1.1-18 \[\]
‣ IsStrongExceptionalCollection( object ) | ( filter ) |
Returns: true or false
The GAP category of exceptional collections.
‣ CreateStrongExceptionalCollection( full, vertices_labels, vertices_labels_latex, cache ) | ( operation ) |
Returns: IsStrongExceptionalCollection
#TODO If the input is full subcategory A in some category C, generated by finite number of objects A!.Objects, then the output is the strong exceptional collection defined by these objects. If the input is a list of objects L of objects in some category C, then the output is CreateStrongExceptionalCollection( A ), where A := FullSubcategoryGeneratedByListOfObjects(C,L).
‣ StrongExceptionalCollection( A ) | ( attribute ) |
Returns: IsStrongExceptionalCollection
If the input is full subcategory A in some category C, generated by finite number of objects A!.Objects, then the output is the strong exceptional collection defined by these objects returned as an attribute for A.
‣ FullSubcategory( E ) | ( attribute ) |
Returns the full subcategory that is generated by the elements of the strong exceptional collection E.
‣ AmbientCategory( E ) | ( attribute ) |
Returns the ambient category of FullSubcategory(E).
‣ EndomorphismAlgebraAttr( E ) | ( attribute ) |
Returns: a quiver algebra
Returns the endomorphism quiver algebra of the exceptional collection E.
‣ EndomorphismAlgebra( E ) | ( operation ) |
Returns: a quiver algebra
delegates to EndomorphismAlgebraAttr(E).
‣ DefiningStrongExceptionalCollection( A ) | ( attribute ) |
Returns: an exceptional collection
Returns the exceptional collection that defines the quiver algebra A.
‣ Algebroid( E ) | ( attribute ) |
Returns: an algebroid
Returns the algebroid defined by the endomorphism quiver algebra of the exceptional collection E.
‣ QuiverRows( E ) | ( attribute ) |
Returns: a category
Returns the category of quiver rows over the endomorphism algebra of E.
‣ TiltingObject( E ) | ( attribute ) |
Returns: an object
Returns the direct sum of the objects of the exceptional collection E.
‣ NumberOfObjects( E ) | ( attribute ) |
Returns: IsInt
Returns the number of objects of the exceptional collection E.
‣ UnderlyingObjects( E ) | ( attribute ) |
Returns: IsList
Returns a list of the objects of the exceptional collection E.
‣ AdditiveClosure( E ) | ( attribute ) |
Returns: an additive category
Returns the additive closure of the FullSubcategory(E).
‣ CategoryOfQuiverRepresentationsOverOppositeAlgebra( E ) | ( attribute ) |
Returns: an abelian category
Returns the category of right quiver representations over the opposite algebra of the endomorphism algebra of the collection.
‣ HomotopyCategory( E ) | ( operation ) |
Returns: a category
Returns the homotopy category of the additive closure of FullSubcategory(E).
‣ AdditiveClosureAsFullSubcategory( E ) | ( attribute ) |
Returns: an additive category
Returns the additive closure of the FullSubcategory(E) as an additive full subcategory in AmbientCategory(E).
1.1-18 \[\]‣ \[\]( E, i ) | ( operation ) |
Returns: an object
Returns the i'th object in E.
‣ PathsOfLengthGreaterThanOne( E, i, j ) | ( operation ) |
Returns: IsList
Returns a generating set for the vector space of morphisms from E_i to E_j that can be factored along at least one object E_l in E with i+1\leq l\leq j-1.
‣ PathsOfLengthOne( E, i, j ) | ( operation ) |
Returns: IsList
Returns a basis for a complementing vector space in Hom(E_i,E_j) to the vector space generated by PathsOfLengthGreaterThanOne(E,i,j).
‣ Arrows( E, i, j ) | ( operation ) |
Returns: IsList
Delegates to PathsOfLengthOne(E,[i,j]).
‣ AllPaths( E, i, j ) | ( operation ) |
Returns: IsList
It returns the union of PathsOfLengthOne and PathsOfLengthGreaterThanOne applied on the same arguments. In other words it returns a generating set for the vector space Hom(E_i,E_j).
‣ BasisOfPaths( E, i, j ) | ( operation ) |
Returns: IsList
It returns a basis for Hom(E_i,E_j) which cosists only from paths.
‣ LabelsForPathsOfLengthOne( E, i, j ) | ( operation ) |
Returns: IsList
It returns labels for PathsOfLengthOne(E,i,j). A label for path of length one gives information about the source, target and position of the path.
‣ LabelsForPathsOfLengthGreaterThanOne( E, i, j ) | ( operation ) |
Returns: a list of lists
It returns labels for PathsOfLengthGreaterThanOne(E,i,j) A label for a path of length greater than one gives information about the labels of the arrows whose composition is the path.
‣ LabelsForAllPaths( E, i, j ) | ( operation ) |
Returns: a list of lists
It returns labels for AllPaths(E,i,j).
‣ LabelsForBasisOfPaths( E, i, j ) | ( operation ) |
Returns: a list of lists
It returns labels for BasisOfPaths(E,i,j).
‣ InterpretListOfMorphismsAsOneMorphismInRangeCategoryOfHomomorphismStructure( A, B, L ) | ( operation ) |
Returns: a morphism in range category of homomorphism structure
The arguments are two objects A, B and a list of morphisms L:=(f_i:A\to B) for i=1,\dots,n that live in a category equipped with homomorphism structure (1,H(-,-),\nu). The output is the morphism \langle \nu(f_1),\nu(f_2),\dots,\nu(f_n)\rangle:\oplus_{i=1}^n 1 \to H(A,B).
‣ RandomQuiverAlgebraWhoseIndecProjectiveRepsAreStrongExceptionalCollection( F, m, n, r ) | ( function ) |
Returns: a quiver path algebra
The arguments are a field F and three non-negative integers m, n and r It returns a quiver algebra with m vertices, n arrows and at most r relations; and whose indecomposable projective or injective objects define an exceptional collection.
‣ RelationsBetweenMorphisms( L ) | ( function ) |
Returns: a list of ring elements
The argument is a list of morphism L in some k-linear category equipped with homomorphism structure. The output is the relations between the morphisms.
‣ FullSubcategoryGeneratedByProjectiveObjects( cat ) | ( attribute ) |
Returns: a full subcategory
The argument is an abelian category C with enouph projectives. The output the full subcategory generated by projective objects in C.
‣ FullSubcategoryGeneratedByInjectiveObjects( cat ) | ( attribute ) |
Returns: a full subcategory
The argument is an abelian category C with enouph injectives. The output the full subcategory generated by injective objects in C.
‣ FullSubcategoryGeneratedByIndecProjectiveObjects( cat ) | ( attribute ) |
Returns: a full subcategory
The argument is a category of quiver representations over some finite dimensional quiver algebra or a category of functors into a matrix category of some homalg field. The output the full subcategory generated by the indecomposable projective objects.
‣ FullSubcategoryGeneratedByIndecInjectiveObjects( cat ) | ( attribute ) |
Returns: a full subcategory
The argument is a category of quiver representations over some finite dimensional quiver algebra or a category of functors into a matrix category of some homalg field. The output the full subcategory generated by the indecomposable injectives objects.
‣ AdditiveClosureAsFullSubcategory( full ) | ( attribute ) |
Returns: an additive full subcategory
The argument is a non-additive full subcategory full whose ambient category C is additive. The output is an additive full subcategory of C generated by all direct sums of objects in full.
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