[GAP Forum] Lie algebra simple root ordering

[Willem Adriaan De Graaf]

Dear Alan,

With the ordering used by GAP the Dynkin diagram of the simple Lie algebra
of type F4 is

1------3==<==4------2

All the best,

Willem

On Tue, 1 Jun 2021 at 08:53, Alan Hylton <agh314 at lehigh.edu> wrote:

> 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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>>