Super Algebras - Category Framework

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# 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 \(1 \in A_0\).

  EXAMPLES:

  Sage

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

  Python

      >>> from sage.all import *
      >>> Algebras(ZZ).Super()
      Category of super algebras over Integer Ring

  class ParentMethods

    Bases: `object`

    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.

    tensor(*parents, **kwargs)

      Return the tensor product of the parents.

      EXAMPLES:

      Sage

          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)

      Python

          >>> 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

          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']

      [Python]

          >>> 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)

    Bases: `SignedTensorProductsCategory`

    extra_super_categories()

      EXAMPLES:

      Sage

          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]

      Python

          >>> 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

    Bases: `object`

    Supercommutative()

      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

          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

      Python

          >>> 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

    alias of `SupercommutativeAlgebras`

  extra_super_categories()

    EXAMPLES:

    Sage

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

    Python

        >>> 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]