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r""" Coalgebras with basis """ #***************************************************************************** # Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr> # Copyright (C) 2008-2011 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.misc.abstract_method import abstract_method from sage.misc.lazy_attribute import lazy_attribute from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring from sage.categories.all import ModulesWithBasis, tensor, Hom from sage.categories.super_modules import SuperModulesCategory
class CoalgebrasWithBasis(CategoryWithAxiom_over_base_ring): """ The category of coalgebras with a distinguished basis.
EXAMPLES::
sage: CoalgebrasWithBasis(ZZ) Category of coalgebras with basis over Integer Ring sage: sorted(CoalgebrasWithBasis(ZZ).super_categories(), key=str) [Category of coalgebras over Integer Ring, Category of modules with basis over Integer Ring]
TESTS::
sage: TestSuite(CoalgebrasWithBasis(ZZ)).run() """
class ParentMethods:
@abstract_method(optional = True) def coproduct_on_basis(self, i): """ The coproduct of the algebra on the basis (optional).
INPUT:
- ``i`` -- the indices of an element of the basis of ``self``
Returns the coproduct of the corresponding basis elements If implemented, the coproduct of the algebra is defined from it by linearity.
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._group.gens() sage: A.coproduct_on_basis(a) B[(1,2,3)] # B[(1,2,3)] """
@lazy_attribute 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)])
""" # 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?
@abstract_method(optional = True) def counit_on_basis(self, i): """ The counit of the algebra on the basis (optional).
INPUT:
- ``i`` -- the indices of an element of the basis of ``self``
Returns the counit of the corresponding basis elements If implemented, the counit of the algebra is defined from it by linearity.
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._group.gens() sage: A.counit_on_basis(a) 1 """
@lazy_attribute def counit(self): """ If :meth:`counit_on_basis` is available, construct the counit morphism from ``self`` to ``self`` `\otimes` ``self`` by extending it by linearity
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.counit(a) (B[(1,2,3)], 1) sage: b, A.counit(b) (B[(1,3)], 1)
"""
class ElementMethods: 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}}) """ raise ValueError("cannot take fewer than 0 coproduct iterations: %s < 0" % str(n))
# Use coassociativity of `\Delta` to perform many coproducts simultaneously.
class Super(SuperModulesCategory): pass |