#18350: Adams operator
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Reporter: elixyre | Owner:
Type: task | Status: needs_review
Priority: major | Milestone: sage-6.7
Component: categories | Resolution:
Keywords: | Merged in:
Authors: Jean-Baptiste | Reviewers:
Priez | Work issues:
Report Upstream: N/A | Commit:
Branch: | fc8726a50725479d84aa9a07f55e28e1f879880a
u/elixyre/ticket/18350 | Stopgaps:
Dependencies: |
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Comment (by alauve):
I will be merging with ticket #18678 over the next few days. In the
meantime, some comments for people in-the-know to weigh-in on (will repeat
in #18678 when updates appear)...
1. `.adams_operator()` should be moved to `bialgebras.py`, then in
`hopf_algebras.py` one should overwrite the bialgebras version, allowing
for negative integer powers. (E.g., the (-2)nd convolution power of the
identity is none other than the 2nd power of the antipode.)
2. In fact, while adams operators naturally belong in `bialgebras.py`, the
present code---in ticket #18678 and #18350---actually belongs in
`bialgebras_with_basis.py`---as it uses `.module_morphism()` and
`.apply_multilinear_morphism()`---but this would require more rewriting
than I feel qualified to handle.
3. More tickets needed! When poking around for an algebra without basis---
on which to test my code---I noticed that Sage doesn't know that `QQ[x]`
is a module over `QQ` (and hence, one cannot build ``QQ[x].tensor(QQ[x])`.
Crazy.
4. Similarly, even though `B = FreeAlgebra(QQ,a,b)` is robust enough that
`B.tensor(B)` doesn't throw errors, quotients are out-of-bounds again.
Putting `C = B.quotient_ring((a*b-b^2,))`, I get an AttributeError when
asking for `C.tensor(C)`.
5. One could also add the following functionality: given linear morphisms
R,S,T for a bialgebra B, create their convolution product, a new morphism,
via `RST = B.convolution_product(R,S,T)` or perhaps, defined only at the
level of a distinguished basis m for B, `RST =
m.convolution_product(R,S,T)`. However, it seems ticket #15832 will have a
lot of overlap with such code, so I'll hold off on implementing it unless
somebody suggests otherwise.
P.S. Any final votes regarding Amy Pang's recent comment?
--
Ticket URL: <http://trac.sagemath.org/ticket/18350#comment:22>
Sage <http://www.sagemath.org>
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