Deleting relations in GAP/QPA

Question

Given a quiver algebra $A$ (example A:=NakayamaAlgebra([4,4,3,2,2,1],GF(3)) )in QPA, one can get the relations of A as follows:

rel:=RelatorsOfFpAlgebra(A);

In the example the output is [ (Z(3)^0)*a1*a2*a3*a4, (Z(3)^0)*a4*a5 ].

Question: How can I delete all relations in RelatorsOfFpAlgebra(A) that contain paths of length at least 3 automatically in a smart way via GAP/QPA?

In the example I would delete (Z(3)^0)*a1*a2*a3*a4 to obtain the desired result [(Z(3)^0)*a4*a5 ].

So anytime arrows get multiplied together more than 1 time I want the result to be deleted from the relations.

One idea might be to use the Filter function of GAP but I do not know how to filter since QPA does not regocnize the relations as paths since they are multiplied with field elements in the beginning.

I hope there is a smart solution for this.

Answer 1

Here is one way of doing this in QPA:

gap> A:=NakayamaAlgebra([4,4,3,2,2,1],GF(3));                
<GF(3)[<quiver with 6 vertices and 5 arrows>]/
<two-sided ideal in <GF(3)[<quiver with 6 vertices and 5 arrows>]>, (2 generators)>>
gap> rels := RelationsOfAlgebra(A);                          
[ (Z(3)^0)*a1*a2*a3*a4, (Z(3)^0)*a4*a5 ]
gap> newrels := Filtered( rels, r -> LengthOfPath(TipMonomial(r)) < 3 );
[ (Z(3)^0)*a4*a5 ]
gap> kQ := OriginalPathAlgebra(A);                                      
<GF(3)[<quiver with 6 vertices and 5 arrows>]>
gap> B := kQ/newrels;             
<GF(3)[<quiver with 6 vertices and 5 arrows>]/
<two-sided ideal in <GF(3)[<quiver with 6 vertices and 5 arrows>]>, (1 generators)>>

Since QPA is using length left lexicographic ordering of the paths, the filter command picks out the relations where all paths in the relations have length at most 2. The algebra $B$ is the new algebra.