In this chapter we will present the notion of crossed modules of commutative algebras and their implementation in this package.
A crossed module is a k-algebra morphism \(\mathcal{X}:=(\partial:S\rightarrow R)\) with a left action of \(R\) on \(S\) satisfying
\[ {\bf XModAlg\ 1} ~:~ \partial(r \cdot s) = r(\partial s), \qquad {\bf XModAlg\ 2} ~:~ (\partial s) \cdot s^{\prime} = ss^{\prime}, \]
for all \(s,s^{\prime }\in S, \ r\in R\). The morphism \(\partial\) is called the boundary map of \(\mathcal{X}\)
Note that, although in this definition we have used a left action, in the category of commutative algebras left and right actions coincide.
‣ XModAlgebra( args ) | ( function ) |
This global function calls one of the following six operations, depending on the arguments supplied.
‣ XModAlgebraByIdeal( A, I ) | ( operation ) |
Let \(A\) be an algebra and \(I\) an ideal of \(A\). Then \(\mathcal{X} = (inc:I\rightarrow A)\) is a crossed module whose action is left multiplication of \(A\) on \(I\). Conversely, given a crossed module \(\mathcal{X} = (\partial : S \rightarrow R)\), it is the case that \({\partial(S)}\) is an ideal of \(R\).
gap> F := GF(5);; gap> one := One(F);; gap> two := Z(5);; gap> z := Zero( F );; gap> l := [ [one,z,z], [z,one,z], [z,z,one] ];; gap> m := [ [z,one,two^3], [z,z,one], [z,z,z] ];; gap> n := [ [z,z,one], [z,z,z], [z,z,z] ];; gap> A := Algebra( F, [l,m] );; gap> SetName( A, "A(l,m)" ); gap> B := Subalgebra( A, [m] );; gap> SetName( B, "A(m)" ); gap> IsIdeal( A, B ); true gap> act := AlgebraActionByMultipliers( A, B );; gap> XAB := XModAlgebraByIdeal( A, B ); [ A(m) -> A(l,m) ] gap> SetName( XAB, "XAB" );
‣ AugmentationXMod( A ) | ( attribute ) |
As a special case of the previous operation, the attribute AugmentationXMod(A) of a group algebra \(A\) is the XModAlgebraByIdeal formed using the AugmentationIdeal of the group algebra.
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 ]
‣ XModAlgebraByMultiplierAlgebra( A ) | ( operation ) |
When \(A\) is an algebra with multiplier algebra \(M\), then the map \(A \to M, ~ a \mapsto \mu_a\) is the boundary of a crossed module in which the action is the identity map on \(M\).
gap> XA := XModAlgebraByMultiplierAlgebra( A ); [ A(l,m) -> <algebra of dimension 3 over GF(5)> ] gap> XModAlgebraAction( XA ); IdentityMapping( <algebra of dimension 3 over GF(5)> )
‣ XModAlgebraBySurjection( f ) | ( operation ) |
Let \(\partial : S\rightarrow R\) be a surjective algebra homomorphism whose kernel lise in the annihilator of \(S\). Define the action of \(R\) on \(S\) by \(r\cdot s = \widetilde{r}s\) where \(\widetilde{r} \in \partial^{-1}(r)\), as described in section AlgebraActionBySurjection (2.2-2) Then \(\mathcal{X}=(\partial : S\rightarrow R)\) is a crossed module with the defined action.
Continuing with the example in that section,
gap> X3 := XModAlgebraBySurjection( nat3 );; gap> Display( X3 ); Crossed module [..->..] :- : Source algebra has generators: [ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ] ] : Range algebra has generators: [ v.1, v.2 ] : Boundary homomorphism maps source generators to: [ v.1 ]
‣ XModAlgebraByBoundaryAndAction( bdy, act ) | ( operation ) |
‣ PreXModAlgebraByBoundaryAndAction( bdy, act ) | ( operation ) |
‣ IsPreXModAlgebra( X0 ) | ( property ) |
An \(R\)-algebra homomorphism \(\mathcal{X} := (\partial : S \rightarrow R)\) which satisfies the condition \({\bf XModAlg\ 1}\) is called a precrossed module. The details of these implementations can be found in [Oda09].
gap> G := SmallGroup( 4, 2 ); <pc group of size 4 with 2 generators> gap> F := GaloisField( 4 ); GF(2^2) gap> R := GroupRing( F, G ); <algebra-with-one over GF(2^2), with 2 generators> gap> Size( R ); 256 gap> SetName( R, "GF(2^2)[k4]" ); gap> e5 := Elements( R )[5]; (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 gap> S := Subalgebra( R, [e5] );; gap> SetName( S, "<e5>" ); gap> RS := Cartesian( R, S );; gap> SetName( RS, "GF(2^2)[k4] x <e5>" ); gap> act := AlgebraAction( R, RS, S );; gap> bdy := AlgebraHomomorphismByFunction( S, R, r->r ); MappingByFunction( <e5>, GF(2^2)[k4], function( r ) ... end ) gap> IsAlgebraAction( act ); true gap> IsAlgebraHomomorphism( bdy ); true gap> XM := PreXModAlgebraByBoundaryAndAction( bdy, act ); [<e5>->GF(2^2)[k4]] gap> IsXModAlgebra( XM ); true gap> Display( XM ); Crossed module [<e5>->GF(2^2)[k4]] :- : Source algebra has generators: [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] : Range algebra GF(2^2)[k4] has generators: [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f2 ] : Boundary homomorphism maps source generators to: [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
‣ XModAlgebraByModule( M, R ) | ( operation ) |
Let \(M\) be a \(R\)-module. Then \(\mathcal{X} = (0:M\rightarrow R)\) is a crossed module. Conversely, given a crossed module \(\mathcal{X} = (\partial :M\rightarrow R)\), one can get that \(\ker\partial\) is a \((R/\partial M)\)-module.
gap> ## example needed
‣ Source( X0 ) | ( attribute ) |
‣ Range( X0 ) | ( attribute ) |
‣ Boundary( X0 ) | ( attribute ) |
‣ XModAlgebraAction( X0 ) | ( attribute ) |
These four attributes are used in the construction of a crossed module \(\mathcal{X}\) where:
Source(X) and Range(X) are the source and the range of the boundary map respectively;
Boundary(X) is the boundary map of the crossed module \(\mathcal{X}\);
XModAlgebraAction(X) is the action used in the crossed module. This is an algebra homomorphism from Range(X) to an algebra of endomorphisms of Source(X).
The following standard GAP operations have special XModAlg implementations:
Display(X) is used to list the components of \(\mathcal{X}\);
Size2d(X) is used for calculating the order of the crossed module \(\mathcal{X}\);
Name(X) is used for giving a name to the crossed module \(\mathcal{X}\) by associating the names of source and range algebras.
In the following example, we construct a crossed module by using the algebra \(GF_{5}D_{4}\) and its augmentation ideal. We also show usage of the attributes listed above.
gap> f := Boundary( XIAk4 ); MappingByFunction( I(GF5[k4]), GF5[k4], function( i ) ... end ) gap> Print( RepresentationsOfObject(XIAk4), "\n" ); [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ] gap> props := [ "CanEasilyCompareElements", "CanEasilySortElements", > "IsDuplicateFree", "IsLeftActedOnByDivisionRing", "IsAdditivelyCommutative", > "IsLDistributive", "IsRDistributive", "IsPreXModDomain", "Is2dAlgebraObject", > "IsPreXModAlgebra", "IsXModAlgebra" ];; gap> known := KnownPropertiesOfObject( XIAk4 );; gap> ForAll( props, p -> (p in known) ); true gap> Print( KnownAttributesOfObject(XIAk4), "\n" ); [ "Name", "Range", "Source", "Boundary", "Size2d", "XModAlgebraAction" ]
‣ SubXModAlgebra( X0 ) | ( operation ) |
‣ IsSubXModAlgebra( X0 ) | ( operation ) |
A crossed module \(\mathcal{X}^{\prime } = (\partial ^{\prime }:S^{\prime}\rightarrow R^{\prime })\) is a subcrossed module of the crossed module \(\mathcal{X} = (\partial :S\rightarrow R)\) if \(S^{\prime }\leq S\), \(R^{\prime}\leq R\), \(\partial^{\prime } = \partial|_{S^{\prime }}\), and the action of \(S^{\prime }\) on \(R^{\prime }\) is induced by the action of \(R\) on \(S\). The operation SubXModAlgebra is used to construct a subcrossed module of a given crossed module.
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> Size( Je4 ); 5 gap> SetName( Je4, "<e4>" ); gap> XJe4 := XModAlgebraByIdeal( Ak4, Je4 ); [ <e4> -> GF5[k4] ] gap> Display( XJe4 ); Crossed module [<e4>->GF5[k4]] :- : 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 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)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] gap> IsSubXModAlgebra( XIAk4, XJe4 ); true
Let \(\mathcal{X} = (\partial:S\rightarrow R)\), \(\mathcal{X}^{\prime} = (\partial^{\prime }:S^{\prime }\rightarrow R^{\prime })\) be (pre)crossed modules and \(\theta :S\rightarrow S^{\prime }\), \(\varphi : R\rightarrow R^{\prime }\) be algebra homomorphisms. If
\[ \varphi \circ \partial = \partial ^{\prime } \circ \theta, \qquad \theta (r\cdot s)=\varphi(r) \cdot \theta (s), \]
for all \(r\in R\), \(s\in S,\) then the pair \((\theta ,\varphi )\) is called a morphism between \(\mathcal{X}\) and \(\mathcal{X}^{\prime } \)
The conditions can be thought as the commutativity of the following diagrams:
\[ \xymatrix@R=40pt@C=40pt{ S \ar[d]_{\partial} \ar[r]^{\theta} & S^{\prime } \ar[d]^{\partial^{\prime }} \\ R \ar[r]_{\varphi} & R^{\prime } } \ \ \ \ \xymatrix@R=40pt@C=40pt{ R \times S \ar[d] \ar[r]^{ \varphi \times \theta } & R^{\prime } \times S^{\prime } \ar[d] \\ S \ar[r]_{ \theta } & S^{\prime }. } \]
In GAP we define the morphisms between algebraic structures such as cat\(^{1}\)-algebras and crossed modules and they are investigated by the function Make2dAlgebraMorphism.
‣ XModAlgebraMorphism( arg ) | ( function ) |
‣ IdentityMapping( X0 ) | ( method ) |
‣ PreXModAlgebraMorphismByHoms( f, g ) | ( operation ) |
‣ XModAlgebraMorphismByHoms( f, g ) | ( operation ) |
‣ IsPreXModAlgebraMorphism( f ) | ( property ) |
‣ IsXModAlgebraMorphism( f ) | ( property ) |
‣ Source( m ) | ( attribute ) |
‣ Range( m ) | ( attribute ) |
‣ IsTotal( m ) | ( method ) |
‣ IsSingleValued( m ) | ( method ) |
‣ Name( m ) | ( method ) |
These operations construct crossed module homomorphisms, which may have the attributes listed.
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 := AllHomsOfAlgebras( IAc4, IBk4 );; gap> theta := homIAIB[3];; gap> homAB := AllHomsOfAlgebras( 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
‣ Kernel( X0 ) | ( method ) |
Let \((\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R) \rightarrow \mathcal{X}^{\prime} = (\partial^{\prime} : S^{\prime} \rightarrow R^{\prime})\) be a crossed module homomorphism. Then the crossed module
\[ \ker(\theta,\varphi) = (\partial| : \ker\theta \rightarrow \ker\varphi ) \]
is called the kernel of \((\theta,\varphi)\). Also, \(\ker(\theta ,\varphi )\) is an ideal of \(\mathcal{X}\). An example is given below.
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
‣ Image( X0 ) | ( operation ) |
Let \((\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R) \rightarrow \mathcal{X}^{\prime} = (\partial^{\prime} : S^{\prime} \rightarrow R^{\prime})\) be a crossed module homomorphism. Then the crossed module
\[ \Im(\theta,\varphi) = (\partial^{\prime}| : \Im\theta \rightarrow \Im\varphi) \]
is called the image of \((\theta,\varphi)\). Further, \(\Im(\theta,\varphi)\) is a subcrossed module of \((S^{\prime},R^{\prime},\partial^{\prime})\).
In this package, the image of a crossed module homomorphism can be obtained by the command ImagesSource. The operation Sub2dAlgObject is effectively used for finding the kernel and image crossed modules induced from a given crossed module homomorphism.
‣ SourceHom( m ) | ( attribute ) |
‣ RangeHom( m ) | ( attribute ) |
‣ IsInjective( m ) | ( property ) |
‣ IsSurjective( m ) | ( property ) |
‣ IsBijjective( m ) | ( property ) |
Let \((\theta,\varphi)\) be a homomorphism of crossed modules. If the homomorphisms \(\theta\) and \(\varphi\) are injective (surjective) then \((\theta,\varphi)\) is injective (surjective).
The attributes SourceHom and RangeHom store the two algebra homomorphisms \(\theta\) and \(\varphi\).
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 ]
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