Dear Russ, > On 08 Feb 2018, at 19:59, Russ Woodroofe <[email protected]> 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 _______________________________________________ Forum mailing list [email protected] https://mail.gap-system.org/mailman/listinfo/forum
