Bases: sage.combinat.free_module.CombinatorialFreeModule
An of a Hopf algebra with basis: the group algebra of a group
This class illustrates a minimal implementation of a Hopf algebra with basis.
The generators of this algebra, as per Algebras.ParentMethods.algebra_generators().
They correspond to the generators of the group.
EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example(); A
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
sage: A.algebra_generators()
Finite family {(1,2,3): B[(1,2,3)], (1,3): B[(1,3)]}
Antipode, on basis elements, as per HopfAlgebrasWithBasis.ParentMethods.antipode_on_basis().
It is given, on basis elements, by
EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: A.antipode_on_basis(a)
B[(1,3,2)]
Coproduct, on basis elements, as per HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis().
The basis elements are group like: .
EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: A.coproduct_on_basis(a)
B[(1,2,3)] # B[(1,2,3)]
Returns the one of the group, which index the one of this algebra, as per AlgebrasWithBasis.ParentMethods.one_basis().
EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: A.one_basis()
()
sage: A.one()
B[()]
Product, on basis elements, as per AlgebrasWithBasis.ParentMethods.product_on_basis().
The product of two basis elements is induced by the product of the corresponding elements of the group.
EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: a*b
(1,2)
sage: A.product_on_basis(a, b)
B[(1,2)]