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>
<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>
<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.
<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>
<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.
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:
<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>
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>.
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