GAP (Algebroids) - Chapter 11: Category of finitely presented associative unital matrix algebras

The input is a category of finitely presented associative unital matrix algebras. The output is the underlying category of finitely presented algebras.

11.1-2 UnderlyingCategoryOfMatrices

The input is a category of finitely presented associative unital matrix \(k\)-algebras. The output is the underlying category of \(k\)-matrices.

11.1-3 CoefficientsRing

The input is a category of finitely presented associative unital matrix \(k\)-algebras. The output is the underlying commutative ring \(k\) of coefficients.

11.1-4 CoefficientsRing

The input is a finitely presented associative unital matrix \(k\)-algebras. The output is the underlying commutative ring \(k\) of coefficients.

11.1-5 DefiningPairOfFinitelyPresentedMatrixAlgebra

The input is a finitely presented associative unital matrix \(k\)-algebras. The output is a pair. Its first entry is the underlying finitely presented \(k\)-algebra Its second entry is a pair with first entry an object in the the underlying category \(V\) of \(k\)-matrices and a list of morphisms in \(V\).

11.1-6 UnderlyingMorphismInCategoryOfFpAlgebras

The input is a finitely presented associative unital matrix algebra. The output is the underlying morphism between the underlying finitely presented algebras.

11.1-7 ListOfMatrixImages

The input is a finitely presented associative unital matrix algebra. The output is the list of matrix images of the generators of the source algebra.

11.1-8 NrGenerators

The input is a finitely presented associative unital matrix algebra. The output is the number of generators.

11.1-9 Generators

The input is a finitely presented associative unital matrix algebra. The output is the list of generators of the underlying finitely presented algebra.

11.1-10 MatrixGenerators

The input is a finitely presented associative unital matrix \(k\)-algebra. The output is the list of matrix generators as morphisms in the underlying category of \(k\)-matrices.

11.1-11 AssociatedLinearClosureOfPathCategory

The input is a finitely presented associative unital matrix \(k\)-algebra. The output is the corresponding ambient free \(k\)-algebra as a \(k\)-linear closure of a path category of loops on a single vertex.

11.1-12 DefiningRelations

The input is a finitely presented associative unital matrix algebra. The output its list of defining relations.

11.1-13 GroebnerBasisOfDefiningRelations

The input is a finitely presented associative unital matrix algebra. The output is the Gröbner basis of its list of defining relations.

11.1-14 AssociatedQuotientCategoryOfLinearClosureOfPathCategory

The input is a finitely presented associative unital matrix \(k\)-algebra. The output is the corresponding underlying finitely presented \(k\)-algebra as a quotient of a \(k\)-linear closure of a path category of loops on a single vertex.

11.1-15 Dimension

The input is a finitely presented associative unital matrix \(k\)-algebra. The output its \(k\)-dimension.

11.1-16 AmbientAlgebra

The input is a finitely presented associative unital matrix \(k\)-algebra. The output is the corresponding ambient free \(k\)-algebra.

11.2 Constructors

11.2-1 CategoryOfFpMatrixAlgebras

The input is a category Alg_k of finitely presented \(k\)-algebras and a category V of \(k\)-matrices. The output is the category of finitely presented associative unital matrix algebras over \(k\).

gap> LoadPackage( "Algebroids", false );
true
gap> F2 := HomalgRingOfIntegersInSingular( 2 );
GF(2)
gap> MatAlg_F2 := CategoryOfFpMatrixAlgebras( F2 );
CategoryOfFpMatrixAlgebras( GF(2) )
gap> Display( MatAlg_F2 );
A CAP category with name CategoryOfFpMatrixAlgebras( GF(2) ):

27 primitive operations were used to derive 76 operations for this category \
which algorithmically
* IsCategoryWithInitialObject
* IsCartesianCategory
* IsSymmetricMonoidalCategory
and not yet algorithmically
* IsCategoryWithCoequalizers
gap> T := TerminalObject( MatAlg_F2 );
<An object in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( T );
GF(2) / [ 1*id(o) ]

f.p. matrix algebra with an empty list of matrix generators
gap> iota := UniversalMorphismFromInitialObject( T );
<A morphism in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> I := Source( iota );
<An object in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( I );
GF(2) / [  ]

f.p. matrix algebra with an empty list of matrix generators
gap> Display( iota );
GF(2) / [ 1*id(o) ]
 ^
 |
[  ]
 |
GF(2) / [  ]
gap> K := TensorUnit( MatAlg_F2 );
<An object in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( K );
GF(2) / [  ]

f.p. matrix algebra with an empty list of matrix generators
gap> K = I;
true
gap> Alg_F2 := UnderlyingCategoryOfFpAlgebras( MatAlg_F2 );
CategoryOfFpAlgebras( GF(2) )
gap> Display( Alg_F2 );
A CAP category with name CategoryOfFpAlgebras( GF(2) ):

31 primitive operations were used to derive 130 operations for this category \
which algorithmically
* IsFiniteCocompleteCategory
* IsBicartesianCategory
* IsSymmetricMonoidalCategory
gap> Mat_F2 := UnderlyingCategoryOfMatrices( MatAlg_F2 );
Rows( GF(2) )
gap> q := FinQuiver( "q(o)[x:o->o,y:o->o]" );
FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" )
gap> P := PathCategory( q );
PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) )
gap> L := F2[P];
GF(2)-LinearClosure( PathCategory( \
FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) )
gap> relations := [ L.x^2 - L.x,  L.y^3 - L.y,  L.xy - L.y * (L.x+L.id_o) ];
[ 1*x^2 + 1*x:(o) -≻ (o), 1*y^3 + 1*y:(o) -≻ (o),\
  1*y⋅x + 1*x⋅y + 1*y:(o) -≻ (o) ]
gap> Q := L / relations;
GF(2)-LinearClosure( PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) ) / \
[ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> Dimension( Q );
6
gap> A := Q / Alg_F2;
<An object in CategoryOfFpAlgebras( GF(2) )>
gap> Dimension( A );
6
gap> Display( A );
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> mx := HomalgDiagonalMatrix( [ 1, 0, 1, 0 ], F2 ) / Mat_F2;
<A morphism in Rows( GF(2) )>
gap> my := DiagMat( [
>               HomalgZeroMatrix( 2, 2, F2 ),
>               CertainRows( HomalgIdentityMatrix( 2, F2 ), [ 2, 1 ] ) ] ) / Mat_F2;
<A morphism in Rows( GF(2) )>
gap> matrix_generators := Pair( 4 / Mat_F2, [ mx, my ] );
[ <A row module over GF(2) of rank 4>,\
  [ <A morphism in Rows( GF(2) )>, <A morphism in Rows( GF(2) )> ] ]
gap> M := ObjectConstructor( MatAlg_F2, Pair( A, matrix_generators ) );
<An object in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( M );
1,0,0,0,
0,0,0,0,
0,0,1,0,
0,0,0,0

0,0,0,0,
0,0,0,0,
0,0,0,1,
0,0,1,0

GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]

f.p. matrix algebra with the above matrix generators
gap> IsWellDefined( M );
true
gap> M.x;
1*x:(o) -≻ (o)
gap> M.y;
1*y:(o) -≻ (o)
gap> Dimension( M );
6
gap> NrGenerators( M );
2
gap> Generators( M );
[ 1*x:(o) -≻ (o), 1*y:(o) -≻ (o) ]
gap> MatrixGenerators( M ) = [ mx, my ];
true
gap> AssociatedLinearClosureOfPathCategory( M );
GF(2)-LinearClosure( PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) )
gap> DefiningRelations( M );
[ 1*x^2 + 1*x:(o) -≻ (o), 1*y^3 + 1*y:(o) -≻ (o), \
  1*y⋅x + 1*x⋅y + 1*y:(o) -≻ (o) ]
gap> AssociatedQuotientCategoryOfLinearClosureOfPathCategory( M );
GF(2)-LinearClosure( PathCategory( FinQuiver( "q(o)[x:o-≻o,y:o-≻o]" ) ) ) \
/ [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> GroebnerBasisOfDefiningRelations( M );
[ 1*x^2 + 1*x:(o) -≻ (o), 1*y^3 + 1*y:(o) -≻ (o), \
  1*y⋅x + 1*x⋅y + 1*y:(o) -≻ (o) ]
gap> P := DirectProduct( I, M );
<An object in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( P );
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

0,0,0,0,0,
0,1,0,0,0,
0,0,0,0,0,
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,
0,0,0,0,1,
0,0,0,1,0

0,0,0,0,0,
0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
0,0,0,0,1

GF(2)<a1,a2,a3,a4> / [ 1*a2^2 + 1*a2, 1*a3^3 + 1*a3, 1*a3⋅a2 + 1*a2⋅a3 + 1*a3, \
... ]

f.p. matrix algebra with the above matrix generators
gap> counit := Counit( M, [ 1, 0 ] );
<A morphism in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( counit );
GF(2) / [  ]
 ^
 |
[ 1*id(o), 0 ]
 |
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> IsWellDefined( counit );
true
gap> o := M.id_o;
1*id(o):(o) -≻ (o)
gap> x := M.x;
1*x:(o) -≻ (o)
gap> y := M.y;
1*y:(o) -≻ (o)
gap> comult := Comultiplication( M,
>                   [ [ [ o, x ], [ x, o ], [ o, o ], [ y^2, y^2 ] ],
>                     [ [ o, y ], [ y, o + y^2 ] ] ] );
<A morphism in CategoryOfFpMatrixAlgebras( GF(2) )>
gap> Display( comult );
GF(2)<a1,a2,a3,a4> / [ 1*a1^2 + 1*a1, 1*a2^3 + 1*a2, 1*a2⋅a1 + 1*a1⋅a2 + 1*a2, \
... ]
 ^
 |
[ 1*a2^2⋅a4^2 + 1*a1 + 1*a3 + 1*id(o), 1*a2⋅a4^2 + 1*a2 + 1*a4 ]
 |
GF(2)<x,y> / [ 1*x^2 + 1*x, 1*y^3 + 1*y, 1*y⋅x + 1*x⋅y + 1*y ]
gap> IsWellDefined( comult );
true
gap> Perform( ListOfMatrixImages( comult ), Display );
Source:
A row module over GF(2) of rank 16

Matrix:
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

Range:
A row module over GF(2) of rank 16

A morphism in Rows( GF(2) )
Source:
A row module over GF(2) of rank 16

Matrix:
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0

Range:
A row module over GF(2) of rank 16

A morphism in Rows( GF(2) )

11.2-2 CategoryOfFpMatrixAlgebras

The input is a commutative ring k. The output is the category of finitely presented associative unital matrix algebras over \(k\).

11.3 Operations

11.3-1 Counit

11.3-2 Comultiplication

11.4 GAP categories

11.4-1 IsCategoryOfFinitelyPresentedMatrixAlgebras

The GAP category of categories of finitely presented associative unital matrix algebras.

11.4-2 IsObjectInCategoryOfFpMatrixAlgebras

11.4-3 IsMorphismInCategoryOfFpMatrixAlgebras

The GAP category of morphisms of finitely presented associative unital matrix algebras.

11 Category of finitely presented associative unital matrix algebras

11.1 Attributes

11.1-1 UnderlyingCategoryOfFpAlgebras