[GAP Forum] Lie Algebra Cohomology Question

[Alan Hylton]

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