#16659: Decomposition of finite dimensional associative algebras
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Reporter: virmaux | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.4
Component: algebra | Resolution:
Keywords: representation | Merged in:
theory, days64, sd67 | Reviewers: Franco Saliola
Authors: Aladin Virmaux | Work issues: merge in develop.
Report Upstream: N/A | Commit:
Branch: u/saliola/t/16659 | 7362f035239f8fdbc992888478a9f4b7d102193c
Dependencies: #11111 | Stopgaps:
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Changes (by saliola):
* keywords: representation theory => representation theory, days64, sd67
Comment:
Here are a few more comments that came up at Sage Days 67.
`FiniteDimensionalAlgebrasWithBasis.ParentMethods.central_orthogonal_idempotents`:
- shouldn't this be `orthogonal_idempotents` since they do not belong
to the center? Or maybe I don't understand the definition: what is meant
by "central"?
- there should be a method to also compute the central orthogonal
idempotents?
`SemisimpleAlgebras.Commutative.ParentMethods._orthogonal_idempotents`:
- these are not idempotents (they are quasi-idempotents); the
documentation string needs to be updated
Documentation string for Pierce decomposition:
- needs the definition of a Pierce decomposition
- `e_i A e_j` are *not* submodules or subalgebras in general, but
subspaces
Other questions:
- is there a method somewhere to test that a family of idempotents is
a minimal family of orthogonal idempotents? if so, then the tests in
`orthogonal_idempotents` need to be updated to use it
- is there a method to test whether an idempotent is primitive?
At this point, I'll have little time to work on these over the next couple
of weeks, but I will answer questions, if you have any.
--
Ticket URL: <http://trac.sagemath.org/ticket/16659#comment:55>
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