Bialgebras

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

Bases: Category_over_base_ring

The category of bialgebras

EXAMPLES:

Sage

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]

Python

>>> 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

Bases: object

is_grouplike()

Return whether self is a grouplike element.

EXAMPLES:

Sage

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

Python

>>> 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()

Return whether self is a primitive element.

EXAMPLES:

Sage

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

Python

>>> 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)

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.

See also

Category.additional_structure()

Todo

This category should be a CategoryWithAxiom.

EXAMPLES:

Sage

sage: Bialgebras(QQ).additional_structure()

Python

>>> from sage.all import *
>>> Bialgebras(QQ).additional_structure()

super_categories()

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> Bialgebras(QQ).super_categories()
[Category of algebras over Rational Field, Category of coalgebras over Rational Field]