Super Algebras#
- class sage.categories.super_algebras.SuperAlgebras(base_category)#
Bases:
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
- class ParentMethods#
Bases:
object
- 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.
- tensor(*parents, **kwargs)#
Return the tensor product of the parents.
EXAMPLES:
sage: # needs sage.combinat sage.modules 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 (github issue #31266):
sage: # needs sage.combinat sage.modules sage: a = SteenrodAlgebra(3).an_element() sage: M = CombinatorialFreeModule(GF(3), ['s', 't', 'u']) sage: s = M.basis()['s'] sage: tensor([a, s]) # needs sage.rings.finite_rings 2*Q_1 Q_3 P(2,1) # B['s']
- class SignedTensorProducts(category, *args)#
Bases:
SignedTensorProductsCategory
- 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
- class SubcategoryMethods#
Bases:
object
- 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
- Supercommutative#
alias of
SupercommutativeAlgebras
- 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]