Ampliación trivial en QPA Software

Question

Para una determinada álgebra del carcaj $A$ se puede calcular el álgebra de extensión trivial $T(A)$ (ver Extensión trivial de un álgebra para la definición) de $A$ con el paquete GAP QPA.

Pregunta: ¿Existe una manera fácil de obtener $A$ como $T(A)$ -módulo en QPA? (tenga en cuenta que $A$ es un cociente de $T(A)$ por $D(A)$ y por lo tanto un $T(A)$ -módulo de esta manera)

Answer 1

Se puede hacer lo siguiente en QPA, donde $A$ es algún cociente admisible de un álgebra de caminos. El código siguiente debería funcionar para cualquier cociente admisible de un álgebra de caminos. La última línea de código da el proyectivo indecomponible $A$ -módulos como $T(A)$ -módulos. De hecho, puede introducir cualquier $A$ -Módulo $M$ en el comando

RestrictionViaAlgebraHomomorphism(f,M)

y obtenerlo como $T(A)$ -módulo.

gap> A; 
<Rationals[<quiver with 3 vertices and 5 arrows>]/
<two-sided ideal in <Rationals[<quiver with 3 vertices and 5 arrows>]>, 
  (5 generators)>>
gap> TA := TrivialExtensionOfQuiverAlgebra(A);
<Rationals[<quiver with 3 vertices and 8 arrows>]/
<two-sided ideal in <Rationals[<quiver with 3 vertices and 8 arrows>]>, 
  (26 generators)>>
gap> Q := QuiverOfPathAlgebra(A);
<quiver with 3 vertices and 5 arrows>
gap> QTA := QuiverOfPathAlgebra(TA);
<quiver with 3 vertices and 8 arrows>
gap> gensA := GeneratorsOfAlgebraWithOne(A);
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e] ]
gap> gensTA := GeneratorsOfAlgebraWithOne(TA);
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [(1)*te_a1_1_1], [(1)*te_a1_1_2], [(1)*te_a1_3_3] ]
gap> n := Length(gensTA) - Length(gensA);
3
gap> gensA := ShallowCopy(gensA);
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e] ]
gap> list := List( [1..n], i -> Zero(A));
[ [<zero> of ...], [<zero> of ...], [<zero> of ...] ]
gap> Append(gensA,list);
gap> gensA;
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [<zero> of ...], [<zero> of ...], [<zero> of ...] ]
gap> f := AlgebraHomomorphismByImages(TA,A,gensTA,gensA);
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [(1)*te_a1_1_1], [(1)*te_a1_1_2], [(1)*te_a1_3_3] ] -> 
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [<zero> of ...], [<zero> of ...], [<zero> of ...] ]
gap> f!.generators := gensTA;
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [(1)*te_a1_1_1], [(1)*te_a1_1_2], [(1)*te_a1_3_3] ]
gap> f!.genimages := gensA;
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e], 
  [<zero> of ...], [<zero> of ...], [<zero> of ...] ]
gap> P := IndecProjectiveModules(A);
[ <[ 1, 4, 3 ]>, <[ 0, 2, 2 ]>, <[ 1, 2, 2 ]> ]
gap> test := List( P, p -> RestrictionViaAlgebraHomomorphism(f,p));
[ <[ 1, 4, 3 ]>, <[ 0, 2, 2 ]>, <[ 1, 2, 2 ]> ]