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Compute radical and idempotents of a quotient algebra

asked 2013-07-12 10:50:17 +0100

Bern gravatar image

updated 2020-10-03 15:16:51 +0100

slelievre gravatar image

I tried the following:

R.<x, y> = PolynomialRing(QQ, 2)
I = Ideal(x^2, y^2)
S = R.quotient(I)

I have the following question:

I would like to compute with Sage the Jacobson radical of the algebra S, all primitive orthogonal idempotents and the central idempotents.

Of course, you can compute this by hand, but I am interested in more complicated examples, too (also in matrix algebras), but wanted to start with this simple example.

Since I am relatively new to Sage, I unfortunately do not know how to compute this.

I would be grateful for any help.

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answered 2018-01-20 15:42:39 +0100

Bern gravatar image

updated 2020-10-03 15:19:07 +0100

slelievre gravatar image

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.

It works as follows:

gap> LoadPackage("qpa");;
gap> k:=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);
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);

If we let $k$ be a finite field, then the command


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

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Thanks for reporting that you found a solution. I turned your comment into an answer.

Could you edit your answer to include code to perform these computations?

It would be interesting to see how you do it in GAP, but also how to do it in Sage.

slelievre gravatar imageslelievre ( 2018-01-21 06:33:21 +0100 )edit

Commutativity is not necessary - sorry that I wrote sth. wrong earlier.

Bern gravatar imageBern ( 2020-10-03 14:29:14 +0100 )edit

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Asked: 2013-07-12 10:50:17 +0100

Seen: 952 times

Last updated: Oct 03 '20