[GAP Forum] Lie algebra simple root ordering

[Alan Hylton]

Hi Willem,

Thank you always for your help. I am sorry I was unclear.

Right now, my goal is to see if I have everything I need to generate the
Dynkin diagrams for these algebras, and know which simple root corresponds
to which dot. It seems like it is almost surely unnecessary to calculate
these out manually.

Given your answer, I played with CartanType on other types and the
enumeration was always in ascending order. Does this mean that in F4, the
fourth simple root is really the first, and so forth? Does simple root 2
"point" to 3 in the Dynkin diagram?

Best,
Alan


On Tue, Jun 1, 2021 at 2:09 AM Willem Adriaan De Graaf <
willem.degraaf at unitn.it> wrote:

> 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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>