Super Algebras

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 \(1 \in A_0\).


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

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.



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