Monoid of ideals in GAP

Question

Given a finite dimensional algebra $A$ and a set of twosided ideals $\{I_1, I_2,...,I_n \}$. Let $G$ be the monoid (or sometimes group) generated by those ideals. Is there a way to obtain this monoid via GAP (or a package like qpa) and check what monoid it is (and wheter it is finite)?

Is there a quick way to get the maximal ideals of the algebra? (at least when its a quiver algebra, this should be easy, but I found no command for that).

what's the operation? – Olexandr Konovalov Jul 14 '17 at 13:36 — Olexandr Konovalov, Jul 14 '17 at 13:36
simply multiplication of twosided ideals. – Mare Jul 14 '17 at 13:38 — Mare, Jul 14 '17 at 13:38

score 2 · Answer 1 · answered Oct 30 '18 at 11:01

In QPA one can do the following:

gap> A := NakayamaAlgebra(Rationals, [3,2]);
<Rationals[<quiver with 2 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 2 vertices and 2 arrows>]>, (1 generators)>>
gap> gens := GeneratorsOfAlgebra(A){[2..5]};
[ [(1)*v1], [(1)*v2], [(1)*a1], [(1)*a2] ]
gap> I1 := Ideal(A, gens{[2..4]});          
<two-sided ideal in <Rationals[<quiver with 2 vertices and 2 arrows>]/
    <two-sided ideal in <Rationals[<quiver with 2 vertices and 2 arrows>]>, (1 generators)>>,
  (3 generators)>
gap> I2 := Ideal(A, gens{[1,3,4]});         
<two-sided ideal in <Rationals[<quiver with 2 vertices and 2 arrows>]/
    <two-sided ideal in <Rationals[<quiver with 2 vertices and 2 arrows>]>, (1 generators)>>,
  (3 generators)>
gap> ProductSpace(I1,I1);                   
<algebra of dimension 4 over Rationals>
gap> ProductSpace(I1,I1) = I1;
true
gap> ProductSpace(I1,I2);     
<vector space of dimension 2 over Rationals>
gap> ProductSpace(I2,ProductSpace(I1,I2));
<vector space of dimension 1 over Rationals>

All the maximal ideals are generated by all the vertices except one and all the arrows. So two vertices give two maximal ideals I1 and I2. The product of two ideals can be computed, for example, the product of I1 and I2 is ProductSpace(I1,I2). Using the above commands one can produce code to generate more ideals.

I hope that this is helpful.

Best regards, the QPA-team.

Monoid of ideals in GAP

1 Answers1