphism <M>\partial</M> is called the <E>boundary map</E>
of <M>\mathcal{X}</M>
<P/>
Note that, although in this definition we have used a left action,
in the category of commutative algebras left and right actions coincide.
<P/>
When only the first axiom is satisfied, it is a <E>precrossed module</E>
which is constructed.
<P/>
The details of these implementations can be found in <Cite Key="aodabas1"/>.
<P/>

<ManSection>
   <Func Name="XModAlgebra"
         Arg="args" />
   <Func Name="PreXModAlgebra"
         Arg="args" />
   <Prop Name="IsXModAlgebra"
         Arg="X0" />
   <Prop Name="IsPreXModAlgebra"
         Arg="X0" />
<Description>
These two global function call one of the following six operations,
depending on the arguments supplied.
The two properties listed are assigned as appropriate to the resulting
structures.
</Description>
</ManSection>

<ManSection>
   <Oper Name="XModAlgebraByIdeal"
         Arg="A I" />
<Description>
Let <M>A</M> be an algebra and <M>I</M> an ideal of <M>A</M>.
Then <M>\mathcal{X} = (inc:I\rightarrow A)</M> is a
crossed module whose action is left multiplication of <M>A</M> on <M>I</M>.
Conversely, given a crossed module
<M>\mathcal{X} = (\partial : S \rightarrow R)</M>,
it is the case that <M>{\partial(S)}</M> is an ideal of <M>R</M>.
<P/>
</Description>
</ManSection>

<Example>
<![CDATA[
gap> F5 := GF(5);;
gap> id5 := One( F5 );;
gap> two := Z(5);;
gap> z5 := Zero( F5 );;
gap> n0 := [ [id5,z5,z5], [z5,id5,z5], [z5,z5,id5] ];;
gap> n1 := [ [z5,id5,two^3], [z5,z5,id5], [z5,z5,z5] ];;
gap> n2 := [ [z5,z5,id5], [z5,z5,z5], [z5,z5,z5] ];;
gap> An := Algebra( F5, [ n0, n1 ] );;
gap> SetName( An, "An" );
gap> Bn := Subalgebra( An, [ n1 ] );;
gap> SetName( Bn, "Bn" );
gap> IsIdeal( An, Bn );
true
gap> actn := AlgebraActionByMultipliers( An, Bn, An );;
gap> Xn := XModAlgebraByIdeal( An, Bn );
[ Bn -> An ]
gap> SetName( Xn, "Xn" );
]]>
</Example>

<ManSection>
   <Attr Name="AugmentationXMod"
         Arg="A" />
<Description>
As a special case of the previous operation, the attribute
<C>AugmentationXMod(A)</C> of a group algebra <M>A</M>
is the <C>XModAlgebraByIdeal</C> formed using the
<C>AugmentationIdeal</C> of the group algebra.
<P/>
</Description>
</ManSection>

<Example>
<![CDATA[
gap> Ak4 := GroupRing( GF(5), DihedralGroup(4) );
<algebra-with-one over GF(5), with 2 generators>
gap> Size( Ak4 );
625
gap> SetName( Ak4, "GF5[k4]" );
gap> IAk4 := AugmentationIdeal( Ak4 );
<two-sided ideal in GF5[k4], (2 generators)>
gap> Size( IAk4 );
125
gap> SetName( IAk4, "I(GF5[k4])" );
gap> XIAk4 := XModAlgebraByIdeal( Ak4, IAk4 );
[ I(GF5[k4]) -> GF5[k4] ]
gap> Display( XIAk4 );
Crossed module [I(GF5[k4])->GF5[k4]] :-
: Source algebra I(GF5[k4]) has generators:
  [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^
    0)*f2 ]
: Range algebra GF5[k4] has generators:
  [ (Z(5)^0)*<identity> of ..., (Z(5)^0)*f1, (Z(5)^0)*f2 ]
: Boundary homomorphism maps source generators to:
  [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^
    0)*f2 ]
gap> Size2d( XIAk4 );
[ 125, 625 ]
]]>
</Example>

<ManSection>
   <Attr Name="Source" Label="for crossed modules of commutative algebras"
         Arg="X0" />
   <Attr Name="Range" Label="for crossed modules of commutative algebras"
         Arg="X0" />
   <Attr Name="Boundary" Label="for crossed modules of commutative algebras"
         Arg="X0" />
   <Attr Name="XModAlgebraAction"
         Arg="X0" />
<Description>
These four attributes are used in the construction of a crossed module
<M>\mathcal{X}</M> where:
<List>
<Item>
<C>Source(X)</C> and <C>Range(X)</C> are the <E>source</E> and the <E>range</E>
of the boundary map respectively;
</Item>
<Item>
<C>Boundary(X)</C> is the boundary map of the crossed module <M>\mathcal{X}</M>;
</Item>
<Item>
<C>XModAlgebraAction(X)</C> is the action used in the crossed module.
This is an algebra homomorphism from <C>Range(X)</C> to an algebra of
endomorphisms of <C>Source(X)</C>.
</Item>
</List>

The following standard &GAP; operations have special &XModAlg; implementations:
<List>
<Item>
<C>Display(X)</C> is used to list the components of <M>\mathcal{X}</M>;
</Item>
<Item>
<C>Size2d(X)</C> for a crossed module <M>\mathcal{X}</M>
returns a <M>2</M>-element list, the sizes of the source and range,
</Item>
<Item>
<C>Dimension(X)</C> for a crossed module <M>\mathcal{X}</M>
returns a <M>2</M>-element list, the dimensions of the source and range,
</Item>
<Item>
<C>Name(X)</C> is used for giving a name to the crossed module
<M>\mathcal{X}</M> by associating the names of source and range algebras.
</Item>
</List>

In the following example, we construct a crossed module by using the algebra
<M>GF_{5}D_{4}</M> and its augmentation ideal.
We also show usage of the attributes listed above.

</Description>
</ManSection>

<Example>
<![CDATA[
gap> RepresentationsOfObject( XIAk4 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ]
gap> KnownPropertiesOfObject( XIAk4 );
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree",
  "IsLeftActedOnByDivisionRing", "IsAdditivelyCommutative", "IsLDistributive",
  "IsRDistributive", "IsPreXModDomain", "Is2dAlgebraObject",
  "IsPreXModAlgebra", "IsXModAlgebra" ]
gap> KnownAttributesOfObject( XIAk4 );
[ "Name", "LeftActingDomain", "Range", "Source", "Boundary", "Size2d",
  "XModAlgebraAction" ]
]]>
</Example>

<ManSection>
   <Oper Name="XModAlgebraByMultiplierAlgebra"
         Arg="A" />
<Description>
When <M>A</M> is an algebra with multiplier algebra <M>M</M>,
then the map <M>A \to M, ~ a \mapsto \mu_a</M> is the boundary
of a crossed module in which the action is the identity map on <M>M</M>.
<P/>
</Description>
</ManSection>

<Example>
<![CDATA[
gap> XAn := XModAlgebraByMultiplierAlgebra( An );
[ An -> <algebra of dimension 3 over GF(5)> ]
gap> XModAlgebraAction( XAn );
IdentityMapping( <algebra of dimension 3 over GF(5)> )
]]>
</Example>

<ManSection>
   <Oper Name="XModAlgebraBySurjection"
         Arg="f" />
<Description>
Let <M>\partial : S\rightarrow R</M> be a surjective algebra homomorphism
whose kernel lies in the annihilator of <M>S</M>.
Define the action of <M>R</M> on <M>S</M> by <M>r\cdot s = \widetilde{r}s</M>
where <M>\widetilde{r} \in \partial^{-1}(r)</M>,
as described in section <Ref Oper="AlgebraActionBySurjection"/>.
Then <M>\mathcal{X}=(\partial : S\rightarrow R)</M>
is a crossed module with the defined action.
<P/>
Continuing with the example in that section,
<P/>
</Description>
</ManSection>

<Example>
<![CDATA[
gap> X2 := XModAlgebraBySurjection( nat2 );;
gap> Display( X2 );
Crossed module [A2->Q2] :-
: Source algebra A2 has generators:
  [ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ] ]
: Range algebra Q2 has generators:
  [ v.1, v.2 ]
: Boundary homomorphism maps source generators to:
  [ v.1 ]
]]>
</Example>

<#Include Label="XModAlgebraByBoundaryAndAction">

<#Include Label="XModAlgebraByModule">

<ManSection>
   <Oper Name="SubXModAlgebra"
         Arg="alg src rng" />
   <Oper Name="IsSubXModAlgebra"
         Arg="alg sub" />
<Description>
A crossed module <M>\mathcal{X}^{\prime }
= (\partial ^{\prime }:S^{\prime}\rightarrow R^{\prime })</M>
is a  subcrossed module of the crossed module
<M>\mathcal{X} = (\partial :S\rightarrow R)</M> if
<M>S^{\prime }\leq S</M>, <M>R^{\prime}\leq R</M>,
<M>\partial^{\prime } = \partial|_{S^{\prime }}</M>,
and the action of <M>S^{\prime }</M> on <M>R^{\prime }</M>
is induced by the action of <M>R</M> on <M>S</M>.
The operation <C>SubXModAlgebra</C> is used to construct a subcrossed module
of a given crossed module.

</Description>
</ManSection>

<Example>
<![CDATA[
gap> e4 := Elements( IAk4 )[4];
(Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2
gap> Je4 := Ideal( IAk4, [e4] );;
gap> SetName( Je4, "" );
gap> Ke4 := Subalgebra( Ak4, [e4] );;
gap> [ Size( Je4 ), Size( Ke4 ) ];
[ 5, 5 ]
gap> Se4 := SubXModAlgebra( XIAk4, Je4, Ke4 );
[ <e4> -> <algebra of dimension 1 over GF(5)> ]
gap> IsSubXModAlgebra( XIAk4, Se4 );
true
gap> Display( Se4 );
Crossed module [<e4> -> ..] :-
: Source algebra <e4> has generators:
[ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ]
: Range algebra has generators:
[ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ]
: Boundary homomorphism maps source generators to:
[ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ]
]]>
</Example>

</Section>

<Section><Heading>(Pre-)Crossed Module Morphisms</Heading>

Let <M>\mathcal{X} = (\partial:S\rightarrow R)</M> and
<M>\mathcal{X}^{\prime} =
(\partial^{\prime }:S^{\prime }\rightarrow R^{\prime })</M>
be (pre)crossed modules and let <M>\theta :S\rightarrow S^{\prime }</M>,
<M>\varphi : R\rightarrow R^{\prime }</M> be algebra homomorphisms.
Then if
<Display>
\varphi \circ \partial = \partial ^{\prime } \circ \theta,
\qquad
\theta (r\cdot s)=\varphi(r) \cdot \theta (s),
</Display>
for all <M>r\in R</M>, <M>s\in S,</M>
the pair <M>(\theta ,\varphi )</M> is called a morphism between
<M>\mathcal{X}</M> and <M>\mathcal{X}^{\prime } </M>
<P/>
The conditions can be thought as the commutativity of the following diagrams:

<Display>
\xymatrix@R=40pt@C=40pt{
  S \ar[d]_{\partial} \ar[r]^{\theta}
                & S^{\prime } \ar[d]^{\partial^{\prime }}  \\
  R  \ar[r]_{\varphi}
                & R^{\prime }
} \ \ \ \
\xymatrix@R=40pt@C=40pt{
  R \times S \ar[d] \ar[r]^{ \varphi \times \theta }
                &  R^{\prime } \times S^{\prime } \ar[d]  \\
  S  \ar[r]_{ \theta } & S^{\prime }.       }
</Display>




<P/>
In &GAP; we define the morphisms between algebraic structures such as
cat<M>^{1}</M>-algebras and crossed modules and they are investigated
by the function <C>Make2dAlgebraMorphism</C>.

<ManSection>
   <Func Name="XModAlgebraMorphism"
         Arg="arg" />
   <Meth Name="IdentityMapping" Label="for crossed modules of algebras"
         Arg="Xalg" />
   <Oper Name="PreXModAlgebraMorphismByHoms"
         Arg="src rng srchom rnghom" />
   <Oper Name="XModAlgebraMorphismByHoms"
         Arg="src rng srchom rnghom" />
   <Prop Name="IsPreXModAlgebraMorphism"
         Arg="mor" />
   <Prop Name="IsXModAlgebraMorphism"
         Arg="mor" />
   <Attr Name="Source" Label="for morphisms of crossed modules of algebras"
         Arg="mor" />
   <Attr Name="Range" Label="for morphisms of crossed modules of algebras"
         Arg="mor" />
   <Meth Name="IsTotal" Label="for morphisms of crossed modules of algebras"
         Arg="mor" />
   <Meth Name="IsSingleValued"
         Label="for morphisms of crossed modules of algebras"
         Arg="m0r" />
   <Meth Name="Name" Label="for morphisms of crossed modules of algebras"
         Arg="mor" />

<Description>
These operations construct crossed module homomorphisms,
which may have the attributes listed.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> c4 := CyclicGroup( 4 );;
gap> Ac4 := GroupRing( GF(2), c4 );
<algebra-with-one over GF(2), with 2 generators>
gap> SetName( Ac4, "GF2[c4]" );
gap> IAc4 := AugmentationIdeal( Ac4 );
<two-sided ideal in GF2[c4], (dimension 3)>
gap> SetName( IAc4, "I(GF2[c4])" );
gap> XIAc4 := XModAlgebra( Ac4, IAc4 );
[ I(GF2[c4]) -> GF2[c4] ]
gap> Bk4 := GroupRing( GF(2), SmallGroup( 4, 2 ) );
<algebra-with-one over GF(2), with 2 generators>
gap> SetName( Bk4, "GF2[k4]" );
gap> IBk4 := AugmentationIdeal( Bk4 );
<two-sided ideal in GF2[k4], (dimension 3)>
gap> SetName( IBk4, "I(GF2[k4])" );
gap> XIBk4 := XModAlgebra( Bk4, IBk4 );
[ I(GF2[k4]) -> GF2[k4] ]
gap> IAc4 = IBk4;
false
gap> homIAIB := AllAlgebraHomomorphisms( IAc4, IBk4 );;
gap> theta := homIAIB[3];;
gap> homAB := AllAlgebraHomomorphisms( Ac4, Bk4 );;
gap> phi := homAB[7];;
gap> mor := XModAlgebraMorphism( XIAc4, XIBk4, theta, phi );
[[I(GF2[c4])->GF2[c4]] => [I(GF2[k4])->GF2[k4]]]
gap> Display( mor );

Morphism of crossed modules :-
: Source = [I(GF2[c4])->GF2[c4]]
:  Range = [I(GF2[k4])->GF2[k4]]
: Source Homomorphism maps source generators to:
  [ <zero> of ..., (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^
    0)*f1*f2, (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^
    0)*f1*f2 ]
: Range Homomorphism maps range generators to:
  [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2,
  (Z(2)^0)*<identity> of ... ]

gap> IsTotal( mor );
true
gap> IsSingleValued( mor );
true
]]>
</Example>

<ManSection>
   <Meth Name="Kernel" Label="for morphisms of crossed modules of algebras"
         Arg="mor" />
<Description>
Let <M>(\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R)
       \rightarrow \mathcal{X}^{\prime} = (\partial^{\prime}
       : S^{\prime} \rightarrow R^{\prime})</M>
be a crossed module homomorphism.
Then the crossed module
<Display>
\ker(\theta,\varphi) = (\partial| : \ker\theta \rightarrow \ker\varphi )
</Display>
is called the <E>kernel</E> of <M>(\theta,\varphi)</M>.
Also, <M>\ker(\theta ,\varphi )</M> is an ideal of <M>\mathcal{X}</M>.
An example is given below.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> Xmor := Kernel( mor );
[ <algebra of dimension 2 over GF(2)> -> <algebra of dimension 2 over GF(2)> ]
gap> IsXModAlgebra( Xmor );
true
gap> Size2d( Xmor );
[ 4, 4 ]
gap> IsSubXModAlgebra( XIAc4, Xmor );
true
]]>
</Example>

<ManSection>
   <Oper Name="Image"
         Arg="mor" />
<Description>
Let <M>(\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R)
        \rightarrow \mathcal{X}^{\prime} = (\partial^{\prime} : S^{\prime}
        \rightarrow R^{\prime})</M>
be a crossed module homomorphism.
Then the crossed module
<Display>
\Im(\theta,\varphi) = (\partial^{\prime}| :
\Im\theta \rightarrow \Im\varphi)
</Display>
is called the image of <M>(\theta,\varphi)</M>.
Further, <M>\Im(\theta,\varphi)</M> is a subcrossed module of
<M>(S^{\prime},R^{\prime},\partial^{\prime})</M>.
<P/>
In this package, the image of a crossed module homomorphism
can be obtained by the command <C>ImagesSource</C>.
The operation <C>Sub2dAlgObject</C> is effectively used
for finding the kernel and image crossed modules
induced from a given crossed module homomorphism.

</Description>
</ManSection>

<ManSection>
   <Attr Name="SourceHom"
         Arg="mor" />
   <Attr Name="RangeHom"
         Arg="mor" />
   <Prop Name="IsInjective"
         Arg="mor" />
   <Prop Name="IsSurjective"
         Arg="mor" />
   <Prop Name="IsBijjective"
         Arg="mor" />
<Description>
Let <M>(\theta,\varphi)</M> be a homomorphism of crossed modules.
If the homomorphisms <M>\theta</M> and <M>\varphi</M>
are injective (surjective) then <M>(\theta,\varphi)</M>
is injective (surjective).
<P/>
The attributes <C>SourceHom</C> and <C>RangeHom</C>
store the two algebra homomorphisms <M>\theta</M> and <M>\varphi</M>.

</Description>
</ManSection>

<Example>
<![CDATA[
gap> ic4 := One( Ac4 );;
gap> e1 := ic4*c4.1 + ic4*c4.2;
(Z(2)^0)*f1+(Z(2)^0)*f2
gap> ImageElm( theta, e1 );
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> e2 := ic4*c4.1;
(Z(2)^0)*f1
gap> ImageElm( phi, e2 );
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> IsInjective( mor );
false
gap> IsSurjective( mor );
false
gap> immor := ImagesSource2DimensionalMapping( mor );;
gap> Display( immor );

Crossed module [..->..] :-
: Source algebra has generators:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
: Range algebra has generators:
  [ (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2, (Z(2)^0)*<identity> of ... ]
: Boundary homomorphism maps source generators to:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]

]]>
</Example>

</Section>

</Chapter>

quality87%

¤ Dauer der Verarbeitung: 0.2 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.