def __init__(self, X, Y):
"""
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W
sage: type(Q.Hom(Q))
<class 'sage.modules.fg_pid.fgp_morphism.FGP_Homset_class_with_category'>
"""
Homset.__init__(self, X, Y)
self._populate_coercion_lists_(element_constructor = FGP_Morphism,
coerce_list = [])
def __call__(self, *args, **kwds):
r"""
Construct an element of ``self`` from the input.
EXAMPLES::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N', structure='topological')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M,N)
sage: f = H.__call__({(X, Y): [x+y, x-y, x*y]}, name='f') ; f
Continuous map f from the 2-dimensional topological manifold M to
the 3-dimensional topological manifold N
sage: f.display()
f: M --> N
(x, y) |--> (u, v, w) = (x + y, x - y, x*y)
There is also the following shortcut for :meth:`one`::
sage: M = Manifold(2, 'M', structure='topological')
sage: H = Hom(M, M)
sage: H(1)
Identity map Id_M of the 2-dimensional topological manifold M
"""
if len(args) == 1:
if self.domain() == self.codomain() and args[0] == 1:
return self.one()
if isinstance(args[0], ContinuousMap):
return Homset.__call__(self, args[0])
return Parent.__call__(self, *args, **kwds)
def __call__(self, *args, **kwds):
r"""
Construct an element of ``self`` from the input.
EXAMPLES::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N', structure='topological')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M,N)
sage: f = H.__call__({(X, Y): [x+y, x-y, x*y]}, name='f') ; f
Continuous map f from the 2-dimensional topological manifold M to
the 3-dimensional topological manifold N
sage: f.display()
f: M --> N
(x, y) |--> (u, v, w) = (x + y, x - y, x*y)
There is also the following shortcut for :meth:`one`::
sage: M = Manifold(2, 'M', structure='topological')
sage: H = Hom(M, M)
sage: H(1)
Identity map Id_M of the 2-dimensional topological manifold M
"""
if len(args) == 1:
if self.domain() == self.codomain() and args[0] == 1:
return self.one()
if isinstance(args[0], ContinuousMap):
return Homset.__call__(self, args[0])
return Parent.__call__(self, *args, **kwds)
def __init__(self, domain, codomain):
"""
The Python constructor
EXAMPLES::
sage: R = QQ['x']['y']['s','t']['X']
sage: p = R.random_element()
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: g = f.section()
sage: g(f(p)) == p
True
::
sage: R = QQ['a','b','x','y']
sage: S = ZZ['a','b']['x','z']
sage: from sage.rings.polynomial.flatten import UnflatteningMorphism
sage: UnflatteningMorphism(R, S)
Traceback (most recent call last):
...
ValueError: rings must have same base ring
::
sage: R = QQ['a','b','x','y']
sage: S = QQ['a','b']['x','z','w']
sage: from sage.rings.polynomial.flatten import UnflatteningMorphism
sage: UnflatteningMorphism(R, S)
Traceback (most recent call last):
...
ValueError: rings must have the same number of variables
"""
if not is_MPolynomialRing(domain):
raise ValueError("domain should be a multivariate polynomial ring")
if not is_PolynomialRing(codomain) and not is_MPolynomialRing(
codomain):
raise ValueError("codomain should be a polynomial ring")
ring = codomain
intermediate_rings = []
while True:
is_polynomial_ring = is_PolynomialRing(ring)
if not (is_polynomial_ring or is_MPolynomialRing(ring)):
break
intermediate_rings.append((ring, is_polynomial_ring))
ring = ring.base_ring()
if domain.base_ring() != intermediate_rings[-1][0].base_ring():
raise ValueError("rings must have same base ring")
if domain.ngens() != sum([R.ngens() for R, _ in intermediate_rings]):
raise ValueError("rings must have the same number of variables")
self._intermediate_rings = intermediate_rings
hom = Homset(domain, codomain, base=ring, check=False)
Morphism.__init__(self, hom)
self._repr_type_str = 'Unflattening'
def __init__(self, X, Y, category=None):
"""
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W
sage: type(Q.Hom(Q))
<class 'sage.modules.fg_pid.fgp_morphism.FGP_Homset_class_with_category'>
"""
if category is None:
from sage.modules.free_module import is_FreeModule
if is_FreeModule(X) and is_FreeModule(Y):
from sage.all import FreeModules
category = FreeModules(X.base_ring())
else:
from sage.all import Modules
category = Modules(X.base_ring())
Homset.__init__(self, X, Y, category)
def __init__(self, X, Y, category=None):
"""
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W
sage: type(Q.Hom(Q))
<class 'sage.modules.fg_pid.fgp_morphism.FGP_Homset_class_with_category'>
"""
if category is None:
from sage.modules.free_module import is_FreeModule
if is_FreeModule(X) and is_FreeModule(Y):
from sage.all import FreeModules
category = FreeModules(X.base_ring())
else:
from sage.all import Modules
category = Modules(X.base_ring())
Homset.__init__(self, X, Y, category)
def __init__(self, X, Y, category=None, base=None, check=True):
"""
Initalize ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x,y,z')
sage: Lyn = L.Lyndon()
sage: H = L.Hall()
sage: HS = Hom(Lyn, H)
We skip the elements test since homsets are not proper parents::
sage: TestSuite(HS).run(skip=["_test_elements"])
"""
if base is None:
base = X.base_ring()
Homset.__init__(self, X, Y, category, base, check)
def __init__(self, X, Y, category=None, base=None, check=True):
"""
Initialize ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x,y,z')
sage: Lyn = L.Lyndon()
sage: H = L.Hall()
sage: HS = Hom(Lyn, H)
We skip the elements test since homsets are not proper parents::
sage: TestSuite(HS).run(skip=["_test_elements"])
"""
if base is None:
base = X.base_ring()
Homset.__init__(self, X, Y, category, base, check)
def __init__(self, domain, codomain, name=None, latex_name=None):
r"""
Initialize ``self``.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N', structure='topological')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M, N) ; H
Set of Morphisms from 2-dimensional topological manifold M to
3-dimensional topological manifold N in Category of manifolds
over Real Field with 53 bits of precision
sage: TestSuite(H).run()
Test for an endomorphism set::
sage: E = Hom(M, M) ; E
Set of Morphisms from 2-dimensional topological manifold M to
2-dimensional topological manifold M in Category of manifolds over
Real Field with 53 bits of precision
sage: TestSuite(E).run()
"""
from sage.manifolds.manifold import TopologicalManifold
if not isinstance(domain, TopologicalManifold):
raise TypeError("domain = {} is not an ".format(domain) +
"instance of TopologicalManifold")
if not isinstance(codomain, TopologicalManifold):
raise TypeError("codomain = {} is not an ".format(codomain) +
"instance of TopologicalManifold")
Homset.__init__(self, domain, codomain)
if name is None:
self._name = "Hom({},{})".format(domain._name, codomain._name)
else:
self._name = name
if latex_name is None:
self._latex_name = r"\mathrm{{Hom}}\left({},{}\right)".format(
domain._latex_name, codomain._latex_name)
else:
self._latex_name = latex_name
def __init__(self, domain, codomain, name=None, latex_name=None):
r"""
Initialize ``self``.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N', structure='topological')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M, N) ; H
Set of Morphisms from 2-dimensional topological manifold M to
3-dimensional topological manifold N in Category of manifolds
over Real Field with 53 bits of precision
sage: TestSuite(H).run()
Test for an endomorphism set::
sage: E = Hom(M, M) ; E
Set of Morphisms from 2-dimensional topological manifold M to
2-dimensional topological manifold M in Category of manifolds over
Real Field with 53 bits of precision
sage: TestSuite(E).run()
"""
from sage.manifolds.manifold import TopologicalManifold
if not isinstance(domain, TopologicalManifold):
raise TypeError("domain = {} is not an ".format(domain) +
"instance of TopologicalManifold")
if not isinstance(codomain, TopologicalManifold):
raise TypeError("codomain = {} is not an ".format(codomain) +
"instance of TopologicalManifold")
Homset.__init__(self, domain, codomain)
if name is None:
self._name = "Hom({},{})".format(domain._name, codomain._name)
else:
self._name = name
if latex_name is None:
self._latex_name = r"\mathrm{{Hom}}\left({},{}\right)".format(
domain._latex_name, codomain._latex_name)
else:
self._latex_name = latex_name
def __init__(self, domain, codomain, name=None, latex_name=None):
r"""
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M, N) ; H
Set of Morphisms from 2-dimensional manifold 'M' to 3-dimensional
manifold 'N' in Category of sets
sage: TestSuite(H).run()
Test for an endomorphism set::
sage: E = Hom(M, M) ; E
Set of Morphisms from 2-dimensional manifold 'M' to 2-dimensional
manifold 'M' in Category of sets
sage: TestSuite(E).run()
"""
from sage.geometry.manifolds.domain import ManifoldOpenSubset
if not isinstance(domain, ManifoldOpenSubset):
raise TypeError("domain = {} is not an ".format(domain) +
"instance of ManifoldOpenSubset")
if not isinstance(codomain, ManifoldOpenSubset):
raise TypeError("codomain = {} is not an ".format(codomain) +
"instance of ManifoldOpenSubset")
Homset.__init__(self, domain, codomain)
if name is None:
self._name = "Hom(" + domain._name + "," + codomain._name + ")"
else:
self._name = name
if latex_name is None:
self._latex_name = \
r"\mathrm{Hom}\left(" + domain._latex_name + "," + \
codomain._latex_name + r"\right)"
else:
self._latex_name = latex_name
self._one = None # to be set by self.one() if self is an endomorphism
def __init__(self, fmodule1, fmodule2, name=None, latex_name=None):
r"""
TESTS::
sage: from sage.tensor.modules.free_module_homset import FreeModuleHomset
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: N = FiniteRankFreeModule(ZZ, 2, name='N')
sage: FreeModuleHomset(M, N)
Set of Morphisms from Rank-3 free module M over the Integer Ring
to Rank-2 free module N over the Integer Ring
in Category of finite dimensional modules over Integer Ring
sage: H = FreeModuleHomset(M, N, name='L(M,N)',
....: latex_name=r'\mathcal{L}(M,N)')
sage: latex(H)
\mathcal{L}(M,N)
"""
from .finite_rank_free_module import FiniteRankFreeModule
if not isinstance(fmodule1, FiniteRankFreeModule):
raise TypeError("fmodule1 = {} is not an ".format(fmodule1) +
"instance of FiniteRankFreeModule")
if not isinstance(fmodule2, FiniteRankFreeModule):
raise TypeError("fmodule2 = {} is not an ".format(fmodule2) +
"instance of FiniteRankFreeModule")
if fmodule1.base_ring() != fmodule2.base_ring():
raise TypeError("the domain and codomain are not defined over " +
"the same ring")
Homset.__init__(self, fmodule1, fmodule2)
if name is None:
self._name = "Hom(" + fmodule1._name + "," + fmodule2._name + ")"
else:
self._name = name
if latex_name is None:
self._latex_name = \
r"\mathrm{Hom}\left(" + fmodule1._latex_name + "," + \
fmodule2._latex_name + r"\right)"
else:
self._latex_name = latex_name
self._one = None # to be set by self.one() if self is an endomorphism
def __init__(self, domain, codomain, name=None, latex_name=None):
r"""
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M, N) ; H
Set of Morphisms from 2-dimensional manifold 'M' to 3-dimensional
manifold 'N' in Category of sets
sage: TestSuite(H).run()
Test for an endomorphism set::
sage: E = Hom(M, M) ; E
Set of Morphisms from 2-dimensional manifold 'M' to 2-dimensional
manifold 'M' in Category of sets
sage: TestSuite(E).run()
"""
from sage.geometry.manifolds.domain import ManifoldOpenSubset
if not isinstance(domain, ManifoldOpenSubset):
raise TypeError("domain = {} is not an ".format(domain) +
"instance of ManifoldOpenSubset")
if not isinstance(codomain, ManifoldOpenSubset):
raise TypeError("codomain = {} is not an ".format(codomain) +
"instance of ManifoldOpenSubset")
Homset.__init__(self, domain, codomain)
if name is None:
self._name = "Hom(" + domain._name + "," + codomain._name + ")"
else:
self._name = name
if latex_name is None:
self._latex_name = \
r"\mathrm{Hom}\left(" + domain._latex_name + "," + \
codomain._latex_name + r"\right)"
else:
self._latex_name = latex_name
self._one = None # to be set by self.one() if self is an endomorphism
def __init__(self, fmodule1, fmodule2, name=None, latex_name=None):
r"""
TESTS::
sage: from sage.tensor.modules.free_module_homset import FreeModuleHomset
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: N = FiniteRankFreeModule(ZZ, 2, name='N')
sage: FreeModuleHomset(M, N)
Set of Morphisms from Rank-3 free module M over the Integer Ring
to Rank-2 free module N over the Integer Ring
in Category of finite dimensional modules over Integer Ring
sage: H = FreeModuleHomset(M, N, name='L(M,N)',
....: latex_name=r'\mathcal{L}(M,N)')
sage: latex(H)
\mathcal{L}(M,N)
"""
from .finite_rank_free_module import FiniteRankFreeModule
if not isinstance(fmodule1, FiniteRankFreeModule):
raise TypeError("fmodule1 = {} is not an ".format(fmodule1) +
"instance of FiniteRankFreeModule")
if not isinstance(fmodule2, FiniteRankFreeModule):
raise TypeError("fmodule2 = {} is not an ".format(fmodule2) +
"instance of FiniteRankFreeModule")
if fmodule1.base_ring() != fmodule2.base_ring():
raise TypeError("the domain and codomain are not defined over " +
"the same ring")
Homset.__init__(self, fmodule1, fmodule2)
if name is None:
self._name = "Hom(" + fmodule1._name + "," + fmodule2._name + ")"
else:
self._name = name
if latex_name is None:
self._latex_name = \
r"\mathrm{Hom}\left(" + fmodule1._latex_name + "," + \
fmodule2._latex_name + r"\right)"
else:
self._latex_name = latex_name
self._one = None # to be set by self.one() if self is an endomorphism
def __init__(self, domain, codomain, category=None):
"""
Initialize ``self``. Type ``QuiverHomSpace?`` for more information.
TESTS::
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1))
sage: TestSuite(H).run()
"""
# The data in the class is stored in the following private variables:
#
# * _base
# The base ring of the representations M and N.
# * _codomain
# The QuiverRep object of the codomain N.
# * _domain
# The QuiverRep object of the domain M.
# * _quiver
# The quiver of the representations M and N.
# * _space
# A free module with ambient space.
#
# The free module _space is the homomorphism space. The ambient space
# is k^n where k is the base ring and n is the sum of the dimensions of
# the spaces of homomorphisms between the free modules attached in M
# and N to the vertices of the quiver. Each coordinate represents a
# single entry in one of those matrices.
# Get the quiver and base ring and check they they are the same for
# both modules
if domain._semigroup != codomain._semigroup:
raise ValueError("representations are not over the same quiver")
self._quiver = domain._quiver
self._semigroup = domain._semigroup
# Check that the bases are compatible, and then initialise the homset:
if codomain.base_ring() != domain.base_ring():
raise ValueError("representations are not over the same base ring")
Homset.__init__(self,
domain,
codomain,
category=category,
base=domain.base_ring())
# To compute the Hom Space we set up a 'generic' homomorphism where the
# maps at each vertex are described by matrices whose entries are
# variables. Then the commutativity of edge diagrams gives us a
# system of equations whose solution space is the Hom Space we're
# looking for. The variables will be numbered consecutively starting
# at 0, ordered first by the vertex the matrix occurs at, then by row
# then by column. We'll have to keep track of which variables
# correspond to which matrices.
# eqs will count the number of equations in our system of equations,
# varstart will be a list whose ith entry is the number of the
# variable located at (0, 0) in the matrix assigned to the
# ith vertex. (So varstart[0] will be 0.)
eqs = 0
verts = domain._quiver.vertices()
varstart = [0] * (len(verts) + 1)
# First assign to varstart the dimension of the matrix assigned to the
# previous vertex.
for v in verts:
varstart[verts.index(v) + 1] = domain._spaces[v].dimension(
) * codomain._spaces[v].dimension()
for e in domain._quiver.edges():
eqs += domain._spaces[e[0]].dimension() * codomain._spaces[
e[1]].dimension()
# After this cascading sum varstart[v] will be the sum of the
# dimensions of the matrices assigned to vertices ordered before v.
# This is equal to the number of the first variable assigned to v.
for i in range(2, len(varstart)):
varstart[i] += varstart[i - 1]
# This will be the coefficient matrix for the system of equations. We
# start with all zeros and will fill in as we go. We think of this
# matrix as acting on the right so the columns correspond to equations,
# the rows correspond to variables, and .kernel() will give a right
# kernel as is needed.
from sage.matrix.constructor import Matrix
coef_mat = Matrix(codomain.base_ring(), varstart[-1], eqs)
# eqn keeps track of what equation we are on. If the maps X and Y are
# assigned to an edge e and A and B are the matrices of variables that
# describe the generic maps at the initial and final vertices of e
# then commutativity of the edge diagram is described by the equation
# AY = XB, or
#
# Sum_k A_ik*Y_kj - Sum_k X_ik*B_kj == 0 for all i and j.
#
# Below we loop through these values of i,j,k and write the
# coefficients of the equation above into the coefficient matrix.
eqn = 0
for e in domain._quiver.edges():
X = domain._maps[e].matrix()
Y = codomain._maps[e].matrix()
for i in range(0, X.nrows()):
for j in range(0, Y.ncols()):
for k in range(0, Y.nrows()):
coef_mat[varstart[verts.index(e[0])] + i * Y.nrows() +
k, eqn] = Y[k, j]
for k in range(0, X.ncols()):
coef_mat[varstart[verts.index(e[1])] + k * Y.ncols() +
j, eqn] = -X[i, k]
eqn += 1
# Now we can create the hom space
self._space = coef_mat.kernel()
# Bind identity if domain = codomain
if domain is codomain:
self.identity = self._identity
def _Hom_(X, Y, category):
# here we overwrite the Rings._Hom_ implementation
return Homset(X, Y, category=category)
def __init__(self, domain):
"""
The Python constructor
EXAMPLES::
sage: R = ZZ['a', 'b', 'c']['x', 'y', 'z']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Multivariate Polynomial Ring in x, y, z over Multivariate Polynomial Ring in a, b, c over Integer Ring
To: Multivariate Polynomial Ring in a, b, c, x, y, z over Integer Ring
::
sage: R = ZZ['a']['b']['c']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Univariate Polynomial Ring in c over Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Integer Ring
To: Multivariate Polynomial Ring in a, b, c over Integer Ring
::
sage: R = ZZ['a']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Multivariate Polynomial Ring in a, b over Univariate Polynomial Ring in a over Integer Ring
To: Multivariate Polynomial Ring in a, a0, b over Integer Ring
::
sage: K.<v> = NumberField(x^3 - 2)
sage: R = K['x','y']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f(R('v*a*x^2 + b^2 + 1/v*y'))
(v)*x^2*a + b^2 + (1/2*v^2)*y
::
sage: R = QQbar['x','y']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f(R('QQbar(sqrt(2))*a*x^2 + b^2 + QQbar(I)*y'))
1.414213562373095?*x^2*a + b^2 + I*y
::
sage: R.<z> = PolynomialRing(QQbar,1)
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f.domain(), f.codomain()
(Multivariate Polynomial Ring in z over Algebraic Field,
Multivariate Polynomial Ring in z over Algebraic Field)
::
sage: R.<z> = PolynomialRing(QQbar)
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f.domain(), f.codomain()
(Univariate Polynomial Ring in z over Algebraic Field,
Univariate Polynomial Ring in z over Algebraic Field)
TESTS::
sage: Pol = QQ['x']['x0']['x']
sage: fl = FlatteningMorphism(Pol)
sage: fl
Flattening morphism:
From: Univariate Polynomial Ring in x over Univariate Polynomial Ring in x0 over Univariate Polynomial Ring in x over Rational Field
To: Multivariate Polynomial Ring in x, x0, x1 over Rational Field
sage: p = Pol([[[1,2],[3,4]],[[5,6],[7,8]]])
sage: fl.section()(fl(p)) == p
True
"""
if not is_PolynomialRing(domain) and not is_MPolynomialRing(domain):
raise ValueError("domain should be a polynomial ring")
ring = domain
variables = []
intermediate_rings = []
while is_PolynomialRing(ring) or is_MPolynomialRing(ring):
intermediate_rings.append(ring)
v = ring.variable_names()
variables.extend(reversed(v))
ring = ring.base_ring()
self._intermediate_rings = intermediate_rings
variables.reverse()
for i, a in enumerate(variables):
if a in variables[:i]:
for index in itertools.count():
b = a + str(index)
if b not in variables: # not just variables[:i]!
break
variables[i] = b
if is_MPolynomialRing(domain):
codomain = PolynomialRing(ring, variables, len(variables))
else:
codomain = PolynomialRing(ring, variables)
hom = Homset(domain, codomain, base=ring, check=False)
Morphism.__init__(self, hom)
self._repr_type_str = 'Flattening'
def __init__(self, domain, codomain, category=None):
"""
Initialize ``self``. Type ``QuiverHomSpace?`` for more information.
TESTS::
sage: Q = DiGraph({1:{2:['a', 'b']}}).path_semigroup()
sage: H = Q.S(QQ, 2).Hom(Q.P(QQ, 1))
sage: TestSuite(H).run()
"""
# The data in the class is stored in the following private variables:
#
# * _base
# The base ring of the representations M and N.
# * _codomain
# The QuiverRep object of the codomain N.
# * _domain
# The QuiverRep object of the domain M.
# * _quiver
# The quiver of the representations M and N.
# * _space
# A free module with ambient space.
#
# The free module _space is the homomorphism space. The ambient space
# is k^n where k is the base ring and n is the sum of the dimensions of
# the spaces of homomorphisms between the free modules attached in M
# and N to the vertices of the quiver. Each coordinate represents a
# single entry in one of those matrices.
# Get the quiver and base ring and check they they are the same for
# both modules
if domain._semigroup != codomain._semigroup:
raise ValueError("representations are not over the same quiver")
self._quiver = domain._quiver
self._semigroup = domain._semigroup
# Check that the bases are compatible, and then initialise the homset:
if codomain.base_ring() != domain.base_ring():
raise ValueError("representations are not over the same base ring")
Homset.__init__(self, domain, codomain, category=category, base = domain.base_ring())
# To compute the Hom Space we set up a 'generic' homomorphism where the
# maps at each vertex are described by matrices whose entries are
# variables. Then the commutativity of edge diagrams gives us a
# system of equations whose solution space is the Hom Space we're
# looking for. The variables will be numbered consecutively starting
# at 0, ordered first by the vertex the matrix occurs at, then by row
# then by column. We'll have to keep track of which variables
# correspond to which matrices.
# eqs will count the number of equations in our system of equations,
# varstart will be a list whose ith entry is the number of the
# variable located at (0, 0) in the matrix assigned to the
# ith vertex. (So varstart[0] will be 0.)
eqs = 0
verts = domain._quiver.vertices()
varstart = [0]*(len(verts) + 1)
# First assign to varstart the dimension of the matrix assigned to the
# previous vertex.
for v in verts:
varstart[verts.index(v) + 1] = domain._spaces[v].dimension()*codomain._spaces[v].dimension()
for e in domain._quiver.edges():
eqs += domain._spaces[e[0]].dimension()*codomain._spaces[e[1]].dimension()
# After this cascading sum varstart[v] will be the sum of the
# dimensions of the matrices assigned to vertices ordered before v.
# This is equal to the number of the first variable assigned to v.
for i in range(2, len(varstart)):
varstart[i] += varstart[i-1]
# This will be the coefficient matrix for the system of equations. We
# start with all zeros and will fill in as we go. We think of this
# matrix as acting on the right so the columns correspond to equations,
# the rows correspond to variables, and .kernel() will give a right
# kernel as is needed.
from sage.matrix.constructor import Matrix
coef_mat = Matrix(codomain.base_ring(), varstart[-1], eqs)
# eqn keeps track of what equation we are on. If the maps X and Y are
# assigned to an edge e and A and B are the matrices of variables that
# describe the generic maps at the initial and final vertices of e
# then commutativity of the edge diagram is described by the equation
# AY = XB, or
#
# Sum_k A_ik*Y_kj - Sum_k X_ik*B_kj == 0 for all i and j.
#
# Below we loop through these values of i,j,k and write the
# coefficients of the equation above into the coefficient matrix.
eqn = 0
for e in domain._quiver.edges():
X = domain._maps[e].matrix()
Y = codomain._maps[e].matrix()
for i in range(0, X.nrows()):
for j in range(0, Y.ncols()):
for k in range(0, Y.nrows()):
coef_mat[varstart[verts.index(e[0])] + i*Y.nrows() + k, eqn] = Y[k, j]
for k in range(0, X.ncols()):
coef_mat[varstart[verts.index(e[1])] + k*Y.ncols() + j, eqn] = -X[i, k]
eqn += 1
# Now we can create the hom space
self._space = coef_mat.kernel()
# Bind identity if domain = codomain
if domain is codomain:
self.identity = self._identity