[GAP Forum] matrix acting on Lie algebra

[Max Horn]

Dear Russ,

On 09 Feb 2018, at 14:42, Russ Woodroofe <rsw9 at cornell.edu> wrote:
>> 

[...]

>> This matrix acts on GF(3)^3, not on L (which is isomorphic as a vector space, but not equal). 
> 	I see now that in the documentation, it indicates that matrix-vector multiplication is only defined for row vectors.

The only definition of matrix-vector multiplication I know of is for row vectors only. For other kinds of vectors, I'd instead apply linear maps to them.

How do you define the multiplication of a matrix with a non-row vector?

> 
> 	Let me make a couple of further suggestions for the documentation, which might prevent some confusion from people trying to do similar things:
> 	OnSubspacesByCanonicalBasis indicates that it operates on the subspaces of a vector space.  And indeed, a Lie algebra has a canonical basis.  Perhaps a sentence like "This function is currently only implemented for row vector spaces" should be added?  Or to simply insert the word "row" before vector space.
> 	It would also be worthwhile to insert the word "row" before vector in the documentation for OnPoints and OnRight.

But OnPoint and OnRight implement actions on many thing -- including non-row vectors with them. As long as the type of the vector, and object acting on it, fit. So for a non-row-vector, that object should be a linear transformation, which in the case of Lie algebras is built-in:

gap> L:=SimpleLieAlgebra("A", 1, GF(3));;
gap> OnRight(L.1, L.1);
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)?
> 	I noticed that the Lie algebra support for GAP didn't have much support for automorphisms, which seems a shame.

For simple Lie algebras in characteristic zero, the SLA package bei Willem De Graaf provides some functionality here.

But there indeed doesn't seem to be anything for the finite field case (hopefully in case I am missing something, others on the forum will add it :-)


Best wishes,
Max