#14161: add a _test_antipode method for Hopf algebras
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Reporter: saliola | Owner: sage-combinat
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-5.8
Component: combinatorics | Resolution:
Keywords: hopf algebras | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Franco Saliola | Merged in:
Dependencies: #14162, #14163 | Stopgaps:
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Description changed by saliola:
Old description:
> Desired doctests:
> {{{
> sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon()
> sage: R._test_antipode()
>
> ::
>
> sage: s = SymmetricFunctions(QQ).schur()
> sage: s._test_antipode()
> }}}
> This would have caught the bug in #14160.
>
> Apply : trac_14161-fs.2.patch
New description:
Add a method that tests the antipode of a Hopf algebra. We know that the
antipode satisfies certain conditions, so check these conditions for the
antipode method using the elements output by {{{some_elements}}}.
{{{
Test the antipode.
An *antipode* `S` of a Hopf algebra is a linear endomorphism of the
Hopf algebra that satisfies the following conditions (see
:wikipedia:`HopfAlgebra`).
- If `\mu` and `\Delta` denote the product and coproduct of the
Hopf algebra, respectively, then `S` satisfies
.. MATH::
\mu \circ (S \tensor 1) \circ \Delta = unit \circ counit
\mu \circ (1 \tensor S) \circ \Delta = unit \circ counit
- `S` is an *anti*-homomorphism
These properties are tested on :meth:`some_elements`.
TESTS::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon()
sage: R._test_antipode()
::
sage: s = SymmetricFunctions(QQ).schur()
sage: s._test_antipode()
}}}
This would have caught the bug in #14160.
Apply : trac_14161-fs.2.patch
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14161#comment:17>
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