Coverage for local/lib/python2.7/site-packages/sage/categories/semisimple_algebras.py : 95%

r""" 

Semisimple Algebras 

""" 

from __future__ import absolute_import 

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

# Copyright (C) 2011-2015 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.bindable_class import BoundClass 

from sage.misc.cachefunc import cached_method 

from sage.misc.lazy_import import LazyImport 

from .category_types import Category_over_base_ring 

from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring 

from .algebras import Algebras 

class SemisimpleAlgebras(Category_over_base_ring): 

""" 

The category of semisimple algebras over a given base ring. 

EXAMPLES:: 

sage: from sage.categories.semisimple_algebras import SemisimpleAlgebras 

sage: C = SemisimpleAlgebras(QQ); C 

Category of semisimple algebras over Rational Field 

This category is best constructed as:: 

sage: D = Algebras(QQ).Semisimple(); D 

Category of semisimple algebras over Rational Field 

sage: D is C 

True 

sage: C.super_categories() 

[Category of algebras over Rational Field] 

Typically, finite group algebras are semisimple:: 

sage: DihedralGroup(5).algebra(QQ) in SemisimpleAlgebras 

True 

Unless the characteristic of the field divides the order of the group:: 

sage: DihedralGroup(5).algebra(IntegerModRing(5)) in SemisimpleAlgebras 

False 

sage: DihedralGroup(5).algebra(IntegerModRing(7)) in SemisimpleAlgebras 

True 

.. SEEALSO:: :wikipedia:`Semisimple_algebra` 

TESTS:: 

sage: TestSuite(C).run() 

""" 

@staticmethod 

def __classget__(cls, base_category, base_category_class): 

""" 

Implement the shorthand ``Algebras(K).Semisimple()`` for ``SemisimpleAlgebras(K)``. 

This magic mimics the syntax of axioms for a smooth transition 

if ``Semisimple`` becomes one. 

EXAMPLES:: 

sage: Algebras(QQ).Semisimple() 

Category of semisimple algebras over Rational Field 

sage: Algebras.Semisimple 

<class 'sage.categories.semisimple_algebras.SemisimpleAlgebras'> 

""" 

if base_category is None: 

return cls 

return BoundClass(cls, base_category.base_ring()) 

@cached_method 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: Algebras(QQ).Semisimple().super_categories() 

[Category of algebras over Rational Field] 

""" 

R = self.base_ring() 

return [Algebras(R)] 

class ParentMethods: 

def radical_basis(self, **keywords): 

r""" 

Return a basis of the Jacobson radical of this algebra. 

- ``keywords`` -- for compatibility; ignored. 

OUTPUT: the empty list since this algebra is semisimple. 

EXAMPLES:: 

sage: A = SymmetricGroup(4).algebra(QQ) 

sage: A.radical_basis() 

() 

TESTS:: 

sage: A.radical_basis.__module__ 

'sage.categories.finite_dimensional_semisimple_algebras_with_basis' 

""" 

return () 

class FiniteDimensional(CategoryWithAxiom_over_base_ring): 

WithBasis = LazyImport('sage.categories.finite_dimensional_semisimple_algebras_with_basis', 'FiniteDimensionalSemisimpleAlgebrasWithBasis')