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 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
andj
is an element of degreei+j
.INPUT:
i
,j
– non-negative integers
OUTPUT:
a basis element indexed by
i+j