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
.