Semisimple Algebras¶
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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()
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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()
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WithBasis
¶ alias of
FiniteDimensionalSemisimpleAlgebrasWithBasis
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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'
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SemisimpleAlgebras.
super_categories
()¶ EXAMPLES:
sage: Algebras(QQ).Semisimple().super_categories() [Category of algebras over Rational Field]
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class