‣ IsQuiverModuleElement( arg ) | ( filter ) |
Returns: true or false
Category for elements of modules over quiver algebras.
‣ IsQuiverModule( arg ) | ( filter ) |
Returns: true or false
Category for modules over quiver algebras.
‣ IsLeftQuiverModuleElement( arg ) | ( filter ) |
Returns: true or false
Category for elements of left modules over quiver algebras.
‣ IsLeftQuiverModule( arg ) | ( filter ) |
Returns: true or false
Category for left modules over quiver algebras.
‣ IsRightQuiverModuleElement( arg ) | ( filter ) |
Returns: true or false
Category for elements of right modules over quiver algebras.
‣ IsRightQuiverModule( arg ) | ( filter ) |
Returns: true or false
Category for right modules over quiver algebras.
‣ IsQuiverBimoduleElement( arg ) | ( filter ) |
Returns: true or false
Category for elements of bimodules over quiver algebras.
‣ IsQuiverBimodule( arg ) | ( filter ) |
Returns: true or false
Category for bimodules over quiver algebras.
‣ LeftQuiverModule( A, dimensions, matrices ) | ( operation ) |
Returns: IsLeftQuiverModule (4.1-4)
Construct a left module over the quiver algebra A. The argument dimensions is a list of positive integers giving the dimension of the vector space in each vertex. The argument matrices is a list of matrices describing the maps corresponding to the arrows. If the quiver of the algebra A is left-oriented, then the maps go in the same directions as the arrows. If the quiver is right-oriented, then the maps go in the opposite directions of the arrows (more precisely, the module is given by a representation over the opposite algebra, as described below).
This operation first constructs a quiver representation R (see the QuiverRepresentation (3.5-1) constructor) and then constructs a left module using R as its UnderlyingRepresentation (???).
If the quiver of the algebra A is left-oriented, then representations over A naturally correspond to left modules; if the quiver is right-oriented, then representations over A naturally correspond to right modules. The underlying representation R must therefore be over A if the quiver is left-oriented, and over the opposite algebra of A if the quiver is right-oriented. The appropriate algebra (A or its opposite) can be obtained by calling AlgebraForLeftModules (???).
‣ LeftQuiverModuleByArrows( A, dimensions, arrows, matrices ) | ( operation ) |
Returns: IsLeftQuiverModule (4.1-4)
Construct a left module over the quiver algebra A, by specifying matrices for certain arrows.
This works like LeftQuiverModule (4.2-1), except that the entries in the list matrices correspond to the arrows in the list arrows. All arrows of the quiver that are not present in this list get zero maps.
‣ LeftZeroModule( A ) | ( attribute ) |
Returns: IsLeftQuiverModule (4.1-4)
Returns the left zero module over the algebra A.
‣ RightQuiverModule( A, dimensions, matrices ) | ( operation ) |
Construct a right module over the quiver algebra A. This operation works like LeftQuiverModule (4.2-1), except that it creates a right module.
‣ RightQuiverModuleByArrows( A, dimensions, matrices ) | ( operation ) |
Construct a right module over the quiver algebra A, by specifying matrices for certain arrows. This operation works like LeftQuiverModuleByArrows (???), except that it creates a right module.
‣ RightZeroModule( A ) | ( attribute ) |
Returns: IsLeftQuiverModule (4.1-4)
Returns the left zero module over the algebra A.
‣ AlgebraForLeftModules( A ) | ( attribute ) |
Returns: IsQuiverAlgebra (2.1-2)
Returns the appropriate algebra to use for representations of left modules over the algebra A. This is the algebra A itself if its quiver is left-oriented, and the opposite algebra of A if its quiver is right-oriented.
‣ AlgebraForRightModules( A ) | ( attribute ) |
Returns: IsQuiverAlgebra (2.1-2)
Returns the appropriate algebra to use for representations of right modules over the algebra A. This is the algebra A itself if its quiver is right-oriented, and the opposite algebra of A if its quiver is left-oriented.
‣ AsModule( R, A ) | ( operation ) |
Returns: IsQuiverModule (4.1-2)
The representation R considered as a (left or right) module over the algebra A. The algebra A must either be the algebra of the representation R, or the opposite of this algebra. This, together with the orientation of the algebra's quiver, determines whether the resulting module is a left module or a right module. Giving any other algebra than the algebra of R or its opposite as the argument A results in an error.
‣ AsLeftModule( R ) | ( operation ) |
Returns: IsLeftQuiverModule (4.1-4)
The representation R considered as a left module.
‣ AsRightModule( R ) | ( operation ) |
Returns: IsRightQuiverModule (4.1-6)
The representation R considered as a right module.
‣ AsBimodule( R, A, B ) | ( operation ) |
Returns: IsLeftQuiverModule (4.1-4)
The representation R considered as a bimodule over the algebras A (left) and B (right). The algebras A and B must have the same base field k, and the representation R must be a representation over the tensor algebra A \otimes_k B^\mathrm{op}.
‣ UnderlyingRepresentation( M ) | ( attribute ) |
Returns: IsQuiverRepresentation (3.1-2)
Returns the quiver representation for the module M. All module operations are delegated to the underlying representation.
‣ QuiverOfModule( M ) | ( attribute ) |
Returns: IsQuiver (1.1-7)
Returns the quiver that the module M represents.
‣ FieldOfModule( M ) | ( attribute ) |
Returns: IsField (Reference: IsField)
Returns the base field that the module M is a vector space over; that is, the base field of the algebra it is a module over. This is also available by the builtin GAP attribute LeftActingDomain (Reference: LeftActingDomain).
‣ VertexDimensions( M ) | ( attribute ) |
Returns: IsDenseList (Reference: IsDenseList)
Returns the dimensions of the vector spaces in the module M, as a list. The entries in the list correspond to the vertices of the quiver.
‣ VertexDimension( M, i ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the dimension of the vector space in the module M at vertex number i, where i is a positive integer.
‣ VertexDimension( M, v ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the dimension of the vector space in the module M at the vertex v.
‣ QuiverModuleElement( M, vectors ) | ( operation ) |
Returns: IsQuiverModuleElement (4.1-1)
Construct the element in the module M given by the vectors in the list vectors. Each entry in vectors corresponds to a vertex in the quiver.
‣ QuiverModuleElementByVertices( M, vertices, vectors ) | ( operation ) |
Returns: IsQuiverModuleElement (4.1-1)
Construct an element in the module M by assigning vectors to certain given vertices. All vertices of the quiver which are not present in the list vertices are assigned the zero vector.
‣ AsModuleElement( e, M ) | ( operation ) |
Returns: IsQuiverModuleElement (4.1-1)
Returns the element of the module M corresponding to the element e in the underlying representation of M.
‣ UnderlyingRepresentationElement( e ) | ( attribute ) |
Returns: IsQuiverRepresentationElement (3.1-1)
Given an element e in a quiver module M, this attribute returns the corresponding element in the underlying representation of M.
‣ ModuleOfElement( e ) | ( attribute ) |
Returns: IsQuiverModule (4.1-2)
Returns the quiver module which the element e belongs to.
‣ ElementVectors( e ) | ( attribute ) |
Returns: IsDenseList (Reference: IsDenseList)
Returns all the vectors of the element e, as a list ordered in the same way as the vertices of the quiver.
‣ ElementVector( e, i ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the vector of the element e at the vertex with number i.
‣ ElementVector( e, v ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the vector of the element e at the vertex v.
‣ \^( a, e ) | ( operation ) |
Returns: IsLeftQuiverModuleElement (4.1-3)
For an element a in a quiver algebra A and an element e in a left A-module, the expression a^e produces the product of a and e.
‣ \^( e, a ) | ( operation ) |
Returns: IsLeftQuiverModuleElement (4.1-3)
For an element a in a quiver algebra A and an element e in a right A-module, the expression e^a produces the product of e and a.
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