def _coerce_map_from_(self, R):
r"""
Return a coerce map from ``R`` to this field.
EXAMPLES::
sage: R.<t> = GF(5)[]
sage: K = R.fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L); f # indirect doctest
Isomorphism:
From: Rational function field in t over Finite Field of size 5
To: Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5
sage: f(~L.gen())
1/t
"""
from sage.rings.function_field.function_field import is_RationalFunctionField
if is_RationalFunctionField(R) and self.variable_name() == R.variable_name() and self.base_ring() is R.constant_base_field():
from sage.categories.all import Hom
parent = Hom(R, self)
from sage.rings.function_field.maps import FunctionFieldToFractionField
return parent.__make_element_class__(FunctionFieldToFractionField)(parent)
return super(FractionField_1poly_field, self)._coerce_map_from_(R)
def internal_coproduct(self):
"""
Compute the internal coproduct of ``self``.
If :meth:`internal_coproduct_on_basis()` is available, construct
the internal coproduct morphism from ``self`` to ``self``
`\otimes` ``self`` by extending it by linearity. Otherwise, this uses
:meth:`internal_coproduct_by_coercion()`, if available.
OUTPUT:
- an element of the tensor squared of ``self``
EXAMPLES::
sage: q = SymmetricFunctionsNonCommutingVariables(QQ).q()
sage: q.internal_coproduct(q[[1,3],[2]] - 2*q[[1]])
-2*q{{1}} # q{{1}} + q{{1, 2, 3}} # q{{1, 3}, {2}} + q{{1, 3}, {2}} # q{{1, 2, 3}}
+ q{{1, 3}, {2}} # q{{1, 3}, {2}}
"""
if self.internal_coproduct_on_basis is not NotImplemented:
# TODO: if self is a coalgebra, then one would want
# to create a morphism of algebras with basis instead
# should there be a method self.coproduct_hom_category?
return Hom(self, tensor([self, self]),
ModulesWithBasis(self.base_ring()))(
on_basis=self.internal_coproduct_on_basis)
elif hasattr(self, "internal_coproduct_by_coercion"):
return self.internal_coproduct_by_coercion
def coproduct(self):
r"""
If :meth:`coproduct_on_basis` is available, construct the
coproduct morphism from ``self`` to ``self`` `\otimes`
``self`` by extending it by linearity. Otherwise, use
:meth:`~Coalgebras.Realizations.ParentMethods.coproduct_by_coercion`,
if available.
EXAMPLES::
sage: A = HopfAlgebrasWithBasis(QQ).example(); A
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
sage: [a,b] = A.algebra_generators()
sage: a, A.coproduct(a)
(B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
sage: b, A.coproduct(b)
(B[(1,3)], B[(1,3)] # B[(1,3)])
"""
if self.coproduct_on_basis is not NotImplemented:
# TODO: if self is a hopf algebra, then one would want
# to create a morphism of algebras with basis instead
# should there be a method self.coproduct_homset_category?
return Hom(self, tensor([self, self]),
ModulesWithBasis(self.base_ring()))(
on_basis=self.coproduct_on_basis)
elif hasattr(self, "coproduct_by_coercion"):
return self.coproduct_by_coercion
def internal_coproduct(self):
"""
Compute the internal coproduct of ``self``.
If :meth:`internal_coproduct_on_basis()` is available, construct
the internal coproduct morphism from ``self`` to ``self``
`\otimes` ``self`` by extending it by linearity. Otherwise, this uses
:meth:`internal_coproduct_by_coercion()`, if available.
OUTPUT:
- an element of the tensor squared of ``self``
EXAMPLES::
sage: cp = SymmetricFunctionsNonCommutingVariables(QQ).cp()
sage: cp.internal_coproduct(cp[[1,3],[2]] - 2*cp[[1]])
-2*cp{{1}} # cp{{1}} + cp{{1, 2, 3}} # cp{{1, 3}, {2}} + cp{{1, 3}, {2}} # cp{{1, 2, 3}}
+ cp{{1, 3}, {2}} # cp{{1, 3}, {2}}
"""
if self.internal_coproduct_on_basis is not NotImplemented:
return Hom(self, tensor([self, self]),
ModulesWithBasis(self.base_ring()))(
on_basis=self.internal_coproduct_on_basis)
elif hasattr(self, "internal_coproduct_by_coercion"):
return self.internal_coproduct_by_coercion
def _apply_functor_to_morphism(self, f):
r"""
Lift a homomorphism of rings to the corresponding homomorphism of the group algebras of ``self.group()``.
INPUT:
- ``f`` - a morphism of rings.
OUTPUT:
A morphism of group algebras.
EXAMPLES::
sage: G = SymmetricGroup(3)
sage: A = GroupAlgebra(G, ZZ)
sage: h = sage.categories.morphism.SetMorphism(Hom(ZZ, GF(5), Rings()), lambda x: GF(5)(x))
sage: hh = A.construction()[0](h)
sage: hh(A.0 + 5 * A.1)
(1,2,3)
"""
codomain = self(f.codomain())
return SetMorphism(
Hom(self(f.domain()), codomain, Rings()), lambda x: sum(
codomain(g) * f(c) for (g, c) in dict(x).iteritems()))
def _coerce_map_from_(self, R):
r"""
Return a coerce map from ``R`` to this field.
EXAMPLES::
sage: R.<t> = GF(5)[]
sage: K = R.fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L); f # indirect doctest
Isomorphism:
From: Rational function field in t over Finite Field of size 5
To: Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5
sage: f(~L.gen())
1/t
"""
from sage.rings.function_field.function_field import RationalFunctionField
if isinstance(R, RationalFunctionField) and self.variable_name(
) == R.variable_name() and self.base_ring() is R.constant_base_field():
from sage.categories.all import Hom
parent = Hom(R, self)
from sage.rings.function_field.maps import FunctionFieldToFractionField
return parent.__make_element_class__(FunctionFieldToFractionField)(
parent)
return super(FractionField_1poly_field, self)._coerce_map_from_(R)
def section(self):
r"""
Return a section of this map.
EXAMPLES::
sage: R.<x> = QQ[]
sage: R.fraction_field().coerce_map_from(R).section()
Section map:
From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
To: Univariate Polynomial Ring in x over Rational Field
"""
from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
from sage.all import Hom
parent = Hom(self.codomain(), self.domain(), SetsWithPartialMaps())
return parent.__make_element_class__(FractionFieldEmbeddingSection)(self)
def section(self):
r"""
Return a section of this map.
EXAMPLES::
sage: R.<x> = QQ[]
sage: R.fraction_field().coerce_map_from(R).section()
Section map:
From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
To: Univariate Polynomial Ring in x over Rational Field
"""
from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
from sage.all import Hom
parent = Hom(self.codomain(), self.domain(), SetsWithPartialMaps())
return parent.__make_element_class__(FractionFieldEmbeddingSection)(self)