[GAP Forum] Lie Algebra Cohomology Question

[Willem Adriaan De Graaf]

Dear Alan,

You can create Z^2(G,G) and also B^2(G,G). In your example:

gap> Z2:= Cocycles( subv, 2 );
<vector space of dimension 2 over GF(2)>
gap> B:=Coboundaries( subv, 2);
<vector space over GF(2), with 4 generators>
gap> Dimension(B);
0

So the H^2 has dimension 2.

For a description of these functions, see
https://www.gap-system.org/Manuals/doc/ref/chap64.html#X7FB815F38143939E

All the best,

Willem

On Mon, 17 May 2021 at 19:56, Alan Hylton <agh314 at lehigh.edu> wrote:

> Howdy,
>
> I am interested in using GAP to do some cohomology calculations towards the
> deformation theory of certain Lie algebras.
>
> Suppose I have the subalgebra G of sl(2) with the basis below:
>
> [1,  0] [0, 1]
> [0, -1] [0, 0]
>
> If the first matrix is e1 and the second e2, I have that [e1, e2] = 2e2. If
> the ground field is Q/R/C or of characteristic zero, then the cohomology
> H^*(G, G) = 0. However, in characteristic 2, this Lie algebra is abelian,
> so it has cohomology. Even though this is the easiest possible place to
> start and no fancy software is needed yet, I'd like to be able to compute
> the cohomology in GAP. Then I can ramp up to more interesting algebras.
>
>  G:=SimpleLieAlgebra("A", 1, GF(2));
> <Lie algebra of dimension 3 over GF(2)>
>
> Looking at the Chevalley basis, I see that my subalgebra above is spanned
> by v.1 and v.3:
> gap> ChevalleyBasis(G);
> [ [ v.1 ], [ v.2 ], [ v.3 ] ]
>
> So I get my subalgebra:
>
> gap> b:= BasisVectors( Basis( G ) );;
> gap> sub_alg:=Subalgebra(G, [b[1], b[3]]);
> <Lie algebra over GF(2), with 2 generators>
>
> And now I can compute my 2-cochains:
>
> gap> subv:=AdjointModule(sub_alg);
> <2-dimensional left-module over <Lie algebra of dimension 2 over GF(2)>>
> gap> subC:=CochainSpace(subv, 2);
> <vector space of dimension 2 over GF(2)>
>
> But these are cochains with coefficients in Z2. I wanted to see if I could
> use something like TensorProduct, but I am stuck. I would not be surprised
> if I have to do something custom with the coboundary, but it would be a
> great step forward to create C^2(G, G) and then reduce to Z^2(G, G).
>
> Any help moving forward with these computations would be greatly
> appreciated! I tried looking into TensorProduct and also
> TensorProductOfAlgebraModules, but could not make these work.
>
> Thanks!
> Alan
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