Semisimple Algebras

class sage.categories.semisimple_algebras.SemisimpleAlgebras(base, name=None)

Bases: sage.categories.category_types.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

TESTS:

sage: TestSuite(C).run()
class FiniteDimensional(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

TESTS:

sage: C = Modules(ZZ).FiniteDimensional(); C
Category of finite dimensional modules over Integer Ring
sage: type(C)
<class 'sage.categories.modules.Modules.FiniteDimensional_with_category'>
sage: type(C).__base__.__base__
<class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'>

sage: TestSuite(C).run()
WithBasis

alias of FiniteDimensionalSemisimpleAlgebrasWithBasis

class SemisimpleAlgebras.ParentMethods
radical_basis(**keywords)

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'
SemisimpleAlgebras.super_categories()

EXAMPLES:

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