Calculating the basic algebra over a finite field in GAP

Question

Assume $A$ is a (nonsemisimple) finite dimensional algebra over a finite field $K$ (for example a group algebra). I want to calculate the basic algebra $B$ of $A$ as a matrix algebra, constructed as follows:

First possibility: Let $e=e_1+e_2+...+e_n$ be the sum of nonisomorphic primitive orthogonal idempotents such that every indecomposable projective modules is isomorphic to $e_iA$ for some $i$. Then $B=eAe$.

Second possibility: Let $P_i, i=1,..,n$ be a list of all nonisomorphic indecomposable projective modules, then $B=End_A(P_1 \oplus P_2 ... \oplus P_n)$.

Is there a way GAP can do this?

Answer 1

To the best of my knowledge there is no turnkey implementation for GAP that computes the basic algebra. As far as algorithmic approaches go, the meataxe-methods are probably the way to g. In particular you might want to look at the commands MTX.Indecomposition, MTX.HomogeneousComponents and MTX.BasisModuleEndomorphisms in GAP. (However the current implementation for endomorphisms is much weaker than that for automorphisms.) Also look at the C-meataxe and the endomorphism computations at: http://math.arizona.edu/~klux/software.php