Super modules

sage.categories.super_modules.SuperModules

The category of super modules.

An \(R\)-super module (where \(R\) is a ring) is an \(R\)-module \(M\) equipped with a decomposition \(M = M_0 \oplus M_1\) into two \(R\)-submodules \(M_0\) and \(M_1\) (called the even part and the odd part of \(M\), respectively).

Thus, an \(R\)-super module automatically becomes a \(\ZZ / 2 \ZZ\)-graded \(R\)-module, with \(M_0\) being the degree-\(0\) component and \(M_1\) being the degree-\(1\) component.

EXAMPLES:

sage: Modules(ZZ).Super()
Category of super modules over Integer Ring
sage: Modules(ZZ).Super().super_categories()
[Category of graded modules over Integer Ring]

The category of super modules defines the super structure which shall be preserved by morphisms:

sage: Modules(ZZ).Super().additional_structure()
Category of super modules over Integer Ring
class sage.categories.super_modules.SuperModulesCategory(base_category)

Bases: sage.categories.covariant_functorial_construction.CovariantConstructionCategory, sage.categories.category_types.Category_over_base_ring

EXAMPLES:

sage: C = Algebras(QQ).Super()
sage: C
Category of super algebras over Rational Field
sage: C.base_category()
Category of algebras over Rational Field
sage: sorted(C.super_categories(), key=str)
[Category of graded algebras over Rational Field,
 Category of super modules over Rational Field]

sage: AlgebrasWithBasis(QQ).Super().base_ring()
Rational Field
sage: HopfAlgebrasWithBasis(QQ).Super().base_ring()
Rational Field
classmethod default_super_categories(category, *args)

Return the default super categories of \(F_{Cat}(A,B,...)\) for \(A,B,...\) parents in \(Cat\).

INPUT:

  • cls – the category class for the functor \(F\)
  • category – a category \(Cat\)
  • *args – further arguments for the functor

OUTPUT:

A join category.

This implements the property that subcategories constructed by the set of whitelisted axioms is a subcategory.

EXAMPLES:

sage: HopfAlgebras(ZZ).WithBasis().FiniteDimensional().Super() # indirect doctest
Category of finite dimensional super hopf algebras with basis over Integer Ring