#16659: Decomposition of finite dimensional associative algebras
-------------------------------------+-------------------------------------
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/virmaux/t/16659 | abcb076cf7ae487d9a34b690b415df40886fd93f
Dependencies: #11111 | Stopgaps:
-------------------------------------+-------------------------------------
Comment (by virmaux):
Hi Franco, thank you for the remarks.
I made some changes after a discussion with Nicolas and Florent:
* Semisimple algebras:
* In the semisimple quotient, the idempotents that we construct are
central as they live in the center of the algebra. Unfortunately, lifting
those idempotents does not preserve this last property. So I left the name
`central_orthogonal_idempotents` for semisimple algebras.
* I updated the documentation for `_orthogonal_decomposition`, but maybe
we should change the name?
* Finite dimensional algebras with basis
* `orthogonal_idempotents` -> `orthogonal_idempotents_central_mod_rad`
because they are obtained by lifting the central_idempotents of the
semisimple quotient.
* As well, we now have a method `lifting_idempotents` instead of a
subfunction in the previous method.
* Peirce (with the correct spelling) is now splitted in two:
`peirce_summand` return a sandwich subspace //e_i A e_j// and
peirce_decomposition return the list of all sandwiches with the orthogonal
idempotents lifted from the central ones of the semisimple quotient.
Other questions:
The idempotents we get for non semisimple algebra are not primitive. The
only primitive idempotents we construct are for commutative semisimple
algebras. To test if an idempotent //e// of //A// is primitive, we may
construct the semisimple quotient of //eA// and test wether it is a simple
module or not. In our finite dimensional case, is it enough to check that
all elements of the basis generate the whole module?
--
Ticket URL: <http://trac.sagemath.org/ticket/16659#comment:58>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
