Super Algebras¶
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class
sage.categories.super_algebras.
SuperAlgebras
(base_category)¶ Bases:
sage.categories.super_modules.SuperModulesCategory
The category of super algebras.
An \(R\)-super algebra is an \(R\)-super module \(A\) endowed with an \(R\)-algebra structure satisfying
\[A_0 A_0 \subseteq A_0, \qquad A_0 A_1 \subseteq A_1, \qquad A_1 A_0 \subseteq A_1, \qquad A_1 A_1 \subseteq A_0\]and \(1 \in A_0\).
EXAMPLES:
sage: Algebras(ZZ).Super() Category of super algebras over Integer Ring
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class
ParentMethods
¶ Bases:
object
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graded_algebra
()¶ Return the associated graded algebra to
self
.Warning
Because a super module \(M\) is naturally \(\ZZ / 2 \ZZ\)-graded, and graded modules have a natural filtration induced by the grading, if \(M\) has a different filtration, then the associated graded module \(\operatorname{gr} M \neq M\). This is most apparent with super algebras, such as the
differential Weyl algebra
, and the multiplication may not coincide.
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tensor
(*parents, **kwargs)¶ Return the tensor product of the parents.
EXAMPLES:
sage: A.<x,y,z> = ExteriorAlgebra(ZZ); A.rename("A") sage: T = A.tensor(A,A); T A # A # A sage: T in Algebras(ZZ).Graded().SignedTensorProducts() True sage: T in Algebras(ZZ).Graded().TensorProducts() False sage: A.rename(None)
This also works when the other elements do not have a signed tensor product (trac ticket #31266):
sage: a = SteenrodAlgebra(3).an_element() sage: M = CombinatorialFreeModule(GF(3), ['s', 't', 'u']) sage: s = M.basis()['s'] sage: tensor([a, s]) 2*Q_1 Q_3 P(2,1) # B['s']
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class
SignedTensorProducts
(category, *args)¶ Bases:
sage.categories.signed_tensor.SignedTensorProductsCategory
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extra_super_categories
()¶ EXAMPLES:
sage: Coalgebras(QQ).Graded().SignedTensorProducts().extra_super_categories() [Category of graded coalgebras over Rational Field] sage: Coalgebras(QQ).Graded().SignedTensorProducts().super_categories() [Category of graded coalgebras over Rational Field]
Meaning: a signed tensor product of coalgebras is a coalgebra
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class
SubcategoryMethods
¶ Bases:
object
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Supercommutative
()¶ Return the full subcategory of the supercommutative objects of
self
.A super algebra \(M\) is supercommutative if, for all homogeneous \(x,y\in M\),
\[x \cdot y = (-1)^{|x||y|} y \cdot x.\]REFERENCES:
Wikipedia article Supercommutative_algebra
EXAMPLES:
sage: Algebras(ZZ).Super().Supercommutative() Category of supercommutative algebras over Integer Ring sage: Algebras(ZZ).Super().WithBasis().Supercommutative() Category of supercommutative algebras with basis over Integer Ring
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Supercommutative
¶ alias of
sage.categories.supercommutative_algebras.SupercommutativeAlgebras
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extra_super_categories
()¶ EXAMPLES:
sage: Algebras(ZZ).Super().super_categories() # indirect doctest [Category of graded algebras over Integer Ring, Category of super modules over Integer Ring]
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class