Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind

### 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 );
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 );;
>               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 );;
>               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 );
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 );
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 );
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)
true
gap> ZsrepG := PositivelyZGradedClosureCategory( srepG );
SymmetricGroup( [ 1 .. 4 ] ) )
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>
>               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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