[GAP Forum] Lie algebra simple root ordering

[Willem Adriaan De Graaf]

Dear Alan,

> When creating a simple Lie algebra with SimpleLieAlgebra, and retrieving
> the simple roots with SimpleSystem, are they given in the Bourbaki
ordering?

Yes, with one exception: F4.

> I want to know, algorithmically, which one is the exceptional one.

Do you mean that you need a function for deciding whether a given simple
Lie algebra is of exceptional type?
One way to do this is to use the function CartanType from the SLA package:

gap> L:= SimpleLieAlgebra("F",4,Rationals);;
gap> CartanType( CartanMatrix( RootSystem(L) ) );
rec( enumeration := [ [ 2, 4, 3, 1 ] ], types := [ [ "F", 4 ] ] )

Best wishes,

Willem



On Mon, 31 May 2021 at 05:12, Alan Hylton <agh314 at lehigh.edu> wrote:

> Howdy,
>
> When creating a simple Lie algebra with SimpleLieAlgebra, and retrieving
> the simple roots with SimpleSystem, are they given in the Bourbaki
> ordering?
>
> I want to know, algorithmically, which one is the exceptional one.
>
> Thanks,
> Alan
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