Path in a quiver to a path in path algebra in GAP

Question

Is it possible to obtain a path in the path algebra without explicitly specifying the arrow?

More specifically, I have a string with letters which are arrows in the quiver. I need to check whether in the path algebra of the quiver, this string is a path. I was able to convert this string into a path in quiver. But is there a way to convert the path in a quiver to a path in its corresponding path algebra?

I am interested in expressing path algebra paths implicitly in terms of generators of quiver. ArrowsOfQuiver(Q) gives the array of arrows of quiver Q. So instead of kQ.a as a path of my algebra, I want to write this something as kQ.ArrowsOfQuiver(Q)[1]. But this syntax isn't working

Answer 1

I am not sure that I fully understand your question, but here is hopefully something that might be useful:

gap> Q := Quiver( 1, [[1,1,"a"],[1,1,"b"]] );
<quiver with 1 vertices and 2 arrows>
gap> kQ := PathAlgebra( Rationals, Q );
<Rationals[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables( kQ );
#I  Assigned the global variables [ v1, a, b ]
gap> a;
(1)*a
gap> b;
(1)*b
gap> p := a*b*a;
(1)*a*b*a
gap> test := CoefficientsAndMagmaElements(p);
[ a*b*a, 1 ]
gap> walk := WalkOfPath( test[1] );                           
[ a, b, a ]
gap> q := test[2]*One(kQ)*Product(walk);                      
(1)*a*b*a
gap> p = q;
true
gap> walk[1]*One(kQ);
(1)*a
gap> walk[2]*One(kQ);
(1)*b
gap> walk[3]*One(kQ);
(1)*a