Coverage for local/lib/python2.7/site-packages/sage/categories/filtered_algebras.py : 100%

r""" 

Filtered Algebras 

""" 

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

# 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.abstract_method import abstract_method 

from sage.categories.filtered_modules import FilteredModulesCategory 

class FilteredAlgebras(FilteredModulesCategory): 

r""" 

The category of filtered algebras. 

An algebra `A` over a commutative ring `R` is *filtered* if 

`A` is endowed with a structure of a filtered `R`-module 

(whose underlying `R`-module structure is identical with 

that of the `R`-algebra `A`) such that the indexing set `I` 

(typically `I = \NN`) is also an additive abelian monoid, 

the unity `1` of `A` belongs to `F_0`, and we have 

`F_i \cdot F_j \subseteq F_{i+j}` for all `i, j \in I`. 

EXAMPLES:: 

sage: Algebras(ZZ).Filtered() 

Category of filtered algebras over Integer Ring 

sage: Algebras(ZZ).Filtered().super_categories() 

[Category of algebras over Integer Ring, 

Category of filtered modules over Integer Ring] 

TESTS:: 

sage: TestSuite(Algebras(ZZ).Filtered()).run() 

REFERENCES: 

- :wikipedia:`Filtered_algebra` 

""" 

class ParentMethods: 

@abstract_method(optional=True) 

def graded_algebra(self): 

""" 

Return the associated graded algebra to ``self``. 

.. TODO:: 

Implement a version of the associated graded algebra 

which does not require ``self`` to have a 

distinguished basis. 

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 

"""