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:
OUTPUT:
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]
alias of QuiverRepHom
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
Return the codomain of the hom space.
OUTPUT:
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
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:
OUTPUT:
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]
Return the dimension of the hom space.
OUTPUT:
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
Return the domain of the hom space.
OUTPUT:
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
Return a list of generators of the hom space (as a \(k\)-vector space).
OUTPUT:
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]
Create the QuiverRep of self as a module over the opposite quiver.
INPUT:
OUTPUT:
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
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
Return the quiver of the representations.
OUTPUT:
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
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