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: from sage.categories.semisimple_algebras import SemisimpleAlgebras sage: C = SemisimpleAlgebras(QQ); C Category of semisimple algebras over Rational Field - >>> 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: 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] - >>> 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: DihedralGroup(5).algebra(QQ) in SemisimpleAlgebras # needs sage.groups True - >>> 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: DihedralGroup(5).algebra(IntegerModRing(5)) in SemisimpleAlgebras # needs sage.groups False sage: DihedralGroup(5).algebra(IntegerModRing(7)) in SemisimpleAlgebras # needs sage.groups True - >>> 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 - class FiniteDimensional(base_category)[source]¶
 - 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: A = SymmetricGroup(4).algebra(QQ) # needs sage.combinat sage.groups sage: A.radical_basis() # needs sage.combinat sage.groups () - >>> from sage.all import * >>> A = SymmetricGroup(Integer(4)).algebra(QQ) # needs sage.combinat sage.groups >>> A.radical_basis() # needs sage.combinat sage.groups ()