Arrow is in a finite linear combinations of paths of a Quiver

Question

I am learning about GAP and QPA. I would like to know when a arrow is in a finite linear combinations of paths of a Quiver. For example:

Q := Quiver(4,[[1,2],[2,4],[1,3],[3,4]]);
KQ := PathAlgebra(Rationals, Q);
AssignGeneratorVariables(KQ);
elem := KQ.a1*KQ.a2-KQ.a3*KQ.a4;

How I know that the arrow a1 is in elem?

If this was about the arrow a3, it's ok:

LeadingMonomial(a3) in WalkOfPath(LeadingMonomial(elem));
> true

Thank you very much!

Answer 1

One can do the following in QPA:

gap> elem := KQ.a1*KQ.a2-KQ.a3*KQ.a4;                                                             
(1)*a1*a2+(-1)*a3*a4
gap> test := CoefficientsAndMagmaElements( elem );                                                
[ a1*a2, 1, a3*a4, -1 ]
gap> n := Length( test ) / 2;
2
gap> not ForAll( [ 1..n ], i -> not ( LeadingMonomial( a1 ) in WalkOfPath( test[ 2 * i - 1 ] ) )
true

We hope these comments are helpful.

The QPA-team.