[GAP Forum] Lie Algebra Cohomology

[Alan Hylton]

Just as a follow-up on how far I got.

Suppose I have a Lie algebra with a one-dimensional center. Then I can find
that Dimension(Cocycles(lie, 0)) = 1, Dimension(Coboundaries(lie, 0)) = 0,
and then I can take this a step further:

z:=Cocycles(lie, 0);
b:=Coboundaries(lie, 0);

gap> Basis(z/b);
Basis( <vector space of dimension 1 over Rationals>, [ <0-cochain> ] )

gap> List(Basis(z/b));
[ <0-cochain> ]

gap> List(Basis(z/b))[1];
<0-cochain>

But now I have no idea how to work with this 0-cochain. I found a function
named CocycleInfo, but cannot seem to figure it out.

Any ideas?

On Mon, Aug 2, 2021 at 9:51 AM Alan Hylton <agh314 at lehigh.edu> wrote:

> Howdy,
>
> For low-dimensional Lie algebras I find myself asking GAP if they have any
> cohomology using the calls I learned on this forum:
> Dimension(Cocycles(my_lie_algebra, n)) -
> Dimension(Coboundaries(my_lie_algebra, n)). Then to get a representative of
> any non-trivial cohomology class I wind up doing all the calculations by
> hand.
>
> I'd like to move into higher dimensions, but I cannot quite figure out how
> to compute cohomology in GAP. Referencing the page
> https://www.gap-system.org/Manuals/doc/ref/chap64.html, I have not seen
> any mention of computing cocycles mod coboundaries. Can anyone point me in
> the right direction?
>
> I also find that I do not understand some of the data types; is there a
> way to view the internal representation of a cochain?
>
> Many thanks!
> Alan
>