# 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.

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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.

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?

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Thank 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.

( 2018-01-20 08:47:45 -0500 )edit