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 \(1 \in A_0\).


sage: Algebras(ZZ).Super()
Category of super algebras over Integer Ring
class ParentMethods#

Bases: object


Return the associated graded algebra to self.


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.


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()
sage: T in Algebras(ZZ).Graded().TensorProducts()
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



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


Return the full subcategory of the supercommutative objects of self.

A super algebra \(M\) is supercommutative if, for all homogeneous \(x,y\in M\),

\[x \cdot y = (-1)^{|x||y|} y \cdot x.\]


Wikipedia article Supercommutative_algebra


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

alias of SupercommutativeAlgebras



sage: Algebras(ZZ).Super().super_categories() # indirect doctest
[Category of graded algebras over Integer Ring,
 Category of super modules over Integer Ring]