[GAP Forum] Computing with projective A-modules of a finite dimensional k-algebra A

[Bernhard Boehmler]

Dear GAP Forum,

I have the following problem(s):

I made some computations in GAP and I now have an "algebra-with-one" R as a
subalgebra of a full matrix algebra.

Moreover, I have five orthogonal primitive idempotents e_1,...,e_5, which
sum up to the identity element of R.

Now, I would like to let GAP calculate some things concerning the
projective right R-modules P_1 and P_2,..., where P_1 = e_1 * R (and
P_2=e_2 * R and so on).

I would like to test, whether P_i and P_j are isomorphic as R-modules for i
unequal to j. I also would like to compute the algebras e_i R e_j for all i
and j.

I can access the generators (as matrices) of the algebra R and I know
e_1,...,e_5 =... (as matrices).


Unfortunately, I wasn't able to find out, how to define P_1 in GAP as a row
module (I would like GAP to look at P_1 as an R-module (R as an algebra) -
is trying to insert P_1 as a row module the right way of procedure?).  I
also would like to compute the radical of P_1 and factor it out.

I, therefore, would we very thankful, if anybody could give me a hint,
given only the "algebra with one" R as a GAP object, how to implement P_1,
P_2, P_1/rad(P_1) and P_2/rad(P_2), the regular module R_R, and so on, as
modules and do calculations with them.

Thank you very much.

Yours sincerely,

Bernhard Boehmler