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In the meantime I have learned that these things can be done with the aid of the GAP-package QPA. This also works in SAGE via letting GAP be the intermediator.

In the meantime I have learned that these things can be done for admissible quotients of path algebras with the aid of the GAP-package QPA. This also works in SAGE via letting GAP be the intermediator.

gap> LoadPackage("qpa");;
gap> k:=Rationals;
Rationals
gap> Q:=Quiver(1,[[1,1,"a"],[1,1,"b"]]);
<quiver with 1 vertices and 2 arrows>
gap> kQ:=PathAlgebra(k,Q);
<Rationals[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables(kQ);
\#I  Assigned the global variables [ v1, a, b ]
gap> rels:=[a^2,b^2, a*b-b*a];
[ (1)*a^2, (1)*b^2, (1)*a*b+(-1)*b*a ]
gap> A:=kQ/rels;
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>, 
  (3 generators)>>
gap> IsAdmissibleQuotientOfPathAlgebra(A);
true
gap> RAD:=RadicalOfAlgebra(A);
<algebra of dimension 3 over Rationals>
gap> Display(RAD);
Algebra( Rationals, [ [(1)*a], [(1)*b], [(1)*a*b] ] )
gap> Dimension(RAD);
3
gap>

In the meantime I have learned that these things can be done for admissible quotients of path algebras with the aid of the GAP-package QPA. This also works in SAGE via letting GAP be the intermediator.

So, if we make the additional restriction that our algebra is commutative, it works as follows:

gap> LoadPackage("qpa");;
gap> k:=Rationals;
Rationals
gap> Q:=Quiver(1,[[1,1,"a"],[1,1,"b"]]);
<quiver with 1 vertices and 2 arrows>
gap> kQ:=PathAlgebra(k,Q);
<Rationals[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables(kQ);
\#I  Assigned the global variables [ v1, a, b ]
gap> rels:=[a^2,b^2, a*b-b*a];
[ (1)*a^2, (1)*b^2, (1)*a*b+(-1)*b*a ]
gap> A:=kQ/rels;
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>, 
  (3 generators)>>
gap> IsAdmissibleQuotientOfPathAlgebra(A);
true
gap> RAD:=RadicalOfAlgebra(A);
<algebra of dimension 3 over Rationals>
gap> Display(RAD);
Algebra( Rationals, [ [(1)*a], [(1)*b], [(1)*a*b] ] )
gap> Dimension(RAD);
3
gap>

If we let $k$ be a finite field, then the command IdempotentsForDecomposition(A); works, but I don't have a computational solution for the case $k=\mathbb{Q}$. I have not yet a solution for the non-commutative case, sorry.case.

In the meantime I have learned that these things can be done for admissible quotients of path algebras with the aid of the GAP-package QPA. This also works in SAGE via letting GAP be the intermediator.

So, if we make the additional restriction that our algebra is commutative, it It works as follows:

gap> LoadPackage("qpa");;
gap> k:=Rationals;
Rationals
gap> Q:=Quiver(1,[[1,1,"a"],[1,1,"b"]]);
<quiver with 1 vertices and 2 arrows>
gap> kQ:=PathAlgebra(k,Q);
<Rationals[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables(kQ);
\#I  Assigned the global variables [ v1, a, b ]
gap> rels:=[a^2,b^2, a*b-b*a];
[ (1)*a^2, (1)*b^2, (1)*a*b+(-1)*b*a ]
gap> A:=kQ/rels;
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>, 
  (3 generators)>>
gap> IsAdmissibleQuotientOfPathAlgebra(A);
true
gap> RAD:=RadicalOfAlgebra(A);
<algebra of dimension 3 over Rationals>
gap> Display(RAD);
Algebra( Rationals, [ [(1)*a], [(1)*b], [(1)*a*b] ] )
gap> Dimension(RAD);
3
gap>

If we let $k$ be a finite field, then the command IdempotentsForDecomposition(A); works, but I don't have a computational solution for the case $k=\mathbb{Q}$. I have not yet a solution for the non-commutative case.

In the meantime I have learned that these things can be done for admissible quotients of path algebras with the aid of the GAP-package GAP package QPA. This also works in SAGE Sage via letting GAP be the intermediator.

It works as follows:

gap> LoadPackage("qpa");;
gap> k:=Rationals;
Rationals
gap> Q:=Quiver(1,[[1,1,"a"],[1,1,"b"]]);
<quiver with 1 vertices and 2 arrows>
gap> kQ:=PathAlgebra(k,Q);
<Rationals[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables(kQ);
\#I  Assigned the global variables [ v1, a, b ]
gap> rels:=[a^2,b^2, a*b-b*a];
[ (1)*a^2, (1)*b^2, (1)*a*b+(-1)*b*a ]
gap> A:=kQ/rels;
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>, 
  (3 generators)>>
gap> IsAdmissibleQuotientOfPathAlgebra(A);
true
gap> RAD:=RadicalOfAlgebra(A);
<algebra of dimension 3 over Rationals>
gap> Display(RAD);
Algebra( Rationals, [ [(1)*a], [(1)*b], [(1)*a*b] ] )
gap> Dimension(RAD);
3
gap>

If we let $k$ be a finite field, then the command IdempotentsForDecomposition(A); command

IdempotentsForDecomposition(A);

works, but I don't have a computational solution for the case $k=\mathbb{Q}$. $k=\mathbb{Q}$. I have do not yet have a solution for the non-commutative case.