Examples of graded connected Hopf algebras with basis¶

sage.categories.examples.graded_connected_hopf_algebras_with_basis.
Example
¶

class
sage.categories.examples.graded_connected_hopf_algebras_with_basis.
GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator
(base_ring)¶ Bases:
sage.combinat.free_module.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.
\[\Delta(P_i) = \sum_{j=0}^i P_{ij} \otimes P_j\]INPUT:
i
– a 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
– a nonnegative integer
OUTPUT:
a nonnegative integer

one_basis
()¶ Returns 0, which index the unit of the Hopf algebra.
OUTPUT:
the nonnegative 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
andj
is an element of degreei+j
.INPUT:
i
,j
– nonnegative integers
OUTPUT:
a basis element indexed by
i+j
