# Ticket #13372(new enhancement)

Opened 9 months ago

## add functionality for duals of algebras, coalgebras, hopf algebras, etc.

Reported by: Owned by: saliola AlexGhitza major sage-5.10 algebra duality, categories, algebras combinat N/A

In this  discussion, we came up with a rough draft of an interface for the method returning the dual of an object. Here is a summary by way of docstrings for the methods:

def dual(self, category=None):
r"""
The dual of ``self``.

By default, the dual is computed in the category
``self.category()``. If the user specifies a category, the dual will
be computed in that category.

INPUT:

- ``category`` -- category (default: the category of ``self``).

OUTPUT:

- The dual of ``self``.

EXAMPLES:

The Hopf algebra of symmetric functions is a self-dual Hopf
algebra::

sage: Sym = SymmetricFunctions(QQ); Sym
Symmetric Functions over Rational Field
sage: Sym.dual()
Symmetric Functions over Rational Field
sage: Sym.dual() is Sym
True

If we view ``Sym`` as an algebra, then its dual is a co-algebra::

sage: C = Sym.dual(category=Algebras(QQ)).category()
Category of duals of algebras over Rational Field
sage: C.super_categories()
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]

The Schur basis for symmetric functions is self-dual and the
homogeneous symmetric functions are dual to the monomial
symmetric functions::

sage: s = Sym.schur()
sage: s.dual() is s
True
sage: h = Sym.homogeneous()
sage: m = Sym.monomial()
sage: h.dual() is m
True

Note that in the above, ``s`` (as well as ``h`` and ``m``) are Hopf
algebras with basis. Hence, their duals are also Hopf algebras with
basis.

The Hopf algebra of quasi-symmetric functions is dual, as a Hopf
algebra, to the Hopf algebra of non-commutative symmetric
functions::

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: NCSF.dual()
Quasisymmetric functions over the Rational Field

::
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.dual()
Non-Commutative Symmetric Functions over the Rational Field

"""
return NotImplemented
def duality_pairing(self, x, y):
r"""
The duality pairing between elements of NSym and elements of QSym.

INPUT:

- ``x`` -- an element of ``self``
- ``y`` -- an element in the dual basis of ``self``

OUTPUT:

- The result of pairing the element ``x`` of ``self`` with the
element ``y`` of the dual of ``self``.

EXAMPLES:

The Schur basis of symmetric functions is self-dual::

sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym(QQ).schur()
sage: s.dual() is s
True
sage: s.duality_pairing(s[2,1,1], s[2,1,1])
1
sage: s.duality_pairing(s[2,1], s[3])
0

The fundamental basis of quasi-symmetric functions is dual to the
ribbon basis of non-commutative symmetric functions::

sage: R = NonCommutativeSymmetricFunctions(QQ).Ribbon()
sage: F = QuasiSymmetricFunctions(QQ).Fundamental()
sage: R.duality_pairing(R[1,1,2], F[1,1,2])
1
sage: R.duality_pairing(R[1,2,1], F[1,1,2])
0
sage: F.duality_pairing(F[1,2,1], R[1,1,2])
0

"""
return NotImplemented

A rudimentary implementation for duality_pairing can be found at #8899, but see also the scalar product code for symmetric functions.

I think a bunch of the code for duality for symmetric functions can be refactored. See sage.combinat.sf.dual.

## Change History

### comment:1 Changed 9 months ago by saliola

Simon raised the following question in the  thread:

Start with an object O in some category C1, take its dual D in C1, and apply the forgetful functor to map it to a sub-category C2; one would not always get the same result as if one first applies the forgetful functor to O and then dualise the result in C2, right?

And hence VectorSpaces?(QQ)(H.dual()) might (perhaps not here, but in other situations) be different from (VectorSpaces?(QQ)(H)).dual(). Would that be a problem?

### comment:2 Changed 9 months ago by saliola

• Description modified (diff)

### comment:3 Changed 9 months ago by saliola

• Description modified (diff)
Note: See TracTickets for help on using tickets.