Let \(k\) be a field and \(I\) be a totally ordered set. We denote the matrix category of \(k\) by \(k\mathrm{-vec}\) (see the package \(\texttt{LinearAlgebraForCAP}\)). The semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\) associated to \(k\) and \(I\) is defined as the full subcategory of the product category \(\prod_{i \in I} k\mathrm{-vec}\) generated by those \(I\)-indexed tuples having only finitely many non-zero entries. By \(\chi^i\), we denote the object which is \(1\) at entry \(i\) and \(0\) otherwise. Thus, an arbitrary object in \(\bigoplus_{i \in I} k\mathrm{-vec}\) can be written as \(\oplus_{i \in I}a_i \chi^i\) for non-negative numbers \(a_i\) for which only finitely many are non-zero.
‣ SemisimpleCategory( k, m, u, s, b, L ) | ( operation ) |
Returns: a category
The arguments are:
a homalg field \(k\),
a membership function \(m\) sending any GAP object to a boolean,
a GAP object \(u\),
a string \(s\) containing a filename in the folder "/gap/AssociatorsDatabase/" of this package,
a boolean \(b\),
a list \(L\) containing 4 entries, where the first 3 are filters and the last one is a string.
The output is a CAP category modelling \(\bigoplus_{i \in I} k\mathrm{-vec}\), where \(I\) is the set defined by the membership function \(m\). Note that objects in \(I\) are expected to be equipped with operations enlisted in the chapter "Irreducible Objects". Furthermore, this CAP category is a rigid symmetric closed monoidal Abelian category. Its tensor product is defined by the data of the file \(s\), where the boolean \(b\) is true if the associator stored in \(s\) was computed for all triples, and false otherwise (cf. chapter "Associators"). Its braiding and duality comes from the additional structure required for \(I\). Its tensor unit is modelled by \(u\). The three filters of the \(L\) are filters for the resulting category, its objects, and its morphisms. \(L_4\) is the name of the resulting category.
‣ SemisimpleCategory( k, m, u, s, b ) | ( operation ) |
Returns: a category
The arguments are:
a homalg field \(k\),
a membership function \(m\) sending any GAP object to a boolean,
a GAP object \(u\),
a string \(s\) containing a filename in the folder "/gap/AssociatorsDatabase/" of this package,
a boolean \(b\).
This function calls SemisimpleCategory on the six arguments [ k, m, u, s, b, [ IsObject, IsObject, IsObject, automatically generated name ] ]
‣ SemisimpleCategoryMorphism( s, L, r ) | ( operation ) |
Returns: a morphism
The arguments are an object \(s\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), a list of pairs \(L = [ [ \phi_1, i_1 ], \dots [ \phi_l, i_l ] ]\) where \(\phi_j\) are morphisms in the Matrix Category \(k\mathrm{-vec}\) and \(i_j \in I\), and another object \(r\) in the same semisimple category. The output is a morphism in \(\bigoplus_{i \in I} k\mathrm{-vec}\) from \(s\) to \(r\) whose \(i\)-th component is given by \(\phi_i\). For this morphism to be well defined, there has to be an \(\phi_i\) for every \(i\) in the support of \(s\) and \(r\).
‣ SemisimpleCategoryMorphismSparse( s, L, r ) | ( operation ) |
Returns: a morphism
The arguments are an object \(s\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), a list of pairs \(L = [ [ \phi_1, i_1 ], \dots [ \phi_l, i_l ] ]\) where \(\phi_j\) are morphisms in the Matrix Category \(k\mathrm{-vec}\) and \(i_j \in I\), and another object \(r\) in the same semisimple category. The output is a morphism in \(\bigoplus_{i \in I} k\mathrm{-vec}\) from \(s\) to \(r\) whose \(i_j\)-th component is given by \(\phi_{i_j}\) for \(j = 1, \dots l\), and by the zero morphism otherwise.
‣ ComponentInclusionMorphism( v, j ) | ( operation ) |
Returns: a morphism
The arguments are an object \(v = \oplus_{i \in I} a_i \chi^{i}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), and an object \(j \in I\). The output is the canonical inclusion \(a_j \chi^j \hookrightarrow \oplus_{i \in I} a_i \chi^{i}\) in \(\bigoplus_{i \in I} k\mathrm{-vec}\).
‣ ComponentProjectionMorphism( v, j ) | ( operation ) |
Returns: a morphism
The arguments are an object \(v = \oplus_{i \in I} a_i \chi^{i}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), and an object \(j \in I\). The output is the canonical projection \(\oplus_{i \in I} a_i \chi^{i} \twoheadrightarrow a_j \chi^j\) in \(\bigoplus_{i \in I} k\mathrm{-vec}\).
‣ SemisimpleCategoryObject( L, C ) | ( operation ) |
Returns: an object
The arguments are a list \(L\) and a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The list \(L\) contains pairs \(L = [ [ a_1, i_1 ], \dots, [ a_l, i_l ] ]\) of non-negative integers \(a_j\) and objects \(i_j \in I\). The output is the object in \(C\) given by \(\oplus_{j=1}^l a_j \chi^{i_j}\).
‣ SemisimpleCategoryObjectConstructorWithFlatList( L, C ) | ( operation ) |
Returns: an object
The arguments are a list \(L\) and a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The list \(L\) contains an even number of elements \(L = [ a_1, i_1, \dots, a_l, i_l ]\) of non-negative integers \(a_j\) and objects \(i_j \in I\). The output is the object in \(C\) given by \(\oplus_{j=1}^l a_j \chi^{i_j}\).
‣ MembershipFunctionForSemisimpleCategory( C ) | ( attribute ) |
Returns: a function
The argument is a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The output is its underlying membership function \(m\) for \(I\).
‣ UnderlyingCategoryForSemisimpleCategory( C ) | ( attribute ) |
Returns: a category
The argument is a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The output is its underlying category \(k\mathrm{-vec}\).
‣ UnderlyingFieldForHomalgForSemisimpleCategory( C ) | ( attribute ) |
Returns: a homalg field
The argument is a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The output is its underlying field \(k\).
‣ GivenObjectFilterForSemisimpleCategory( C ) | ( attribute ) |
Returns: a filter
The argument is a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The output is its object filter which could be specified in the constructor of \(C\).
‣ GivenMorphismFilterForSemisimpleCategory( C ) | ( attribute ) |
Returns: a filter
The argument is a semisimple category \(C = \bigoplus_{i \in I} k\mathrm{-vec}\). The output is its morphism filter which could be specified in the constructor of \(C\).
‣ SemisimpleCategoryMorphismList( alpha ) | ( attribute ) |
Returns: a list
The argument is a morphism \(\alpha = ( \alpha_i )_{i \in I}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is the list of pairs \([ [ \alpha_{i_1}, i_1 ], \dots, [ \alpha_{i_l}, i_l ] ]\) where \(i_j\) ranges through the support of the source and range of \(\alpha\).
‣ UnderlyingFieldForHomalg( alpha ) | ( attribute ) |
Returns: a homalg field
The argument is a morphism \(\alpha = ( \alpha_i )_{i \in I}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is the homalg field \(k\).
‣ SemisimpleCategoryObjectList( v ) | ( attribute ) |
Returns: a list
The argument is an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category. The output is the list \([ [ a_1, i_1 ], \dots [ a_l, i_l ] ]\).
‣ SemisimpleCategoryObjectListWithActualObjects( v ) | ( attribute ) |
Returns: a list
The argument is an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category. The output is the list \([ [ a_1, \chi^{i_1} ], \dots [ a_l, \chi^{i_l} ] ]\).
‣ Support( v ) | ( attribute ) |
Returns: a list
The argument is an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category. The output is the list \([ i_1, \dots, i_l ]\).
‣ UnderlyingFieldForHomalg( v ) | ( attribute ) |
Returns: a homalg field
The argument is an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is the homalg field \(k\).
‣ Dimension( v ) | ( attribute ) |
Returns: an integer
The argument is an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is the integer \(\sum_{j=1}^l a_j \cdot \mathrm{dim}( i_j )\). This functions works under the assumption that there is a notion of dimension on the objects in \(I\).
‣ Component( alpha, i ) | ( operation ) |
Returns: a vector space morphism
The argument is a morphism \(\alpha = ( \alpha_i )_{i \in I}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\) and an object \(i \in I\). The output is \(\alpha_i\).
‣ NormalizeSemisimpleCategoryObjectList( L ) | ( operation ) |
Returns: a list
The argument is a list \(L = [ [ a_1, i_1 ], \dots, [ a_l, i_l ] ]\) of non-negative integers \(a_j\) and objects \(i_j \in I\), where \(I\) correspond to irreducible objects in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is again a list of pairs consisting of integers an elements in \(I\), but with the following normalization:
Each \(a_j\) is positive,
\(i_j\) is strictly less than \(i_{j+1}\).
‣ Multiplicity( v, i ) | ( operation ) |
Returns: an integer
The arguments are an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), and an element \(i \in I\). The output is the integer \(a_i\).
‣ Component( v, i ) | ( operation ) |
Returns: a vector space object
The arguments are an object \(v = \oplus_{j=1}^l a_j \chi^{i_j}\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\), and an element \(i \in I\). The output is the \(k\)-vector space object \(k^{a_i}\) in Cap's Matrix Category.
‣ TestPentagonIdentity( v_1, v_2, v_3, v_4 ) | ( operation ) |
Returns: a boolean
This is a debug operation. The arguments are 4 objects \(v_1, v_2, v_3, v_4\) in a category. The output is true if the pentagon identity holds for those 4 objects, false otherwise.
‣ TestPentagonIdentityForAllQuadruplesInList( L ) | ( operation ) |
Returns: a boolean
This is a debug operation. The argument is a list \(L\) consisting of quadruples of objects in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is true if the pentagon identity holds for all those quadruples, false otherwise.
‣ TestBraidingCompatability( v_1, v_2, v_3 ) | ( operation ) |
Returns: a boolean
This is a debug operation. The arguments are 3 objects \(v_1, v_2, v_3\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is true if the braiding compatabilities with the associator hold, false otherwise.
‣ TestBraidingCompatabilityForAllTriplesInList( L ) | ( operation ) |
Returns: a boolean
This is a debug operation. The argument is a list \(L\) consisting of triples of objects in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is true if the braiding compatabilities with the associator hold for all those triples false otherwise.
‣ TestZigZagIdentitiesForDual( v ) | ( operation ) |
Returns: a boolean
This is a debug operation. The argument is an object \(v\) in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is true if the zig zag identity for duals hold, false otherwise.
‣ TestZigZagIdentitiesForDualForAllObjectsInList( L ) | ( operation ) |
Returns: a boolean
This is a debug operation. The argument is a list \(L\) consisting of objects in a semisimple category \(\bigoplus_{i \in I} k\mathrm{-vec}\). The output is true if the zig zag identity for duals hold for all those objects, false otherwise.
‣ IsSemisimpleCategoryMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a semisimple category.
‣ IsSemisimpleCategoryObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in a semisimple category.
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