Graded Hopf algebras

sage.categories.graded_hopf_algebras.GradedHopfAlgebras(base_ring)

The category of graded Hopf algebras.

EXAMPLES:

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

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.