def coproduct(self):
"""
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 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.coproduct_on_basis)
elif hasattr(self, "coproduct_by_coercion"):
return self.coproduct_by_coercion
def coproduct(self):
"""
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 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:
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 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_on_basis(self, g):
r"""
Coproduct, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis`.
The basis elements are group like: `\Delta(g) = g \otimes g`.
EXAMPLES::
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: A.coproduct_on_basis(a)
B[(1,2,3)] # B[(1,2,3)]
"""
g = self.monomial(g)
return tensor([g, g])
def coproduct_on_basis(self, g):
r"""
Coproduct, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis`.
The basis elements are group like: `\Delta(g) = g \otimes g`.
EXAMPLES::
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: A.coproduct_on_basis(a)
B[(1,2,3)] # B[(1,2,3)]
"""
g = self.monomial(g)
return tensor([g, g])
def coproduct_on_basis(self, g):
r"""
Coproduct, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis`.
The basis elements are group-like: `\Delta(g) = g \otimes g`.
EXAMPLES::
sage: A = GroupAlgebra(DihedralGroup(3), QQ)
sage: (a, b) = A._group.gens()
sage: A.coproduct_on_basis(a)
(1,2,3) # (1,2,3)
"""
from sage.categories.all import tensor
g = self.monomial(g)
return tensor([g, g])
def internal_coproduct_by_coercion(self, x):
r"""
Return the internal coproduct by coercing the element to the powersum basis.
INPUT:
- ``x`` -- an element of ``self``
OUTPUT:
- an element of the tensor squared of ``self``
EXAMPLES::
sage: h = SymmetricFunctionsNonCommutingVariables(QQ).h()
sage: h[[1,3],[2]].internal_coproduct() # indirect doctest
2*h{{1}, {2}, {3}} # h{{1}, {2}, {3}} - h{{1}, {2}, {3}} # h{{1, 3}, {2}}
- h{{1, 3}, {2}} # h{{1}, {2}, {3}} + h{{1, 3}, {2}} # h{{1, 3}, {2}}
"""
R = self.realization_of().a_realization()
return self.tensor_square().sum(coeff * tensor([self(R[A]), self(R[B])])
for ((A, B), coeff) in R(x).internal_coproduct())
def internal_coproduct_by_coercion(self, x):
r"""
Return the internal coproduct by coercing the element to the powersum basis.
INPUT:
- ``x`` -- an element of ``self``
OUTPUT:
- an element of the tensor squared of ``self``
EXAMPLES::
sage: h = SymmetricFunctionsNonCommutingVariables(QQ).h()
sage: h[[1,3],[2]].internal_coproduct() # indirect doctest
2*h{{1}, {2}, {3}} # h{{1}, {2}, {3}} - h{{1}, {2}, {3}} # h{{1, 3}, {2}}
- h{{1, 3}, {2}} # h{{1}, {2}, {3}} + h{{1, 3}, {2}} # h{{1, 3}, {2}}
"""
R = self.realization_of().a_realization()
return self.tensor_square().sum(coeff * tensor([self(R[A]), self(R[B])])
for ((A, B), coeff) in R(x).internal_coproduct())
def coproduct_iterated(self, n=1):
r"""
Apply ``n`` coproducts to ``self``.
.. TODO::
Remove dependency on ``modules_with_basis`` methods.
EXAMPLES::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,2].coproduct_iterated(0)
Psi[2, 2]
sage: Psi[2,2].coproduct_iterated(2)
Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[2] # Psi[2]
+ Psi[] # Psi[2, 2] # Psi[] + 2*Psi[2] # Psi[] # Psi[2]
+ 2*Psi[2] # Psi[2] # Psi[] + Psi[2, 2] # Psi[] # Psi[]
TESTS::
sage: p = SymmetricFunctions(QQ).p()
sage: p[5,2,2].coproduct_iterated()
p[] # p[5, 2, 2] + 2*p[2] # p[5, 2] + p[2, 2] # p[5]
+ p[5] # p[2, 2] + 2*p[5, 2] # p[2] + p[5, 2, 2] # p[]
sage: p([]).coproduct_iterated(3)
p[] # p[] # p[] # p[]
::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,2].coproduct_iterated(0)
Psi[2, 2]
sage: Psi[2,2].coproduct_iterated(3)
Psi[] # Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[] # Psi[2] # Psi[2]
+ Psi[] # Psi[] # Psi[2, 2] # Psi[] + 2*Psi[] # Psi[2] # Psi[] # Psi[2]
+ 2*Psi[] # Psi[2] # Psi[2] # Psi[] + Psi[] # Psi[2, 2] # Psi[] # Psi[]
+ 2*Psi[2] # Psi[] # Psi[] # Psi[2] + 2*Psi[2] # Psi[] # Psi[2] # Psi[]
+ 2*Psi[2] # Psi[2] # Psi[] # Psi[] + Psi[2, 2] # Psi[] # Psi[] # Psi[]
::
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: m[[1,3],[2]].coproduct_iterated(2)
m{} # m{} # m{{1, 3}, {2}} + m{} # m{{1}} # m{{1, 2}}
+ m{} # m{{1, 2}} # m{{1}} + m{} # m{{1, 3}, {2}} # m{}
+ m{{1}} # m{} # m{{1, 2}} + m{{1}} # m{{1, 2}} # m{}
+ m{{1, 2}} # m{} # m{{1}} + m{{1, 2}} # m{{1}} # m{}
+ m{{1, 3}, {2}} # m{} # m{}
sage: m[[]].coproduct_iterated(3), m[[1,3],[2]].coproduct_iterated(0)
(m{} # m{} # m{} # m{}, m{{1, 3}, {2}})
"""
if n < 0:
raise ValueError(
"cannot take fewer than 0 coproduct iterations: %s < 0" %
str(n))
if n == 0:
return self
if n == 1:
return self.coproduct()
from sage.functions.all import floor, ceil
from sage.rings.all import Integer
# Use coassociativity of `\Delta` to perform many coproducts simultaneously.
fn = floor(Integer(n - 1) / 2)
cn = ceil(Integer(n - 1) / 2)
split = lambda a, b: tensor(
[a.coproduct_iterated(fn),
b.coproduct_iterated(cn)])
return self.coproduct().apply_multilinear_morphism(split)
def coproduct_iterated(self, n=1):
r"""
Apply ``n`` coproducts to ``self``.
.. TODO::
Remove dependency on ``modules_with_basis`` methods.
EXAMPLES::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,2].coproduct_iterated(0)
Psi[2, 2]
sage: Psi[2,2].coproduct_iterated(2)
Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[2] # Psi[2]
+ Psi[] # Psi[2, 2] # Psi[] + 2*Psi[2] # Psi[] # Psi[2]
+ 2*Psi[2] # Psi[2] # Psi[] + Psi[2, 2] # Psi[] # Psi[]
TESTS::
sage: p = SymmetricFunctions(QQ).p()
sage: p[5,2,2].coproduct_iterated()
p[] # p[5, 2, 2] + 2*p[2] # p[5, 2] + p[2, 2] # p[5]
+ p[5] # p[2, 2] + 2*p[5, 2] # p[2] + p[5, 2, 2] # p[]
sage: p([]).coproduct_iterated(3)
p[] # p[] # p[] # p[]
::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,2].coproduct_iterated(0)
Psi[2, 2]
sage: Psi[2,2].coproduct_iterated(3)
Psi[] # Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[] # Psi[2] # Psi[2]
+ Psi[] # Psi[] # Psi[2, 2] # Psi[] + 2*Psi[] # Psi[2] # Psi[] # Psi[2]
+ 2*Psi[] # Psi[2] # Psi[2] # Psi[] + Psi[] # Psi[2, 2] # Psi[] # Psi[]
+ 2*Psi[2] # Psi[] # Psi[] # Psi[2] + 2*Psi[2] # Psi[] # Psi[2] # Psi[]
+ 2*Psi[2] # Psi[2] # Psi[] # Psi[] + Psi[2, 2] # Psi[] # Psi[] # Psi[]
::
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: m[[1,3],[2]].coproduct_iterated(2)
m{} # m{} # m{{1, 3}, {2}} + m{} # m{{1}} # m{{1, 2}}
+ m{} # m{{1, 2}} # m{{1}} + m{} # m{{1, 3}, {2}} # m{}
+ m{{1}} # m{} # m{{1, 2}} + m{{1}} # m{{1, 2}} # m{}
+ m{{1, 2}} # m{} # m{{1}} + m{{1, 2}} # m{{1}} # m{}
+ m{{1, 3}, {2}} # m{} # m{}
sage: m[[]].coproduct_iterated(3), m[[1,3],[2]].coproduct_iterated(0)
(m{} # m{} # m{} # m{}, m{{1, 3}, {2}})
"""
if n < 0:
raise ValueError("cannot take fewer than 0 coproduct iterations: %s < 0" % str(n))
if n == 0:
return self
if n == 1:
return self.coproduct()
from sage.functions.all import floor, ceil
from sage.rings.all import Integer
# Use coassociativity of `\Delta` to perform many coproducts simultaneously.
fn = floor(Integer(n-1)/2); cn = ceil(Integer(n-1)/2)
split = lambda a,b: tensor([a.coproduct_iterated(fn), b.coproduct_iterated(cn)])
return self.coproduct().apply_multilinear_morphism(split)
