Examples of Hopf algebras with basis

class sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra(R, G)

Bases: CombinatorialFreeModule

An example of a Hopf algebra with basis: the group algebra of a group

This class illustrates a minimal implementation of a Hopf algebra with basis.

algebra_generators()

Return the generators of this algebra, as per 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(g)

Antipode, on basis elements, as per HopfAlgebrasWithBasis.ParentMethods.antipode_on_basis().

It is given, on basis elements, by \(\nu(g) = g^{-1}\)

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(g)

Coproduct, on basis elements, as per HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis().

The basis elements are group like: \(\Delta(g) = g \otimes g\).

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)]

counit_on_basis(g)

Counit, on basis elements, as per HopfAlgebrasWithBasis.ParentMethods.counit_on_basis().

The counit on the basis elements is 1.

EXAMPLES:

sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: (a, b) = A._group.gens()
sage: A.counit_on_basis(a)
1

one_basis()

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(g1, g2)

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)]