[GAP Forum] Lie Algebra Commutator Matrix Question

[Alan Hylton]

Hi Dima,

And thanks for the reply!

Yes, this is a bit weird - I admit it. Following some research I've seen
into certain index computations, I need to essentially create my
multiplication table, and essentially view the vectors as indeterminates.
So if I had a Lie algebra spanned by e1 and e2, and [e1, e2] = 2e2, then
the multiplication table would be a 2x2 matrix with E_12 = 2e2 and 0's
elsewhere. If I view e2 as an indeterminate, then, this matrix has rank 1.

Thanks!

On Sat, May 15, 2021 at 9:01 AM Dima Pasechnik <dima at pasechnik.info> wrote:

> On Fri, May 14, 2021 at 11:17:53PM -0400, Alan Hylton wrote:
> > I had a question about Lie algebras. Following some papers I've read, I'd
> > like to see how I could calculate the rank of the commutator matrix. That
> > is, if I build the multiplication table into a matrix, can GAP get its
> > rank? I can do this in Maple, but it is very time consuming. I'd like to
> > see how this works across platforms.
>
> the multiplication table of an algebra is not really a numeric matrix,
> it's a matrix of linear forms, with variables corresponding to generators.
> (it's (i,j) entry is [g_i,g_j]=\sum_{k=1}^n c_{ij}^k g_k). So it's n x n x
> n tensor,
> that you can variously slice into nxn matrices.
>
> Could you be more specific in explaing  rank of what  you'd like to
> compute?
>
> Dima
>