# [GAP Forum] RootSystem returning fail?

degraaf at science.unitn.it degraaf at science.unitn.it
Wed Nov 18 15:26:46 GMT 2009

```Dear Dima,

Here the problem is that your Lie algebra is not semisimple;
you would need to take its derived subalgebra:

D:= LieDerivedSubalgebra(ll);

But then it returns fail as well, as the Cartan subalgebra is
not split over the rationals (and as far as I am aware GAP does
not have methods to factor polynomials over extensions):

h:= Basis( CartanSubalgebra(D) )[1];;
ad:= AdjointMatrix( Basis(ll), h );;
gap> MinimalPolynomial(Rationals,ad);
x^3-7*x

Best wishes,

Willem

Dear all,
is the following a bug, or a feature?
gap> aa;
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
[ [ 0, 2, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 1 ] ],
[ [ -2, 0, -4, -4 ], [ 0, 0, 0, 4 ], [ 1, 1, 0, 2 ], [ 0, -1, 0, -2 ] ],
[ [ 0, 0, 0, 8 ], [ -2, 0, -4, 0 ], [ 0, -2, 0, -4 ], [ 1, 0, 0, 0 ] ] ]
gap> ll:=LieAlgebra(Rationals,aa);
<Lie algebra over Rationals, with 4 generators>
gap> RootSystem(ll);
fail
gap>

(In fact, I know that this algebra is isomorphic to gl(2,C), and I would like
to find its representation in M_2(C))

Thanks,
Dmitrii

```

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