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