Bialgebras¶
- class sage.categories.bialgebras.Bialgebras(base, name=None)[source]¶
- Bases: - Category_over_base_ring- The category of bialgebras. - EXAMPLES: - sage: Bialgebras(ZZ) Category of bialgebras over Integer Ring sage: Bialgebras(ZZ).super_categories() [Category of algebras over Integer Ring, Category of coalgebras over Integer Ring] - >>> from sage.all import * >>> Bialgebras(ZZ) Category of bialgebras over Integer Ring >>> Bialgebras(ZZ).super_categories() [Category of algebras over Integer Ring, Category of coalgebras over Integer Ring] - class ElementMethods[source]¶
- Bases: - object- is_grouplike()[source]¶
- Return whether - selfis a grouplike element.- EXAMPLES: - sage: s = SymmetricFunctions(QQ).schur() # needs sage.modules sage: s([5]).is_grouplike() # needs lrcalc_python sage.modules False sage: s([]).is_grouplike() # needs lrcalc_python sage.modules True - >>> from sage.all import * >>> s = SymmetricFunctions(QQ).schur() # needs sage.modules >>> s([Integer(5)]).is_grouplike() # needs lrcalc_python sage.modules False >>> s([]).is_grouplike() # needs lrcalc_python sage.modules True 
 - is_primitive()[source]¶
- Return whether - selfis a primitive element.- EXAMPLES: - sage: # needs sage.modules sage: s = SymmetricFunctions(QQ).schur() sage: s([5]).is_primitive() # needs lrcalc_python False sage: p = SymmetricFunctions(QQ).powersum() sage: p([5]).is_primitive() True - >>> from sage.all import * >>> # needs sage.modules >>> s = SymmetricFunctions(QQ).schur() >>> s([Integer(5)]).is_primitive() # needs lrcalc_python False >>> p = SymmetricFunctions(QQ).powersum() >>> p([Integer(5)]).is_primitive() True 
 
 - class Super(base_category)[source]¶
- Bases: - SuperModulesCategory
 - WithBasis[source]¶
- alias of - BialgebrasWithBasis
 - additional_structure()[source]¶
- Return - None.- Indeed, the category of bialgebras defines no additional structure: a morphism of coalgebras and of algebras between two bialgebras is a bialgebra morphism. - See also - Todo - This category should be a - CategoryWithAxiom.- EXAMPLES: - sage: Bialgebras(QQ).additional_structure() - >>> from sage.all import * >>> Bialgebras(QQ).additional_structure() 
 - super_categories()[source]¶
- EXAMPLES: - sage: Bialgebras(QQ).super_categories() [Category of algebras over Rational Field, Category of coalgebras over Rational Field] - >>> from sage.all import * >>> Bialgebras(QQ).super_categories() [Category of algebras over Rational Field, Category of coalgebras over Rational Field]