Bialgebras¶
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class
sage.categories.bialgebras.
Bialgebras
(base, name=None)¶ Bases:
sage.categories.category_types.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]
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class
ElementMethods
¶ Bases:
object
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is_grouplike
()¶ Return whether
self
is a grouplike element.EXAMPLES:
sage: s = SymmetricFunctions(QQ).schur() sage: s([5]).is_grouplike() False sage: s([]).is_grouplike() True
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is_primitive
()¶ Return whether
self
is a primitive element.EXAMPLES:
sage: s = SymmetricFunctions(QQ).schur() sage: s([5]).is_primitive() False sage: p = SymmetricFunctions(QQ).powersum() sage: p([5]).is_primitive() True
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class
Super
(base_category)¶
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WithBasis
¶ alias of
sage.categories.bialgebras_with_basis.BialgebrasWithBasis
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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
Todo
This category should be a
CategoryWithAxiom
.EXAMPLES:
sage: Bialgebras(QQ).additional_structure()
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super_categories
()¶ EXAMPLES:
sage: Bialgebras(QQ).super_categories() [Category of algebras over Rational Field, Category of coalgebras over Rational Field]
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class