[GAP Forum] matrix acting on Lie algebra

[Max Horn]

Dear Russ,

> On 08 Feb 2018, at 19:59, Russ Woodroofe <rsw9 at cornell.edu> wrote:
> 
> 
> 
> 	Dear GAP Forum,
> 	I'm trying to get GAP to act on a finite Lie algebra, without much luck.
> 
> 	I tried the following:
> gap> L:=SimpleLieAlgebra("A", 1, GF(3));
> <Lie algebra of dimension 3 over GF(3)>
> gap> e:=Elements(L)[4];
> v.2
> gap> mat:=AdjointMatrix(Basis(L), e);
> [ [ 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3) ], [ Z(3), 0*Z(3), 0*Z(3) ] ]
> gap> e*mat;
> [ [ 0*v.1, 0*v.1, 0*v.1 ], [ 0*v.1, 0*v.1, (Z(3))*v.2 ], [ (Z(3))*v.2, 0*v.1, 0*v.1 ] ]
> gap> mat*e;
> [ [ 0*v.1, 0*v.1, 0*v.1 ], [ 0*v.1, 0*v.1, (Z(3))*v.2 ], [ (Z(3))*v.2, 0*v.1, 0*v.1 ] ]
> 
> 	Unless I'm mistaken, e*mat should return the same thing as e*e, which is to say 0 (=0*v.1).

I am afraid you are mistaken :-). You asked for a representation matrix of the adjoint action of e on L.

This matrix acts on GF(3)^3, not on L (which is isomorphic as a vector space, but not equal). 

So, the matrix/vector computation corresponding to e*e actually would be this:

# convert to coordinates in GF(3)^3
gap> v:=Coefficients(Basis(L), e);
[ 0*Z(3), Z(3)^0, 0*Z(3) ]

# apply the matrix; per manual, "[t]he adjoint map is the left multiplication by x."
gap> w := mat*v;
[ 0*Z(3), 0*Z(3), 0*Z(3) ]

# convert back to L
gap> LinearCombination(Basis(L), w);
0*v.1
# or shorter:
gap> Basis(L) * w;
0*v.1



>  Perhaps I'm missing something obvious?  Is there some workaround?  I did look through the "AsObject" methods, and didn't see anything that looked helpful.
> 
> 	(I'd ultimately like to find orbits of the action on subspaces, but I think this is the level at which it is breaking down.)

What exactly do you mean by "orbit" here? L is not a group action, after all. Perhaps you mean the action of the corresponding finite group of Lie type (so eg. SL_2(F_3) in this example)?


Cheers,
Max