def graded_algebra(self):
r"""
Return the associated graded module to ``self``.
See :class:`~sage.algebras.associated_graded.AssociatedGradedAlgebra`
for the definition and the properties of this.
If the filtered module ``self`` with basis is called `A`,
then this method returns `\operatorname{gr} A`. The method
:meth:`to_graded_conversion` returns the canonical
`R`-module isomorphism `A \to \operatorname{gr} A` induced
by the basis of `A`, and the method
:meth:`from_graded_conversion` returns the inverse of this
isomorphism. The method :meth:`projection` projects
elements of `A` onto `\operatorname{gr} A` according to
their place in the filtration on `A`.
.. WARNING::
When not overridden, this method returns the default
implementation of an associated graded module --
namely, ``AssociatedGradedAlgebra(self)``, where
``AssociatedGradedAlgebra`` is
:class:`~sage.algebras.associated_graded.AssociatedGradedAlgebra`.
But some instances of :class:`FilteredModulesWithBasis`
override this method, as the associated graded module
often is (isomorphic) to a simpler object (for instance,
the associated graded module of a graded module can be
identified with the graded module itself). Generic code
that uses associated graded modules (such as the code
of the :meth:`induced_graded_map` method below) should
make sure to only communicate with them via the
:meth:`to_graded_conversion`,
:meth:`from_graded_conversion` and
:meth:`projection` methods (in particular,
do not expect there to be a conversion from ``self``
to ``self.graded_algebra()``; this currently does not
work for Clifford algebras). Similarly, when
overriding :meth:`graded_algebra`, make sure to
accordingly redefine these three methods, unless their
definitions below still apply to your case (this will
happen whenever the basis of your :meth:`graded_algebra`
has the same indexing set as ``self``, and the partition
of this indexing set according to degree is the same as
for ``self``).
EXAMPLES::
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: A.graded_algebra()
Graded Module of An example of a filtered module with basis:
the free module on partitions over Integer Ring
"""
from sage.algebras.associated_graded import AssociatedGradedAlgebra
return AssociatedGradedAlgebra(self)
def graded_algebra(self):
r"""
Return the associated graded module to ``self``.
See :class:`~sage.algebras.associated_graded.AssociatedGradedAlgebra`
for the definition and the properties of this.
.. SEEALSO::
:meth:`~sage.categories.filtered_modules_with_basis.ParentMethods.graded_algebra`
EXAMPLES::
sage: W.<x,y> = algebras.DifferentialWeyl(QQ)
sage: W.graded_algebra()
Graded Algebra of Differential Weyl algebra of
polynomials in x, y over Rational Field
"""
from sage.algebras.associated_graded import AssociatedGradedAlgebra
return AssociatedGradedAlgebra(self)
def graded_algebra(self):
r"""
Return the associated graded algebra to ``self``.
See :class:`~sage.algebras.associated_graded.AssociatedGradedAlgebra`
for the definition and the properties of this.
If the filtered algebra ``self`` with basis is called `A`,
then this method returns `\operatorname{gr} A`. The method
:meth:`to_graded_conversion` returns the canonical
`R`-module isomorphism `A \to \operatorname{gr} A` induced
by the basis of `A`, and the method
:meth:`from_graded_conversion` returns the inverse of this
isomorphism. The method :meth:`projection` projects
elements of `A` onto `\operatorname{gr} A` according to
their place in the filtration on `A`.
.. WARNING::
When not overridden, this method returns the default
implementation of an associated graded algebra --
namely, ``AssociatedGradedAlgebra(self)``, where
``AssociatedGradedAlgebra`` is
:class:`~sage.algebras.associated_graded.AssociatedGradedAlgebra`.
But many instances of :class:`FilteredAlgebrasWithBasis`
override this method, as the associated graded algebra
often is (isomorphic) to a simpler object (for instance,
the associated graded algebra of a graded algebra can be
identified with the graded algebra itself). Generic code
that uses associated graded algebras (such as the code
of the :meth:`induced_graded_map` method below) should
make sure to only communicate with them via the
:meth:`to_graded_conversion`,
:meth:`from_graded_conversion`, and
:meth:`projection` methods (in particular,
do not expect there to be a conversion from ``self``
to ``self.graded_algebra()``; this currently does not
work for Clifford algebras). Similarly, when
overriding :meth:`graded_algebra`, make sure to
accordingly redefine these three methods, unless their
definitions below still apply to your case (this will
happen whenever the basis of your :meth:`graded_algebra`
has the same indexing set as ``self``, and the partition
of this indexing set according to degree is the same as
for ``self``).
.. TODO::
Maybe the thing about the conversion from ``self``
to ``self.graded_algebra()`` on the Clifford at least
could be made to work? (I would still warn the user
against ASSUMING that it must work -- as there is
probably no way to guarantee it in all cases, and
we shouldn't require users to mess with
element constructors.)
EXAMPLES::
sage: A = AlgebrasWithBasis(ZZ).Filtered().example()
sage: A.graded_algebra()
Graded Algebra of An example of a filtered algebra with basis:
the universal enveloping algebra of
Lie algebra of RR^3 with cross product over Integer Ring
"""
from sage.algebras.associated_graded import AssociatedGradedAlgebra
return AssociatedGradedAlgebra(self)