Examples of Hopf algebras with basis¶
- class sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra(R, G)[source]¶
- 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()[source]¶
- 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)]} - >>> from sage.all import * >>> 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 >>> A.algebra_generators() Finite family {(1,2,3): B[(1,2,3)], (1,3): B[(1,3)]} 
 - antipode_on_basis(g)[source]¶
- 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)] - >>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() >>> (a, b) = A._group.gens() >>> A.antipode_on_basis(a) B[(1,3,2)] 
 - coproduct_on_basis(g)[source]¶
- 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)] - >>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() >>> (a, b) = A._group.gens() >>> A.coproduct_on_basis(a) B[(1,2,3)] # B[(1,2,3)] 
 - counit_on_basis(g)[source]¶
- 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 - >>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() >>> (a, b) = A._group.gens() >>> A.counit_on_basis(a) 1 
 - one_basis()[source]¶
- Return 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[()] - >>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() >>> A.one_basis() () >>> A.one() B[()] 
 - product_on_basis(g1, g2)[source]¶
- 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)] - >>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() >>> (a, b) = A._group.gens() >>> a*b (1,2) >>> A.product_on_basis(a, b) B[(1,2)]