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
TESTS:
sage: TestSuite(Algebras(ZZ).Super()).run()
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class
ParentMethods
¶ -
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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SuperAlgebras.
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