# Examples of graded connected Hopf algebras with basis¶

This class illustrates an implementation of a graded Hopf algebra with basis that has one primitive generator of degree 1 and basis elements indexed by non-negative integers.

This Hopf algebra example differs from what topologists refer to as a graded Hopf algebra because the twist operation in the tensor rule satisfies

$(\mu \otimes \mu) \circ (id \otimes \tau \otimes id) \circ (\Delta \otimes \Delta) = \Delta \circ \mu$

where $$\tau(x\otimes y) = y\otimes x$$.

coproduct_on_basis(i)

The coproduct of a basis element.

$\Delta(P_i) = \sum_{j=0}^i P_{i-j} \otimes P_j$

INPUT:

• i – a non-negative integer

OUTPUT:

• an element of the tensor square of self

degree_on_basis(i)

The degree of a non-negative integer is itself

INPUT:

• i – a non-negative integer

OUTPUT:

• a non-negative integer

one_basis()

Returns 0, which index the unit of the Hopf algebra.

OUTPUT:

• the non-negative integer 0

EXAMPLES:

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
sage: H.one_basis()
0
sage: H.one()
P0

product_on_basis(i, j)

The product of two basis elements.

The product of elements of degree i and j is an element of degree i+j.

INPUT:

• i, j – non-negative integers

OUTPUT:

• a basis element indexed by i+j