Examples of Hopf algebras with basis¶
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class
sage.categories.examples.hopf_algebras_with_basis.
MyGroupAlgebra
(R, G)¶ Bases:
sage.combinat.free_module.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.
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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)]}
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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)]
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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)]
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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
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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[()]
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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)]
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