Graded Hopf algebras - Category Framework

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# Graded Hopf algebras

sage.categories.graded_hopf_algebras.GradedHopfAlgebras(base_ring)

  The category of graded Hopf algebras.

  EXAMPLES:

  Sage

      sage: C = GradedHopfAlgebras(QQ); C
      Join of Category of Hopf algebras over Rational Field
          and Category of graded algebras over Rational Field
          and Category of graded coalgebras over Rational Field
      sage: C is HopfAlgebras(QQ).Graded()
      True

  Python

      >>> from sage.all import *
      >>> C = GradedHopfAlgebras(QQ); C
      Join of Category of Hopf algebras over Rational Field
          and Category of graded algebras over Rational Field
          and Category of graded coalgebras over Rational Field
      >>> C is HopfAlgebras(QQ).Graded()
      True

  Note

  This is not a graded Hopf algebra as is typically defined in algebraic topology as the product in the tensor square \((x \otimes y) (a \otimes b) = (xa) \otimes (yb)\) does not carry an additional sign. For this, instead use `super Hopf algebras`.