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1 Introduction
 1.1 Examples for the internal symmetric algebra
 1.2 Examples for the internal exterior algebra

1 Introduction

1.1 Examples for the internal symmetric algebra

The following examples demonstrate the internalization of the symmetric algera and its modules in the positively graded closure of various tensor categories.

1.1-1 The tensor category of finite dimensional GF(2)-vector spaces
gap> F2 := HomalgRingOfIntegers( 2 );;
gap> F2vec := MatrixCategory( F2 );;
gap> v := VectorSpaceObject( 3, F2 );;
gap> ZF2vec := PositivelyZGradedClosureCategory( F2vec );
PositivelyZGradedClosureCategory( Category of matrices over GF(2) )
gap> V := ObjectInZGradedClosureCategoryWithBounds( ZF2vec, v, 1 );;
gap> SVMod := CategoryOfLeftSModules( v );
Abelian category of left modules over the internal symmetric algebra of
A vector space object over GF(2) of dimension 3
with undecidable (mathematical) equality of morphisms and
uncomputable lifts and colifts
gap> ModSV := CategoryOfRightSModules( v );;
gap> SV := UnderlyingActingObject( SVMod );
<An object in PositivelyZGradedClosureCategory(
 Category of matrices over GF(2) )>
gap> S := SymmetricAlgebraAsLeftModule( v );
<An object in Abelian category of
 left modules over the internal symmetric algebra of
 A vector space object over GF(2) of dimension 3
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> u := VectorSpaceObject( 3, F2 );;
gap> U := ObjectInZGradedClosureCategoryWithBounds( ZF2vec, u, 3 );;
gap> F := FreeInternalModule( U, SVMod );
<An object in Abelian category of
 left modules over the internal symmetric algebra of
 A vector space object over GF(2) of dimension 3
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> H := FreeInternalModule( U, ModSV );;
gap> e1 := MorphismInZGradedClosureCategoryWithBounds(
>               ObjectInZGradedClosureCategoryWithBounds( ZF2vec, SV[0], 3 ),
>               VectorSpaceMorphism(
>                   SV[0],
>                   CertainRows( HomalgIdentityMatrix( 10, F2 ), [ 1 ] ),
>                   SV[3]
>               ),
>               3,
>               SV );;
gap> e1 := InternalElement( e1 );
degree: 3

 1 . . . . . . . . .

A morphism in Category of matrices over GF(2)

1.1-2 The tensor category of finite dimensional Q-vector spaces
gap> Q := HomalgFieldOfRationals( );;
gap> Qvec := MatrixCategory( Q );;
gap> v := VectorSpaceObject( 3, Q );;
gap> ZQvec := PositivelyZGradedClosureCategory( Qvec );
PositivelyZGradedClosureCategory( Category of matrices over Q )
gap> V := ObjectInZGradedClosureCategoryWithBounds( ZQvec, v, 1 );;
gap> SVMod := CategoryOfLeftSModules( v );
Abelian category of left modules over the internal symmetric algebra of
A vector space object over Q of dimension 3
with undecidable (mathematical) equality of morphisms and
uncomputable lifts and colifts
gap> ModSV := CategoryOfRightSModules( v );;
gap> SV := UnderlyingActingObject( SVMod );
<An object in PositivelyZGradedClosureCategory( Category of matrices over Q )>
gap> S := SymmetricAlgebraAsLeftModule( v );
<An object in Abelian category of
 left modules over the internal symmetric algebra of
 A vector space object over Q of dimension 3
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> u := VectorSpaceObject( 3, Q );;
gap> U := ObjectInZGradedClosureCategoryWithBounds( ZQvec, u, 3 );;
gap> F := FreeInternalModule( U, SVMod );
<An object in Abelian category of
 left modules over the internal symmetric algebra of
 A vector space object over Q of dimension 3
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> H := FreeInternalModule( U, ModSV );;
gap> e1 := MorphismInZGradedClosureCategoryWithBounds(
>               ObjectInZGradedClosureCategoryWithBounds( ZQvec, SV[0], 3 ),
>               VectorSpaceMorphism(
>                   SV[0],
>                   CertainRows( HomalgIdentityMatrix( 10, Q ), [ 1 ] ),
>                   SV[3]
>               ),
>               3,
>               SV );;
gap> e1 := InternalElement( e1 );
degree: 3

[ [  1,  0,  0,  0,  0,  0,  0,  0,  0,  0 ] ]

A morphism in Category of matrices over Q

1.1-3 The tensor category of finite dimensional Q-representations of the symmetric group Sym(4).
gap> G := SymmetricGroup( 4 );;
gap> RepG := RepresentationCategory( G );;
gap> irr := Irr( G );;
gap> one := RepresentationCategoryObject( irr[5], RepG, "1" );;
gap> sigma := RepresentationCategoryObject( irr[1], RepG, "sigma" );;
gap> rho := RepresentationCategoryObject( irr[3], RepG, "rho" );;
gap> nu := RepresentationCategoryObject( irr[4], RepG, "nu" );;
gap> chi := RepresentationCategoryObject( irr[2], RepG, "chi" );;
gap> v := chi;;
gap> ZRepG := PositivelyZGradedClosureCategory( RepG );
PositivelyZGradedClosureCategory( The representation category of
SymmetricGroup( [ 1 .. 4 ] ) )
gap> V := ObjectInZGradedClosureCategoryWithBounds( ZRepG, v, 1 );;
gap> SVMod := CategoryOfLeftSModules( v );
Abelian category of left modules over the internal symmetric algebra of
1*(chi)
with undecidable (mathematical) equality of morphisms and
uncomputable lifts and colifts
gap> ModSV := CategoryOfRightSModules( v );;
gap> SV := UnderlyingActingObject( SVMod );
<An object in PositivelyZGradedClosureCategory( The representation category of
 SymmetricGroup( [ 1 .. 4 ] ) )>
gap> S := SymmetricAlgebraAsLeftModule( v );
<An object in Abelian category of
 left modules over the internal symmetric algebra of 1*(chi)
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> u := chi;;
gap> U := ObjectInZGradedClosureCategoryWithBounds( ZRepG, u, 3 );;
gap> F := FreeInternalModule( U, SVMod );
<An object in Abelian category of
 left modules over the internal symmetric algebra of 1*(chi)
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> H := FreeInternalModule( U, ModSV );;
gap> c1 := UniversalMorphismFromFreeModule( F, Support( F[6] )[4], 6, 1 );;
gap> c2 := UniversalMorphismFromFreeModule( F, Support( F[6] )[4], 6, 2 );;
gap> c := 2 * c1 - 3 * c2;;
gap> Display( c[6] );
Component: (nu)

2,-3,0,0

A morphism in Category of matrices over Q

------------------------
gap> a := InternalElement( SV, Support( SV[1] )[1], 1, 1 );;
gap> b := InternalElement( SV, Support( SV[2] )[1], 2, 1 );;
gap> c := InternalElement( SV, Support( SV[2] )[2], 2, 1 );;
gap> d := InternalElement( SV, Support( SV[3] )[3], 3, 1 );;
gap> cc := UniversalMorphismFromFreeModule( c );;
gap> dd := UniversalMorphismFromFreeModule( d );;
gap> pp := ProjectionInFactorOfFiberProduct( [ cc, dd ], 1 );;
gap> qq := ProjectionInFactorOfFiberProduct( [ cc, dd ], 2 );;
gap> ff := PreCompose( pp, cc ) - PreCompose( qq, dd );;
gap> Set( List( Sublist( ff, [ 0 .. 5 ] ), IsZero ) );
[ true ]
gap> ss := UniversalMorphismIntoDirectSum( [ pp, -qq ] );;
gap> tt := UniversalMorphismFromDirectSum( [ cc, dd ] );;
gap> homology := HomologyObject( ss, tt );;
gap> Set( List( Sublist( homology, [ 0 .. 5 ] ), IsZero ) );
[ true ]
gap> e1 := InternalElement( SV, sigma, 3, 1 );;
gap> e2 := InternalElement( SV, chi, 3, 1 );;
gap> e3 := InternalElement( SV, chi, 3, 2 );;
gap> e4 := InternalElement( SV, nu, 3, 1 );;
gap> e := e1 + 2 * e2 - 1/3 * e3 + e4;
degree: 3

Component: (sigma)

1

A morphism in Category of matrices over Q

------------------------
Component: (chi)

2,-1/3

A morphism in Category of matrices over Q

------------------------
Component: (nu)

1

A morphism in Category of matrices over Q

------------------------
gap> e2 * e3 = BraidedMultiplication( e3, e2 );
true
gap> m1 := InternalElement( F, nu, 6, 1 );;
gap> m2 := InternalElement( F, nu, 6, 2 );;
gap> m := 2 * m1 - 1/3 * m2;
degree: 6

Component: (nu)

2,-1/3,0,0

A morphism in Category of matrices over Q

------------------------

1.1-4 The tensor category of finite dimensional Z-graded Q-representations of the symmetric group Sym(4).
gap> G := SymmetricGroup( 4 );;
gap> RepG := RepresentationCategoryZGraded( G );;
gap> irr := Irr( G );;
gap> v := RepresentationCategoryZGradedObject( 1, irr[2], RepG );;
gap> ZRepG := PositivelyZGradedClosureCategory( RepG );
PositivelyZGradedClosureCategory( The skeletal Z-graded
representation category of SymmetricGroup( [ 1 .. 4 ] ) )
gap> V := ObjectInZGradedClosureCategoryWithBounds( ZRepG, v );;
gap> SVMod := CategoryOfLeftSModules( v );
Abelian category of left modules over the internal symmetric algebra of
1*(x_[1, 2])
with undecidable (mathematical) equality of morphisms and
uncomputable lifts and colifts
gap> ModSV := CategoryOfRightSModules( v );;
gap> SV := UnderlyingActingObject( SVMod );
<An object in PositivelyZGradedClosureCategory( The skeletal Z-graded
 representation category of SymmetricGroup( [ 1 .. 4 ] ) )>
gap> S := SymmetricAlgebraAsLeftModule( v );
<An object in Abelian category of
 left modules over the internal symmetric algebra of 1*(x_[1, 2])
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> u := RepresentationCategoryZGradedObject( 3, irr[2], RepG );;
gap> U := ObjectInZGradedClosureCategoryWithBounds( ZRepG, u );;
gap> F := FreeInternalModule( U, SVMod );
<An object in Abelian category
 of left modules over the internal symmetric algebra of 1*(x_[1, 2])
 with undecidable (mathematical) equality of morphisms and
 uncomputable lifts and colifts>
gap> H := FreeInternalModule( U, ModSV );;
gap> nu6 := Support( F[6] )[4];
<x_[6, 4]>
gap> c1 := UniversalMorphismFromFreeModule( F, nu6, 1 );;
gap> c2 := UniversalMorphismFromFreeModule( F, nu6, 2 );;
gap> c := 2 * c1 - 3 * c2;;
gap> Display( c[6] );
Component: (x_[6, 4])

2,-3,0,0

A morphism in Category of matrices over Q

------------------------
gap> a := InternalElement( SV, Support( SV[1] )[1], 1 );;
gap> b := InternalElement( SV, Support( SV[2] )[1], 1 );;
gap> c := InternalElement( SV, Support( SV[2] )[2], 1 );;
gap> d := InternalElement( SV, Support( SV[3] )[3], 1 );;
gap> cc := UniversalMorphismFromFreeModule( c );;
gap> dd := UniversalMorphismFromFreeModule( d );;
gap> pp := ProjectionInFactorOfFiberProduct( [ cc, dd ], 1 );;
gap> qq := ProjectionInFactorOfFiberProduct( [ cc, dd ], 2 );;
gap> ff := PreCompose( pp, cc ) - PreCompose( qq, dd );;
gap> Set( List( Sublist( ff, [ 0 .. 5 ] ), IsZero ) );
[ true ]
gap> ss := UniversalMorphismIntoDirectSum( [ pp, -qq ] );;
gap> tt := UniversalMorphismFromDirectSum( [ cc, dd ] );;
gap> homology := HomologyObject( ss, tt );;
gap> Set( List( Sublist( homology, [ 0 .. 5 ] ), IsZero ) );
[ true ]
gap> sigma3 := Support( SV[3] )[1];
<x_[3, 1]>
gap> chi3 := Support( SV[3] )[2];
<x_[3, 2]>
gap> nu3 := Support( SV[3] )[3];
<x_[3, 4]>
gap> e1 := InternalElement( SV, sigma3, 1 );;
gap> e2 := InternalElement( SV, chi3, 1 );;
gap> e3 := InternalElement( SV, chi3, 2 );;
gap> e4 := InternalElement( SV, nu3, 1 );;
gap> e := e1 + 2 * e2 - 1/3 * e3 + e4;
degree: 3

Component: (x_[3, 1])

1

A morphism in Category of matrices over Q

------------------------
Component: (x_[3, 2])

2,-1/3

A morphism in Category of matrices over Q

------------------------
Component: (x_[3, 4])

1

A morphism in Category of matrices over Q

------------------------
gap> e2 * e3 = BraidedMultiplication( e3, e2 );
true
gap> m1 := InternalElement( F, nu6, 1 );;
gap> m2 := InternalElement( F, nu6, 2 );;
gap> m := 2 * m1 - 1/3 * m2;
degree: 6

Component: (x_[6, 4])

2,-1/3,0,0

A morphism in Category of matrices over Q

------------------------

1.1-5 Futher computations in the symmetric algebra

This is an example for explicit computations in the symmetric algebra \mathrm{S}W internal to the positively graded closure \mathrm{srep}_k(G)_+ of the tensor category \mathrm{srep}_k(G), where G := S_4 is the symmetric group on four points and k = \mathbb{Q}, its minimal splitting field.

gap> LoadPackage( "InternalModules" : OnlyNeeded );
true
gap> srepG := RepresentationCategory( SymmetricGroup( 4 ) );
The representation category of SymmetricGroup( [ 1 .. 4 ] )
gap> Display( srepG );
A CAP category with name The representation category
of SymmetricGroup( [ 1 .. 4 ] ):

44 primitive operations were used to derive 357 operations for this category
which algorithmically
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRing
* IsAbelianCategory
* IsRigidSymmetricClosedMonoidalCategory
and furthermore mathematically
* IsSkeletalCategory
gap> G := UnderlyingGroupForRepresentationCategory( srepG );
Sym( [ 1 .. 4 ] )
gap> irr := Irr( G );;
gap> one := RepresentationCategoryObject( irr[5], srepG, "𝟙" );
1*(𝟙)
gap> sigma := RepresentationCategoryObject( irr[1], srepG, "σ" );
1*(σ)
gap> rho := RepresentationCategoryObject( irr[3], srepG, "ρ" );
1*(ρ)
gap> nu := RepresentationCategoryObject( irr[4], srepG, "ν" );
1*(ν)
gap> chi := RepresentationCategoryObject( irr[2], srepG, "χ" );
1*(χ)
gap> TensorProduct( rho, sigma, rho );
1*(σ) + 1*(ρ) + 1*(𝟙)
gap> TensorProduct( rho, rho );
1*(σ) + 1*(ρ) + 1*(𝟙)
gap> Display( AssociatorRightToLeft( rho, sigma, rho ) );
Component: (σ)

-1

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

1

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

-1

A morphism in Category of matrices over Q

------------------------
gap> Display( Braiding( rho, rho ) );
Component: (σ)

-1

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

1

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

1

A morphism in Category of matrices over Q

------------------------
gap> W := chi;
1*(χ)
gap> LoadPackage( "GradedCategories" );
true
gap> ZsrepG := PositivelyZGradedClosureCategory( srepG );
PositivelyZGradedClosureCategory( The representation category of
SymmetricGroup( [ 1 .. 4 ] ) )
gap> LoadPackage( "InternalModules" );
true
gap> SWMod := CategoryOfLeftSModules( W );
Abelian category of left modules over the internal symmetric algebra of 1*(χ)
with undecidable (mathematical) equality of morphisms
and uncomputable lifts and colifts
gap> SW := UnderlyingActingObject( SWMod );
<An object in PositivelyZGradedClosureCategory( The representation category of
SymmetricGroup( [ 1 .. 4 ] ) )>
gap> SW[0];
1*(𝟙)
gap> SW[1];
1*(χ)
gap> SW[2];
1*(ρ) + 1*(ν) + 1*(𝟙)
gap> SW[3];
1*(σ) + 2*(χ) + 1*(ν)
gap> SW[4];
1*(χ) + 2*(ρ) + 2*(ν) + 2*(𝟙)
gap> chi1 := InternalElement( SW, chi, 1, 1 );
degree: 1

Component: (χ)

1

A morphism in Category of matrices over Q

------------------------

gap> rho2 := InternalElement( SW, rho, 2, 1 );
degree: 2

Component: (ρ)

1

A morphism in Category of matrices over Q

------------------------

gap> nu2 := InternalElement( SW, nu, 2, 1 );
degree: 2

Component: (ν)

1

A morphism in Category of matrices over Q

------------------------

gap> nu3 := InternalElement( SW, nu, 3, 1 );
degree: 3

Component: (ν)

1

A morphism in Category of matrices over Q

------------------------

gap> chi1 * chi1;
degree: 2

Component: (ρ)

1

A morphism in Category of matrices over Q

------------------------
Component: (ν)

1

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

1

A morphism in Category of matrices over Q

------------------------

gap> chi1 * (chi1 * chi1);
degree: 3

Component: (σ)

1

A morphism in Category of matrices over Q

------------------------
Component: (χ)

5,3

A morphism in Category of matrices over Q

------------------------
Component: (ν)

3

A morphism in Category of matrices over Q

------------------------

gap> chi1 * (chi1 * chi1) = (chi1 * chi1) * chi1;
true
gap> chi1 * rho2;
degree: 3

Component: (χ)

0,2

A morphism in Category of matrices over Q

------------------------
Component: (ν)

1

A morphism in Category of matrices over Q

------------------------

gap> chi1 * rho2 = rho2 * chi1;
true
gap> chi1 * nu2;
degree: 3

Component: (σ)

1

A morphism in Category of matrices over Q

------------------------
Component: (χ)

8,0

A morphism in Category of matrices over Q

------------------------
Component: (ν)

2

A morphism in Category of matrices over Q

------------------------

gap> chi1 * nu2 = nu2 * chi1;
true
gap> chi1 * (chi1 * (chi1 * chi1));
degree: 4

Component: (χ)

-2

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

3,-31

A morphism in Category of matrices over Q

------------------------
Component: (ν)

5,6

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

5,3

A morphism in Category of matrices over Q

------------------------

gap> (chi1 * chi1) * (chi1 * chi1);
degree: 4

Component: (χ)

8

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

9/4,43/4

A morphism in Category of matrices over Q

------------------------
Component: (ν)

-16,-6

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

-7,9/8

A morphism in Category of matrices over Q

------------------------

gap> ((chi1 * chi1) * chi1) * chi1;
degree: 4

Component: (χ)

26

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

3,-31

A morphism in Category of matrices over Q

------------------------
Component: (ν)

5,6

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

5,3

A morphism in Category of matrices over Q

------------------------

gap> (chi1 * chi1) * rho2;
degree: 4

Component: (χ)

4

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

5/4,-9/4

A morphism in Category of matrices over Q

------------------------
Component: (ν)

-4,-4

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

0,1/8

A morphism in Category of matrices over Q

------------------------

gap> chi1 * (chi1 * rho2);
degree: 4

Component: (χ)

-2

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

2,-12

A morphism in Category of matrices over Q

------------------------
Component: (ν)

-1,-1

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

0,2

A morphism in Category of matrices over Q

------------------------

gap> (chi1 * chi1) * nu2;
degree: 4

Component: (χ)

4

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

0,16

A morphism in Category of matrices over Q

------------------------
Component: (ν)

-8,-1

A morphism in Category of matrices over Q

------------------------
Component: (𝟙)

-4,0

A morphism in Category of matrices over Q

------------------------

gap> chi1 * nu3;
degree: 4

Component: (χ)

4

A morphism in Category of matrices over Q

------------------------
Component: (ρ)

0,-12

A morphism in Category of matrices over Q

------------------------
Component: (ν)

1,1

A morphism in Category of matrices over Q

------------------------

gap> chi1 * nu3 = nu3 * chi1;
true
gap> chi1 * (chi1 * (chi1 * chi1)) <> ((chi1 * chi1) * chi1) * chi1;
true

1.2 Examples for the internal exterior algebra

The following examples demonstrate the internalization of the exterior algera and its modules in the finitely graded closure of various tensor categories.

1.2-1 The tensor category of finite dimensional Q-vector spaces
gap> Q := HomalgFieldOfRationals( );;
gap> Qvec := MatrixCategory( Q );;
gap> v := VectorSpaceObject( 6, Q );;
gap> ZQvec := FinitelyZGradedClosureCategory( Qvec );
FinitelyZGradedClosureCategory( Category of matrices over Q )
gap> V := ObjectInZGradedClosureCategoryWithBounds( ZQvec, v, 1 );;
gap> ExtVMod := CategoryOfLeftEModules( v );
Abelian category of left modules over the internal exterior algebra of
A vector space object over Q of dimension 6
gap> ModExtV := CategoryOfRightEModules( v );;
gap> ExtV := UnderlyingActingObject( ExtVMod );
<An object in FinitelyZGradedClosureCategory( Category of matrices over Q )>
gap> ext := ExteriorAlgebraAsLeftModule( v );
<An object in Abelian category of
 left modules over the internal exterior algebra of
 A vector space object over Q of dimension 6>
gap> e1 := MorphismInZGradedClosureCategoryWithBounds(
>               ObjectInZGradedClosureCategoryWithBounds( ZQvec, ExtV[0], 3 ),
>               VectorSpaceMorphism(
>                   ExtV[0],
>                   CertainRows( HomalgIdentityMatrix( 20, Q ), [ 6 ] ),
>                   ExtV[3]
>               ),
>               3,
>               ExtV );;
gap> e1 := InternalElement( e1 );
degree: 3

[ [  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0 ] ]

A morphism in Category of matrices over Q
gap> u := VectorSpaceObject( 3, Q );;
gap> U := ObjectInZGradedClosureCategoryWithBounds( ZQvec, u, 3 );;
gap> F := FreeInternalModule( U, ExtVMod );
<An object in Abelian category of
 left modules over the internal exterior algebra of
 A vector space object over Q of dimension 6>
gap> H := FreeInternalModule( U, ModExtV );;
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