[GAP Forum] Lie Algebra Cohomology Question

[Alan Hylton]

Oh Geez - I missed the "V" in the definition of cochains: c: L × ⋯ × L → V ,
I thought that the coefficients were in the field.

This is very helpful!

Thanks,
Alan

On Mon, May 17, 2021 at 4:04 PM Willem Adriaan De Graaf <
willem.degraaf at unitn.it> wrote:

> 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
>> _______________________________________________
>> Forum mailing list
>> Forum at gap-system.org
>> https://mail.gap-system.org/mailman/listinfo/forum
>>
>