Semisimple Algebras#

class sage.categories.semisimple_algebras.SemisimpleAlgebras(base, name=None)[source]#

Bases: Category_over_base_ring

The category of semisimple algebras over a given base ring.

EXAMPLES:

Sage

sage: from sage.categories.semisimple_algebras import SemisimpleAlgebras
sage: C = SemisimpleAlgebras(QQ); C
Category of semisimple algebras over Rational Field

Python

>>> from sage.all import *
>>> from sage.categories.semisimple_algebras import SemisimpleAlgebras
>>> C = SemisimpleAlgebras(QQ); C
Category of semisimple algebras over Rational Field

This category is best constructed as:

Sage

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]

Python

>>> from sage.all import *
>>> D = Algebras(QQ).Semisimple(); D
Category of semisimple algebras over Rational Field
>>> D is C
True

>>> C.super_categories()
[Category of algebras over Rational Field]

Typically, finite group algebras are semisimple:

Sage

sage: DihedralGroup(5).algebra(QQ) in SemisimpleAlgebras                        # needs sage.groups
True

Python

>>> from sage.all import *
>>> DihedralGroup(Integer(5)).algebra(QQ) in SemisimpleAlgebras                        # needs sage.groups
True

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

Sage

sage: DihedralGroup(5).algebra(IntegerModRing(5)) in SemisimpleAlgebras         # needs sage.groups
False

sage: DihedralGroup(5).algebra(IntegerModRing(7)) in SemisimpleAlgebras         # needs sage.groups
True

Python

>>> from sage.all import *
>>> DihedralGroup(Integer(5)).algebra(IntegerModRing(Integer(5))) in SemisimpleAlgebras         # needs sage.groups
False

>>> DihedralGroup(Integer(5)).algebra(IntegerModRing(Integer(7))) in SemisimpleAlgebras         # needs sage.groups
True

See also

Wikipedia article Semisimple_algebra

class FiniteDimensional(base_category)[source]#

Bases: CategoryWithAxiom_over_base_ring

WithBasis[source]#: alias of FiniteDimensionalSemisimpleAlgebrasWithBasis

class ParentMethods[source]#

Bases: object

radical_basis(**keywords)[source]#

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

sage: A = SymmetricGroup(4).algebra(QQ)                                 # needs sage.combinat sage.groups
sage: A.radical_basis()                                                 # needs sage.combinat sage.groups
()

Python

>>> from sage.all import *
>>> A = SymmetricGroup(Integer(4)).algebra(QQ)                                 # needs sage.combinat sage.groups
>>> A.radical_basis()                                                 # needs sage.combinat sage.groups
()

super_categories()[source]#

EXAMPLES:

Sage

sage: Algebras(QQ).Semisimple().super_categories()
[Category of algebras over Rational Field]

Python

>>> from sage.all import *
>>> Algebras(QQ).Semisimple().super_categories()
[Category of algebras over Rational Field]