Examples of graded connected Hopf algebras with basis

sage.categories.examples.graded_connected_hopf_algebras_with_basis.Example

alias of GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator

class sage.categories.examples.graded_connected_hopf_algebras_with_basis.GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator(base_ring)

Bases: CombinatorialFreeModule

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 nonnegative 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 – nonnegative integer

OUTPUT: an element of the tensor square of self

degree_on_basis(i)

The degree of a nonnegative integer is itself.

INPUT:

• i – nonnegative integer

OUTPUT: nonnegative integer

one_basis()

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

OUTPUT: the nonnegative integer 0

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
>>> H.one_basis()
0
>>> 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 – nonnegative integers

OUTPUT: a basis element indexed by i+j