ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 20 Jan 2018 15:47:45 +0100- Can SAGE calculate with projective (indecomposable) A-modules (A is a finite dimensional Q-algebra)?https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/Hi,
I have the following question:
Given a $\mathbb{Q}$-algebra A by generators (the generators are matrices) and knowing 5 primitive orthogonal idempotents $e_1$,...,$e_5$ (as matrices), which sum up to $1_A$ (the identity matrix), is there a way / procedure in SAGE, that can compute the projective indecomposable modules $P_1=e_1\cdot A$,...,$P_5=e_5\cdot A$ and then test, whether $P_i$ and $P_j$ are isomorphic as $A$-modules for $i\neq j$?
Thank you very much.Tue, 13 Aug 2013 08:21:34 +0200https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/
- Answer by Nicolas M ThiĆ©ry for <p>Hi,
I have the following question:</p>
<p>Given a $\mathbb{Q}$-algebra A by generators (the generators are matrices) and knowing 5 primitive orthogonal idempotents $e_1$,...,$e_5$ (as matrices), which sum up to $1_A$ (the identity matrix), is there a way / procedure in SAGE, that can compute the projective indecomposable modules $P_1=e_1\cdot A$,...,$P_5=e_5\cdot A$ and then test, whether $P_i$ and $P_j$ are isomorphic as $A$-modules for $i\neq j$?</p>
<p>Thank you very much.</p>
https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/?answer=15338#post-id-15338Aladin Virmaux has written some code (ported from Florent Hivert's implementation in MuPAD) that computes the Cartan matrix of a finite dimensional algebra by constructing orthogonal idempotents and building the sandwiches e_i A e_j. The code is meant to go into Sage at some point, but it still needs polishing and has a couple dependencies.
Probably something could be extracted out of it for your need, but I don't expect it to do better than the naive implementation: building the a basis of the projective module e_i A by repeated multiplication on the right by the generators and so on.
If your algebra is the algebra of a monoid or semigroup, let us know for there we have much more under hand :-)
What kind of dimension do you have in mind?Tue, 13 Aug 2013 09:54:59 +0200https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/?answer=15338#post-id-15338
- Comment by Bern for <p>Aladin Virmaux has written some code (ported from Florent Hivert's implementation in MuPAD) that computes the Cartan matrix of a finite dimensional algebra by constructing orthogonal idempotents and building the sandwiches e_i A e_j. The code is meant to go into Sage at some point, but it still needs polishing and has a couple dependencies.</p>
<p>Probably something could be extracted out of it for your need, but I don't expect it to do better than the naive implementation: building the a basis of the projective module e_i A by repeated multiplication on the right by the generators and so on.</p>
<p>If your algebra is the algebra of a monoid or semigroup, let us know for there we have much more under hand :-)</p>
<p>What kind of dimension do you have in mind?</p>
https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/?comment=40716#post-id-40716Thank you very much for your answer. I was thinking of the k-dimension of the algebra. I should have written that in the question. Sorry for my late reply. As a comment, I would like to say that in the meantime I have learned that these things can be done with the aid of the GAP-package QPA. This also works in SAGE via letting GAP be the intermediator.Sat, 20 Jan 2018 15:47:45 +0100https://ask.sagemath.org/question/10424/can-sage-calculate-with-projective-indecomposable-a-modules-a-is-a-finite-dimensional-q-algebra/?comment=40716#post-id-40716