Message: 3 Date: Wed, 6 Aug 2014 12:27:42 +0200 From: Bernhard Boehmler<bernhard.boehm...@googlemail.com> To:fo...@gap-system.org Subject: [GAP Forum] question concerning the GAP package qpa (find all ideals with a certain property) Message-ID: <cad1scqynfowtdpfmhussusb50-glpeehotin5f_yzjp4syj...@mail.gmail.com> Content-Type: text/plain; charset=UTF-8Dear 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. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
