[GAP Forum] Simple Lie algebras of Cartan type

[Willem De Graaf]

Dear prof. Bois,

You asked about Witt algebras:

>
> I am currently experimenting with computations in simple modular Lie 
> algebras of Cartan type, but I am a bit confused with what basis GAP 
> considers. Let me explain more precisely.
>
> Say I wish to consider the restricted Lie algebra W_2, whose elements 
> I view for instance as derivations of the truncated polynomial ring 
> F[x,y]/(x^p,y^p). Now if I give the following code to GAP:
>
> F:=GF(5);
> W:=SimpleLieAlgebra("W",[1,1],F);
> b:=BasisVectors(Basis(L));
>
> to what elements of W_2 do the vectors b[1],b[2],...,b[50] correspond? 
> And what about more general Cartan type Lie algebras?

For the Lie algebra of type W you can do the following:

gap> w:=SimpleLieAlgebraTypeW( [1,1], GF(5) );;

This will return a list containing the Lie algebra and a as second
element a list of 50 entries. Each of those entries is a list describing
the corresponding basis element, for example:

gap> w[2][32];
[ [ 1, 1 ], 2 ]

This means that the 32-nd basis element is x^aD_2, where a is the multi
index a=[1,1].
The Lie algebras of types S and H are constructed as subalgebras
of a Lie algebra of type W; there the correspondence is not so clear.

I hope that this helps you. If you have further questions, then please
ask.

All the best,

Willem de Graaf