Hopf algebras with basis#

class sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis(base_category)[source]#

Bases: CategoryWithAxiom_over_base_ring

The category of Hopf algebras with a distinguished basis

EXAMPLES:

sage: C = HopfAlgebrasWithBasis(QQ)
sage: C
Category of Hopf algebras with basis over Rational Field
sage: C.super_categories()
[Category of Hopf algebras over Rational Field,
 Category of bialgebras with basis over Rational Field]

>>> from sage.all import *
>>> C = HopfAlgebrasWithBasis(QQ)
>>> C
Category of Hopf algebras with basis over Rational Field
>>> C.super_categories()
[Category of Hopf algebras over Rational Field,
 Category of bialgebras with basis over Rational Field]

We now show how to use a simple Hopf algebra, namely the group algebra of the dihedral group (see also AlgebrasWithBasis):

sage: A = C.example(); A                                                        # needs sage.groups
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.rename("A")                                                             # needs sage.groups
sage: A.category()                                                              # needs sage.groups
Category of finite dimensional Hopf algebras with basis over Rational Field

sage: A.one_basis()                                                             # needs sage.groups
()
sage: A.one()                                                                   # needs sage.groups
B[()]

sage: A.base_ring()                                                             # needs sage.groups
Rational Field
sage: A.basis().keys()                                                          # needs sage.groups
Dihedral group of order 6 as a permutation group

sage: # needs sage.groups
sage: [a,b] = A.algebra_generators()
sage: a, b
(B[(1,2,3)], B[(1,3)])
sage: a^3, b^2
(B[()], B[()])
sage: a*b
B[(1,2)]

sage: A.product           # todo: not quite ...                                 # needs sage.groups
<bound method MagmaticAlgebras.WithBasis.ParentMethods._product_from_product_on_basis_multiply of A>
sage: A.product(b, b)                                                           # needs sage.groups
B[()]

sage: A.zero().coproduct()                                                      # needs sage.groups
0
sage: A.zero().coproduct().parent()                                             # needs sage.groups
A # A
sage: a.coproduct()                                                             # needs sage.groups
B[(1,2,3)] # B[(1,2,3)]

sage: TestSuite(A).run(verbose=True)                                            # needs sage.groups
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_antipode() . . . pass
running ._test_associativity() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_characteristic() . . . pass
running ._test_construction() . . . pass
running ._test_distributivity() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_monomial_coefficients() . . . pass
  running ._test_new() . . . pass
  running ._test_nonzero_equal() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass
sage: A.__class__                                                               # needs sage.groups
<class 'sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category'>
sage: A.element_class                                                           # needs sage.groups
<class 'sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category.element_class'>

>>> from sage.all import *
>>> A = C.example(); A                                                        # needs sage.groups
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.rename("A")                                                             # needs sage.groups
>>> A.category()                                                              # needs sage.groups
Category of finite dimensional Hopf algebras with basis over Rational Field

>>> A.one_basis()                                                             # needs sage.groups
()
>>> A.one()                                                                   # needs sage.groups
B[()]

>>> A.base_ring()                                                             # needs sage.groups
Rational Field
>>> A.basis().keys()                                                          # needs sage.groups
Dihedral group of order 6 as a permutation group

>>> # needs sage.groups
>>> [a,b] = A.algebra_generators()
>>> a, b
(B[(1,2,3)], B[(1,3)])
>>> a**Integer(3), b**Integer(2)
(B[()], B[()])
>>> a*b
B[(1,2)]

>>> A.product           # todo: not quite ...                                 # needs sage.groups
<bound method MagmaticAlgebras.WithBasis.ParentMethods._product_from_product_on_basis_multiply of A>
>>> A.product(b, b)                                                           # needs sage.groups
B[()]

>>> A.zero().coproduct()                                                      # needs sage.groups
0
>>> A.zero().coproduct().parent()                                             # needs sage.groups
A # A
>>> a.coproduct()                                                             # needs sage.groups
B[(1,2,3)] # B[(1,2,3)]

>>> TestSuite(A).run(verbose=True)                                            # needs sage.groups
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_antipode() . . . pass
running ._test_associativity() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_characteristic() . . . pass
running ._test_construction() . . . pass
running ._test_distributivity() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_monomial_coefficients() . . . pass
  running ._test_new() . . . pass
  running ._test_nonzero_equal() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass
>>> A.__class__                                                               # needs sage.groups
<class 'sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category'>
>>> A.element_class                                                           # needs sage.groups
<class 'sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category.element_class'>

Let us look at the code for implementing A:

sage: A??                               # not implemented                       # needs sage.groups

>>> from sage.all import *
>>> A??                               # not implemented                       # needs sage.groups
class ElementMethods[source]#

Bases: object

Filtered[source]#

alias of FilteredHopfAlgebrasWithBasis

FiniteDimensional[source]#

alias of FiniteDimensionalHopfAlgebrasWithBasis

Graded[source]#

alias of GradedHopfAlgebrasWithBasis

class ParentMethods[source]#

Bases: object

antipode()[source]#

The antipode of this Hopf algebra.

If antipode_basis() is available, this constructs the antipode morphism from self to self by extending it by linearity. Otherwise, self.antipode_by_coercion() is used, if available.

EXAMPLES:

sage: # needs sage.groups
sage: A = HopfAlgebrasWithBasis(ZZ).example(); A
An example of Hopf algebra with basis: the group algebra of the
 Dihedral group of order 6 as a permutation group over Integer Ring
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: [a,b] = A.algebra_generators()
sage: a, A.antipode(a)
(B[(1,2,3)], B[(1,3,2)])
sage: b, A.antipode(b)
(B[(1,3)], B[(1,3)])

>>> from sage.all import *
>>> # needs sage.groups
>>> A = HopfAlgebrasWithBasis(ZZ).example(); A
An example of Hopf algebra with basis: the group algebra of the
 Dihedral group of order 6 as a permutation group over Integer Ring
>>> A = HopfAlgebrasWithBasis(QQ).example()
>>> [a,b] = A.algebra_generators()
>>> a, A.antipode(a)
(B[(1,2,3)], B[(1,3,2)])
>>> b, A.antipode(b)
(B[(1,3)], B[(1,3)])
antipode_on_basis(x)[source]#

The antipode of the Hopf algebra on the basis (optional)

INPUT:

  • x – an index of an element of the basis of self

Returns the antipode of the basis element indexed by x.

If this method is implemented, then antipode() is defined from this by linearity.

EXAMPLES:

sage: # needs sage.groups
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: W = A.basis().keys(); W
Dihedral group of order 6 as a permutation group
sage: w = W.gen(0); w
(1,2,3)
sage: A.antipode_on_basis(w)
B[(1,3,2)]

>>> from sage.all import *
>>> # needs sage.groups
>>> A = HopfAlgebrasWithBasis(QQ).example()
>>> W = A.basis().keys(); W
Dihedral group of order 6 as a permutation group
>>> w = W.gen(Integer(0)); w
(1,2,3)
>>> A.antipode_on_basis(w)
B[(1,3,2)]
Super[source]#

alias of SuperHopfAlgebrasWithBasis

class TensorProducts(category, *args)[source]#

Bases: TensorProductsCategory

The category of Hopf algebras with basis constructed by tensor product of Hopf algebras with basis

class ElementMethods[source]#

Bases: object

class ParentMethods[source]#

Bases: object

extra_super_categories()[source]#

EXAMPLES:

sage: C = HopfAlgebrasWithBasis(QQ).TensorProducts()
sage: C.extra_super_categories()
[Category of Hopf algebras with basis over Rational Field]
sage: sorted(C.super_categories(), key=str)
[Category of Hopf algebras with basis over Rational Field,
 Category of tensor products of Hopf algebras over Rational Field,
 Category of tensor products of algebras with basis over Rational Field]

>>> from sage.all import *
>>> C = HopfAlgebrasWithBasis(QQ).TensorProducts()
>>> C.extra_super_categories()
[Category of Hopf algebras with basis over Rational Field]
>>> sorted(C.super_categories(), key=str)
[Category of Hopf algebras with basis over Rational Field,
 Category of tensor products of Hopf algebras over Rational Field,
 Category of tensor products of algebras with basis over Rational Field]
example(G=None)[source]#

Returns an example of algebra with basis:

sage: HopfAlgebrasWithBasis(QQ['x']).example()                              # needs sage.groups
An example of Hopf algebra with basis: the group algebra of the
Dihedral group of order 6 as a permutation group
over Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> HopfAlgebrasWithBasis(QQ['x']).example()                              # needs sage.groups
An example of Hopf algebra with basis: the group algebra of the
Dihedral group of order 6 as a permutation group
over Univariate Polynomial Ring in x over Rational Field

An other group can be specified as optional argument:

sage: HopfAlgebrasWithBasis(QQ).example(SymmetricGroup(4))                  # needs sage.groups
An example of Hopf algebra with basis: the group algebra of the
Symmetric group of order 4! as a permutation group over Rational Field

>>> from sage.all import *
>>> HopfAlgebrasWithBasis(QQ).example(SymmetricGroup(Integer(4)))                  # needs sage.groups
An example of Hopf algebra with basis: the group algebra of the
Symmetric group of order 4! as a permutation group over Rational Field