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r""" 

Associated Graded Algebras To Filtered Algebras 

 

AUTHORS: 

 

- Travis Scrimshaw (2014-10-08): Initial version 

""" 

 

#***************************************************************************** 

# Copyright (C) 2014 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from sage.misc.cachefunc import cached_method 

from copy import copy 

 

from sage.categories.modules_with_basis import ModulesWithBasis 

from sage.sets.family import Family 

from sage.combinat.free_module import CombinatorialFreeModule 

 

class AssociatedGradedAlgebra(CombinatorialFreeModule): 

r""" 

The associated graded algebra/module `\operatorname{gr} A` 

of a filtered algebra/module with basis `A`. 

 

Let `A` be a filtered module over a commutative ring `R`. 

Let `(F_i)_{i \in I}` be the filtration of `A`, with `I` being 

a totally ordered set. Define 

 

.. MATH:: 

 

G_i = F_i / \sum_{j < i} F_j 

 

for every `i \in I`, and then 

 

.. MATH:: 

 

\operatorname{gr} A = \bigoplus_{i \in I} G_i. 

 

There are canonical projections `p_i : F_i \to G_i` for 

every `i \in I`. Moreover `\operatorname{gr} A` is naturally a 

graded `R`-module with `G_i` being the `i`-th graded component. 

This graded `R`-module is known as the *associated graded module* 

(or, for short, just *graded module*) of `A`. 

 

Now, assume that `A` (endowed with the filtration 

`(F_i)_{i \in I}`) is not just a filtered `R`-module, but also 

a filtered `R`-algebra. 

Let `u \in G_i` and `v \in G_j`, and let `u' \in F_i` and 

`v' \in F_j` be lifts of `u` and `v`, respectively (so that 

`u = p_i(u')` and `v = p_j(v')`). Then, we define a 

multiplication `*` on `\operatorname{gr} A` (not to be mistaken 

for the multiplication of the original algebra `A`) by 

 

.. MATH:: 

 

u * v = p_{i+j} (u' v'). 

 

The *associated graded algebra* (or, for short, just 

*graded algebra*) of `A` is the graded algebra 

`\operatorname{gr} A` (endowed with this multiplication). 

 

Now, assume that `A` is a filtered `R`-algebra with basis. 

Let `(b_x)_{x \in X}` be the basis of `A`, 

and consider the partition `X = \bigsqcup_{i \in I} X_i` of 

the set `X`, which is part of the data of a filtered 

algebra with basis. We know (see 

:class:`~sage.categories.filtered_modules_with_basis.FilteredModulesWithBasis`) 

that `A` (being a filtered `R`-module with basis) is canonically 

(when the basis is considered to be part of the data) 

isomorphic to `\operatorname{gr} A` as an `R`-module. Therefore 

the `k`-th graded component `G_k` can be identified with 

the span of `(b_x)_{x \in X_k}`, or equivalently the 

`k`-th homogeneous component of `A`. Suppose 

that `u' v' = \sum_{k \leq i+j} m_k` where `m_k \in G_k` (which 

has been identified with the `k`-th homogeneous component of `A`). 

Then `u * v = m_{i+j}`. We also note that the choice of 

identification of `G_k` with the `k`-th homogeneous component 

of `A` depends on the given basis. 

 

The basis `(b_x)_{x \in X}` of `A` gives rise to a basis 

of `\operatorname{gr} A`. This latter basis is still indexed 

by the elements of `X`, and consists of the images of the 

`b_x` under the `R`-module isomorphism from `A` to 

`\operatorname{gr} A`. It makes `\operatorname{gr} A` into 

a graded `R`-algebra with basis. 

 

In this class, the `R`-module isomorphism from `A` to 

`\operatorname{gr} A` is implemented as 

:meth:`to_graded_conversion` and also as the default 

conversion from `A` to `\operatorname{gr} A`. Its 

inverse map is implemented as 

:meth:`from_graded_conversion`. 

The projection `p_i : F_i \to G_i` is implemented as 

:meth:`projection` ``(i)``. 

 

INPUT: 

 

- ``A`` -- a filtered module (or algebra) with basis 

 

OUTPUT: 

 

The associated graded module of `A`, if `A` is just a filtered 

`R`-module. 

The associated graded algebra of `A`, if `A` is a filtered 

`R`-algebra. 

 

EXAMPLES: 

 

Associated graded module of a filtered module:: 

 

sage: A = Modules(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA.category() 

Category of graded modules with basis over Rational Field 

sage: x = A.basis()[Partition([3,2,1])] 

sage: grA(x) 

Bbar[[3, 2, 1]] 

 

Associated graded algebra of a filtered algebra:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA.category() 

Category of graded algebras with basis over Rational Field 

sage: x,y,z = [grA.algebra_generators()[s] for s in ['x','y','z']] 

sage: x 

bar(U['x']) 

sage: y * x + z 

bar(U['x']*U['y']) + bar(U['z']) 

sage: A(y) * A(x) + A(z) 

U['x']*U['y'] 

 

We note that the conversion between ``A`` and ``grA`` is 

the canonical ``QQ``-module isomorphism stemming from the 

fact that the underlying ``QQ``-modules of ``A`` and 

``grA`` are isomorphic:: 

 

sage: grA(A.an_element()) 

bar(U['x']^2*U['y']^2*U['z']^3) + 2*bar(U['x']) + 3*bar(U['y']) + bar(1) 

sage: elt = A.an_element() + A.algebra_generators()['x'] + 2 

sage: grelt = grA(elt); grelt 

bar(U['x']^2*U['y']^2*U['z']^3) + 3*bar(U['x']) + 3*bar(U['y']) + 3*bar(1) 

sage: A(grelt) == elt 

True 

 

.. TODO:: 

 

The algebra ``A`` must currently be an instance of (a subclass of) 

:class:`CombinatorialFreeModule`. This should work with any 

filtered algebra with a basis. 

 

.. TODO:: 

 

Implement a version of associated graded algebra for 

filtered algebras without a distinguished basis. 

 

REFERENCES: 

 

- :wikipedia:`Filtered_algebra#Associated_graded_algebra` 

""" 

def __init__(self, A, category=None): 

""" 

Initialize ``self``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: TestSuite(grA).run(elements=[prod(grA.algebra_generators())]) 

""" 

if A not in ModulesWithBasis(A.base_ring().category()).Filtered(): 

raise ValueError("the base algebra must be filtered and with basis") 

self._A = A 

 

base_ring = A.base_ring() 

base_one = base_ring.one() 

 

category = A.category().Graded().or_subcategory(category) 

try: 

opts = copy(A.print_options()) 

if not opts['prefix'] and not opts['bracket']: 

opts['bracket'] = '(' 

opts['prefix'] = opts['prefix'] + 'bar' 

except AttributeError: 

opts = {'prefix': 'Abar'} 

 

CombinatorialFreeModule.__init__(self, base_ring, A.basis().keys(), 

category=category, **opts) 

 

# Setup the conversion back 

phi = self.module_morphism(diagonal=lambda x: base_one, codomain=A) 

self._A.register_conversion(phi) 

 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().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 Rational Field 

""" 

from sage.categories.algebras_with_basis import AlgebrasWithBasis 

if self in AlgebrasWithBasis: 

return "Graded Algebra of {}".format(self._A) 

return "Graded Module of {}".format(self._A) 

 

def _latex_(self): 

r""" 

Return a latex representation of ``self``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: latex(A.graded_algebra()) 

\operatorname{gr} ... 

""" 

from sage.misc.latex import latex 

return "\\operatorname{gr} " + latex(self._A) 

 

def _element_constructor_(self, x): 

r""" 

Construct an element of ``self`` from ``x``. 

 

If ``self`` `= \operatorname{gr} A` for a filtered algebra 

`A` with basis, and if ``x`` is an element of `A`, then 

this returns the image of `x` under the canonical `R`-module 

isomorphism `A \to \operatorname{gr} A`. (In this case, 

this is equivalent to calling 

``self.to_graded_conversion()(x)``.) 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA(A.an_element()) 

bar(U['x']^2*U['y']^2*U['z']^3) 

+ 2*bar(U['x']) + 3*bar(U['y']) + bar(1) 

sage: grA(A.an_element() + A.algebra_generators()['x'] + 2) 

bar(U['x']^2*U['y']^2*U['z']^3) 

+ 3*bar(U['x']) + 3*bar(U['y']) + 3*bar(1) 

""" 

if isinstance(x, CombinatorialFreeModule.Element): 

if x.parent() is self._A: 

return self._from_dict(dict(x)) 

return super(AssociatedGradedAlgebra, self)._element_constructor_(x) 

 

def gen(self, *args, **kwds): 

""" 

Return a generator of ``self``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA.gen('x') 

bar(U['x']) 

""" 

try: 

x = self._A.gen(*args, **kwds) 

except AttributeError: 

x = self._A.algebra_generators()[args[0]] 

return self(x) 

 

@cached_method 

def algebra_generators(self): 

""" 

Return the algebra generators of ``self``. 

 

This assumes that the algebra generators of `A` provided by 

its ``algebra_generators`` method are homogeneous. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA.algebra_generators() 

Finite family {'y': bar(U['y']), 'x': bar(U['x']), 'z': bar(U['z'])} 

""" 

G = self._A.algebra_generators() 

return Family(G.keys(), lambda x: self(G[x]), name="generator") 

 

def degree_on_basis(self, x): 

""" 

Return the degree of the basis element indexed by ``x``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: all(A.degree_on_basis(x) == grA.degree_on_basis(x) 

....: for g in grA.algebra_generators() for x in g.support()) 

True 

""" 

return self._A.degree_on_basis(x) 

 

@cached_method 

def one_basis(self): 

""" 

Return the basis index of the element `1` of 

`\operatorname{gr} A`. 

 

This assumes that the unity `1` of `A` belongs to `F_0`. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: grA.one_basis() 

1 

""" 

return self._A.one_basis() 

 

def product_on_basis(self, x, y): 

""" 

Return the product on basis elements given by the 

indices ``x`` and ``y``. 

 

EXAMPLES:: 

 

sage: A = Algebras(QQ).WithBasis().Filtered().example() 

sage: grA = A.graded_algebra() 

sage: G = grA.algebra_generators() 

sage: x,y,z = G['x'], G['y'], G['z'] 

sage: x * y # indirect doctest 

bar(U['x']*U['y']) 

sage: y * x 

bar(U['x']*U['y']) 

sage: z * y * x 

bar(U['x']*U['y']*U['z']) 

""" 

ret = self._A.product_on_basis(x, y) 

deg = self._A.degree_on_basis(x) + self._A.degree_on_basis(y) 

return self.sum_of_terms([(i,c) for i,c in ret 

if self._A.degree_on_basis(i) == deg], 

distinct=True)