Super Algebras¶
- class sage.categories.super_algebras.SuperAlgebras(base_category)[source]¶
- 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 - >>> from sage.all import * >>> Algebras(ZZ).Super() Category of super algebras over Integer Ring - class ParentMethods[source]¶
- Bases: - object- graded_algebra()[source]¶
- 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)[source]¶
- 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) - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> A = ExteriorAlgebra(ZZ, names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3); A.rename('A') >>> T = A.tensor(A,A); T A # A # A >>> T in Algebras(ZZ).Graded().SignedTensorProducts() True >>> T in Algebras(ZZ).Graded().TensorProducts() False >>> A.rename(None) - This also works when the other elements do not have a signed tensor product (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'] - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> a = SteenrodAlgebra(Integer(3)).an_element() >>> M = CombinatorialFreeModule(GF(Integer(3)), ['s', 't', 'u']) >>> s = M.basis()['s'] >>> tensor([a, s]) # needs sage.rings.finite_rings 2*Q_1 Q_3 P(2,1) # B['s'] 
 
 - class SignedTensorProducts(category, *args)[source]¶
- Bases: - SignedTensorProductsCategory- extra_super_categories()[source]¶
- 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] - >>> from sage.all import * >>> Coalgebras(QQ).Graded().SignedTensorProducts().extra_super_categories() [Category of graded coalgebras over Rational Field] >>> 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[source]¶
- Bases: - object- Supercommutative()[source]¶
- 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 - >>> from sage.all import * >>> Algebras(ZZ).Super().Supercommutative() Category of supercommutative algebras over Integer Ring >>> Algebras(ZZ).Super().WithBasis().Supercommutative() Category of supercommutative algebras with basis over Integer Ring 
 
 - Supercommutative[source]¶
- alias of - SupercommutativeAlgebras
 - extra_super_categories()[source]¶
- EXAMPLES: - sage: Algebras(ZZ).Super().super_categories() # indirect doctest [Category of graded algebras over Integer Ring, Category of super modules over Integer Ring] - >>> from sage.all import * >>> Algebras(ZZ).Super().super_categories() # indirect doctest [Category of graded algebras over Integer Ring, Category of super modules over Integer Ring]