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#############################################################################
##
#W alg2obj.gd The XMODALG package Zekeriya Arvasi
#W & Alper Odabas
#Y Copyright (C) 2014-2025, Zekeriya Arvasi & Alper Odabas,
##
############################## 2d-algebras ###########################
DeclareCategory( "Is2dAlgebra", Is2DimensionalDomain and IsAlgebra );
DeclareCategoryCollections( "Is2dAlgebra" );
BindGlobal( "Family2dAlgebra",
NewFamily( "Family2dAlgebra", Is2dAlgebra, CanEasilySortElements,
CanEasilySortElements ) );
DeclareProperty( "Is2dAlgebraObject", Is2DimensionalDomain );
DeclareOperation( "Sub2dAlgebra", [ Is2dAlgebra, IsAlgebra, IsAlgebra ] );
######################### (pre-)crossed modules ######################
DeclareRepresentation( "IsPreXModAlgebraObj",
Is2dAlgebra and IsAttributeStoringRep, [ "boundary", "action" ] );
DeclareProperty( "IsPreXModAlgebra", Is2dAlgebra );
DeclareProperty( "IsXModAlgebra", Is2dAlgebra );
InstallTrueMethod( IsPreXModAlgebra, IsXModAlgebra );
BindGlobal( "PreXModAlgebraObjType",
NewType( Family2dAlgebra, IsPreXModAlgebraObj ) );
DeclareAttribute( "XModAlgebraAction", IsPreXModAlgebra );
DeclareOperation( "XModAlgebraObj",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareOperation( "XModAlgebraObjNC",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareGlobalFunction( "XModAlgebra" );
## section 4.1.7
############################################################################
##
## XModAlgebraByBoundaryAndAction( <hom> <alg> )
##
## <#GAPDoc Label="XModAlgebraByBoundaryAndAction">
## <ManSection>
## <Oper Name="XModAlgebraByBoundaryAndAction"
## Arg="bdy act" />
## <Oper Name="PreXModAlgebraByBoundaryAndAction"
## Arg="bdy,act" />
## <Description>
## When a suitable pair of algebra homomorphisms are available,
## these operations may be used.
## The example uses the algebra action created in section
## <Ref Sect="AlgebraActionByHomomorphism" />.
## </Description>
## </ManSection>
## <Example>
## <![CDATA[
## gap> bdy3 := AlgebraHomomorphismByImages( Rc3, A3, [ g3 ], [ m3 ] );
## [ (1)*(1,2,3) ] -> [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
## gap> X3 := XModAlgebraByBoundaryAndAction( bdy3, actg3 );
## [ GR(c3) -> A3 ]
## gap> Display( X3 );
## Crossed module [GR(c3) -> A3] :-
## : Source algebra GR(c3) has generators:
## [ (1)*(), (1)*(1,2,3) ]
## : Range algebra A3 has generators:
## [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
## : Boundary homomorphism maps source generators to:
## [ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
## [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
## ]]>
## </Example>
## <#/GAPDoc>
##
DeclareOperation( "PreXModAlgebraByBoundaryAndAction",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareOperation( "PreXModAlgebraByBoundaryAndActionNC",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareOperation( "XModAlgebraByBoundaryAndAction",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareOperation( "XModAlgebraByBoundaryAndActionNC",
[ IsAlgebraHomomorphism, IsAlgebraAction ] );
DeclareOperation( "XModAlgebraBySurjection", [ IsAlgebraHomomorphism ] );
DeclareOperation( "XModAlgebraByMultiplierAlgebra",
[ IsAlgebra ] );
DeclareOperation( "XModAlgebraByIdeal",
[ IsAlgebra, IsAlgebra ] );
DeclareAttribute( "AugmentationXMod", IsGroupAlgebra );
DeclareOperation( "SubPreXModAlgebra",
[ IsPreXModAlgebra, IsAlgebra, IsAlgebra ] );
DeclareOperation( "SubXModAlgebra", [ IsXModAlgebra, IsAlgebra, IsAlgebra] );
DeclareOperation( "IsSubPreXModAlgebra", [ Is2dAlgebraObject, Is2dAlgebraObject ] );
DeclareOperation( "IsSubXModAlgebra", [ Is2dAlgebraObject, Is2dAlgebraObject ] );
############################# cat1-algebras ##########################
DeclareRepresentation( "IsPreCat1AlgebraObj",
Is2dAlgebra and IsAttributeStoringRep,
[ "tailMap", "headMap", "rangeEmbedding" ] );
BindGlobal( "PreCat1AlgebraObjType",
NewType( Family2dAlgebra, IsPreCat1AlgebraObj ) );
DeclareProperty( "IsPreCat1Algebra", Is2dAlgebra );
DeclareProperty( "IsCat1Algebra", Is2dAlgebra );
DeclareGlobalFunction( "PreCat1Algebra" );
DeclareGlobalFunction( "Cat1Algebra" );
DeclareOperation( "PreCat1AlgebraObj",
[ IsAlgebraHomomorphism, IsAlgebraHomomorphism, IsAlgebraHomomorphism ] );
DeclareOperation( "PreCat1AlgebraByTailHeadEmbedding",
[ IsAlgebraHomomorphism, IsAlgebraHomomorphism, IsAlgebraHomomorphism ] );
DeclareAttribute( "HeadMap", IsPreCat1Algebra );
DeclareAttribute( "TailMap", IsPreCat1Algebra );
DeclareAttribute( "RangeEmbedding", IsPreCat1Algebra );
DeclareAttribute( "KernelEmbedding", IsPreCat1Algebra );
DeclareOperation( "SubPreCat1Algebra",
[ IsPreCat1Algebra, IsAlgebra, IsAlgebra ] );
DeclareOperation( "SubCat1Algebra",
[ IsCat1Algebra, IsAlgebra, IsAlgebra ] );
DeclareOperation( "IsSubPreCat1Algebra",
[ Is2dAlgebraObject, Is2dAlgebraObject ] );
DeclareOperation( "IsSubCat1Algebra",
[ Is2dAlgebraObject, Is2dAlgebraObject ] );
DeclareOperation( "PreCat1AlgebraByEndomorphisms",
[ IsAlgebraHomomorphism, IsAlgebraHomomorphism ] );
DeclareOperation( "Cat1AlgebraSelect", [ IsInt, IsInt, IsInt, IsInt ] );
DeclareProperty( "IsIdentityCat1Algebra", IsCat1Algebra );
DeclareOperation( "AllCat1Algebras", [ IsField, IsGroup ] );
DeclareOperation( "IsIsomorphicCat1Algebra", [ IsCat1Algebra, IsCat1Algebra ] );
DeclareOperation( "IsomorphicCat1AlgebraFamily", [ IsCat1Algebra, IsList ] );
DeclareOperation( "AllCat1AlgebrasUpToIsomorphism", [ IsList ] );
########################## conversion functions #######################
DeclareAttribute( "PreXModAlgebraOfPreCat1Algebra", IsPreCat1Algebra );
DeclareAttribute( "PreCat1AlgebraOfPreXModAlgebra", IsPreXModAlgebra );
DeclareAttribute( "XModAlgebraOfCat1Algebra", IsCat1Algebra );
DeclareAttribute( "Cat1AlgebraOfXModAlgebra", IsXModAlgebra );
## addition to Cat1AlgebraOfXModAlgebra, section 5.1.1
############################################################################
##
## <#GAPDoc Label="Cat1AlgebraOfXModAlgebra">
## <P/>
## As a second example, we convert the crossed module <M>X4</M>
## constructed in section <Ref Sect="XModAlgebraByModule"/>
##
## <Example>
## <![CDATA[
## gap> C3 := Cat1AlgebraOfXModAlgebra( X3 );
## [A3 |X GR(c3) -> A3]
## gap> Display( C3 );
## Cat1-algebra [A3 |X GR(c3) => A3] :-
## : range algebra has generators:[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
## : tail homomorphism maps source generators to:
## [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
## [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
## : head homomorphism maps source generators to:
## [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
## [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
## [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
## [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ] ]
## : range embedding maps range generators to: [ v.1 ]
## : kernel has generators: [ v.4, v.5, v.6 ]
## ]]>
## </Example>
## <#/GAPDoc>
##
DeclareOperation( "SDproduct", [ Is2dAlgebraObject ] );
[ Dauer der Verarbeitung: 0.21 Sekunden
(vorverarbeitet)
]
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