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.degr...@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 <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
