Bialgebras#

class sage.categories.bialgebras.Bialgebras(base, name=None)#

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]
class ElementMethods#

Bases: object

is_grouplike()#

Return whether self is a grouplike element.

EXAMPLES:

sage: s = SymmetricFunctions(QQ).schur()                                # needs sage.modules
sage: s([5]).is_grouplike()                                             # needs sage.modules
False
sage: s([]).is_grouplike()                                              # needs sage.modules
True
is_primitive()#

Return whether self is a primitive element.

EXAMPLES:

sage: # needs sage.modules
sage: s = SymmetricFunctions(QQ).schur()
sage: s([5]).is_primitive()
False
sage: p = SymmetricFunctions(QQ).powersum()
sage: p([5]).is_primitive()
True
class Super(base_category)#

Bases: SuperModulesCategory

WithBasis#

alias of BialgebrasWithBasis

additional_structure()#

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.

Todo

This category should be a CategoryWithAxiom.

EXAMPLES:

sage: Bialgebras(QQ).additional_structure()
super_categories()#

EXAMPLES:

sage: Bialgebras(QQ).super_categories()
[Category of algebras over Rational Field, Category of coalgebras over Rational Field]