Quiver Homspace¶
- class sage.quivers.homspace.QuiverHomSpace(domain, codomain, category=None)¶
Bases:
sage.categories.homset.Homset
A homomorphism of quiver representations (of one and the same quiver) is given by specifying, for each vertex of the quiver, a homomorphism of the spaces assigned to this vertex such that these homomorphisms commute with the edge maps. This class handles the set of all such maps, \(Hom_Q(M, N)\).
INPUT:
domain
– the domain of the homomorphism spacecodomain
– the codomain of the homomorphism space
OUTPUT:
QuiverHomSpace
, the homomorphism spaceHom_Q(domain, codomain)
Note
The quivers of the domain and codomain must be equal or a
ValueError
is raised.EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1)) sage: H.dimension() 2 sage: H.gens() [Homomorphism of representations of Multi-digraph on 2 vertices, Homomorphism of representations of Multi-digraph on 2 vertices]
- Element¶
alias of
sage.quivers.morphism.QuiverRepHom
- base_ring()¶
Return the base ring of the representations.
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1)) sage: H.base_ring() Rational Field
- codomain()¶
Return the codomain of the hom space.
OUTPUT:
QuiverRep
, the codomain of the Hom space
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: P = Q.P(QQ, 1) sage: H = Q.S(QQ, 2).Hom(P) sage: H.codomain() is P True
- coordinates(hom)¶
Return the coordinates of the map when expressed in terms of the generators (i. e., the output of the
gens
method) of the hom space.INPUT:
hom
–QuiverRepHom
OUTPUT:
list, the coordinates of the given map when written in terms of the generators of the
QuiverHomSpace
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: S = Q.S(QQ, 2) sage: P = Q.P(QQ, 1) sage: H = S.Hom(P) sage: f = S.hom({2: [[1,-1]]}, P) sage: H.coordinates(f) [1, -1]
- dimension()¶
Return the dimension of the hom space.
OUTPUT:
integer, the dimension
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1)) sage: H.dimension() 2
- domain()¶
Return the domain of the hom space.
OUTPUT:
QuiverRep
, the domain of the Hom space
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: S = Q.S(QQ, 2) sage: H = S.Hom(Q.P(QQ, 1)) sage: H.domain() is S True
- gens()¶
Return a list of generators of the hom space (as a \(k\)-vector space).
OUTPUT:
list of
QuiverRepHom
objects, the generators
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1)) sage: H.gens() [Homomorphism of representations of Multi-digraph on 2 vertices, Homomorphism of representations of Multi-digraph on 2 vertices]
- left_module(basis=False)¶
Create the QuiverRep of
self
as a module over the opposite quiver.INPUT:
basis
- bool. IfFalse
, then only the module is returned. IfTrue
, then a tuple is returned. The first element is the QuiverRep and the second element is a dictionary which associates to each vertex a list. The elements of this list are the homomorphisms which correspond to the basis elements of that vertex in the module.
OUTPUT:
QuiverRep
or tuple
Warning
The codomain of the Hom space must be a left module.
Note
The left action of a path \(e\) on a map \(f\) is given by \((ef)(m) = ef(m)\). This gives the Hom space its structure as a left module over the path algebra. This is then converted to a right module over the path algebra of the opposite quiver
Q.reverse()
and returned.EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b'], 3: ['c', 'd']}, 2:{3:['e']}}).path_semigroup() sage: P = Q.P(GF(3), 3) sage: A = Q.free_module(GF(3)) sage: H = P.Hom(A) sage: H.dimension() 6 sage: M, basis_dict = H.left_module(true) sage: M.dimension_vector() (4, 1, 1) sage: Q.reverse().P(GF(3), 3).dimension_vector() (4, 1, 1)
As lists start indexing at 0 the \(i\)-th vertex corresponds to the \((i-1)\)-th entry of the dimension vector:
sage: len(basis_dict[2]) == M.dimension_vector()[1] True
- natural_map()¶
The natural map from domain to codomain.
This is the zero map.
EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}, 2:{3:['c']}}).path_semigroup() sage: spaces = {1: QQ^2, 2: QQ^2, 3:QQ^1} sage: maps = {(1, 2, 'a'): [[1, 0], [0, 0]], (1, 2, 'b'): [[0, 0], [0, 1]], (2, 3, 'c'): [[1], [1]]} sage: M = Q.representation(QQ, spaces, maps) sage: spaces2 = {2: QQ^1, 3: QQ^1} sage: S = Q.representation(QQ, spaces2) sage: S.hom(M) # indirect doctest Homomorphism of representations of Multi-digraph on 3 vertices sage: S.hom(M) == S.Hom(M).natural_map() True
- quiver()¶
Return the quiver of the representations.
OUTPUT:
DiGraph
, the quiver of the representations
EXAMPLES:
sage: P = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = P.S(QQ, 2).Hom(P.P(QQ, 1)) sage: H.quiver() is P.quiver() True
- zero()¶
Return the zero morphism.
Note
It is needed to override the method inherited from the category of modules, because it would create a morphism that is of the wrong type and does not comply with
QuiverRepHom
.EXAMPLES:
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup() sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1)) sage: H.zero() + H.an_element() == H.an_element() True sage: isinstance(H.zero(), H.element_class) True