Coverage for local/lib/python2.7/site-packages/sage/categories/super_algebras.py : 90%

r""" 

Super Algebras 

""" 

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

# Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

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

# http://www.gnu.org/licenses/ 

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

from sage.categories.super_modules import SuperModulesCategory 

from sage.categories.algebras import Algebras 

from sage.categories.modules import Modules 

from sage.misc.lazy_import import LazyImport 

class SuperAlgebras(SuperModulesCategory): 

""" 

The category of super algebras. 

An `R`-*super algebra* is an `R`-super module `A` endowed with an 

`R`-algebra structure satisfying 

.. MATH:: 

A_0 A_0 \subseteq A_0, \qquad 

A_0 A_1 \subseteq A_1, \qquad 

A_1 A_0 \subseteq A_1, \qquad 

A_1 A_1 \subseteq A_0 

and `1 \in A_0`. 

EXAMPLES:: 

sage: Algebras(ZZ).Super() 

Category of super algebras over Integer Ring 

TESTS:: 

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

""" 

def extra_super_categories(self): 

""" 

EXAMPLES:: 

sage: Algebras(ZZ).Super().super_categories() # indirect doctest 

[Category of graded algebras over Integer Ring, 

Category of super modules over Integer Ring] 

""" 

return [self.base_category().Graded()] 

class ParentMethods: 

def graded_algebra(self): 

r""" 

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

.. WARNING:: 

Because a super module `M` is naturally `\ZZ / 2 \ZZ`-graded, and 

graded modules have a natural filtration induced by the grading, if 

`M` has a different filtration, then the associated graded module 

`\operatorname{gr} M \neq M`. This is most apparent with super 

algebras, such as the :class:`differential Weyl algebra 

<sage.algebras.weyl_algebra.DifferentialWeylAlgebra>`, and the 

multiplication may not coincide. 

""" 

raise NotImplementedError