Examples of graded connected Hopf algebras with basis¶
- sage.categories.examples.graded_connected_hopf_algebras_with_basis.Example[source]¶
- alias of - GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator
- class sage.categories.examples.graded_connected_hopf_algebras_with_basis.GradedConnectedCombinatorialHopfAlgebraWithPrimitiveGenerator(base_ring)[source]¶
- 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)[source]¶
- 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)[source]¶
- The degree of a nonnegative integer is itself. - INPUT: - i– nonnegative integer
 - OUTPUT: nonnegative integer 
 - one_basis()[source]¶
- Return 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 - >>> from sage.all import * >>> H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() >>> H.one_basis() 0 >>> H.one() P0