[GAP Forum] [GAP FORUM]: question concerning the GAP package qpa (find all, ideals with a certain property)

[Øyvind Solberg]

> Message: 3
> Date: Wed, 6 Aug 2014 12:27:42 +0200
> From: Bernhard Boehmler<bernhard.boehmler at googlemail.com>
> To:forum at gap-system.org
> Subject: [GAP Forum] question concerning the GAP package qpa (find all
> 	ideals with a certain property)
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> 	<CAD1scQYNFowTDPfmHUsSUSb50-gLPEehotiN5f_Yzjp4SyJH1A at mail.gmail.com>
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>
> Dear GAP forum,
>
> I have the following question concerning the GAP package qpa.
>
> Let k be a fixed finite field and let Q be a fixed quiver. Let kQ denote
> the associated path algebra.
>
> Since k is finite, there are only finitely many admissible ideals I of kQ
> with the property I^u=0 for some fixed natural number u.
>
> I would like to know, if there is a way to tell qpa to find all such
> ideals, and, if so, how to do this.
>
> With other words, my input is: [k,Q,u] and the output should be a list
> containing all quiver algebras kQ/I as entries.
>
> Thanks for the help!
Dear Bernhard and the GAP Forum,
I am a little bit confused concerning "... admissible ideals  I  of
kQ  with the property  I^u=0 for some fixed natural number  u".
Are you restricting to quivers without oriented cycles? Or are you
thinking of quotients  A=kQ/I, where  I  is an admissible ideal such
that the radical of A, rad A, satisfies  (rad A)^u = 0?  I assume that
you are considering the latter case.

Let J be the ideal generated by the arrows in the path algebra kQ.
Consider the ring R = kQ/J^u.  This is a finite dimensional algebra.
You are asking for all twosided ideals of R which is contained in
J^2/J^u, or equivalently, all sub-bimodules of J^2/J^u, that is, all
submodules of J^2/J^u as a module over the enveloping algebra R^e of R
(that is, R\otimes_k R^\op, which can be constructed in QPA).  So
actually you are asking: given a module M over a finite dimensional
quotient (in your case R^e) of a path algebra, find all the submodules
of M.

This, I think, can very soon become a CPU and a memory intensive
undertaking, given the "correct" examples.  However, in this situation
one knows the simple modules, so one could inductively build all
submodules from either constructing all maximal submodules of  M
or all simple submodules of  M, where the situation which creates
"problems" would be when the top or the socle of  M  is not a basic
module.  It should however be possible to write such an algorithm
within QPA, but I don't know anybody having done it yet.

So, to my knowledge there is no way to find all such ideals.

I hope that these comments are helpful.

Best wishes, Oeyvind Solberg.