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
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