Obtaining centraliser algebras $eAe$ for a given quiver algebra in QPA

Question

Let $A$ be a connected quiver algebra and $e=e_{k_1}+...+e_{k_r}$ a sum of basic primitive orthogonal idemponts. Is there a quick way to obtain the quiver algebra $eAe$ (by quiver and relations) in the GAP-package QPA when one has $A$ and $e$? One can assume that $eAe$ is connected in case this helps.

I know that one can use the command EndOfModuleAsQuiver algebra to calculate $End_A(eA)=eAe$ but this takes too long in practise, so I wonder whether there is a better/quicker method?

Answer 1

As of today, one can do the following in QPA. Here $A$ is an admissible quotient of a path algebra. The second argument in the function

QuiverAlgebraOfeAe( A, e ) 

is an idempotent in the algebra $A$.

gap> gensA := GeneratorsOfAlgebraWithOne(A);
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e] ]
gap> eAe := QuiverAlgebraOfeAe(A,gensA[2]); 
<Rationals[<quiver with 1 vertices and 1 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 1 arrows>]>, 
  (1 generators)>>