[GAP Forum] Re: representations of Lie algebras

[degraaf at science.unitn.it]

Dear R. N. Tsai,

You wrote about a problem with a factor 2:

>  B1:=M(A1*A2-A2*A1);
>  B2:=M(A1)*M(A2)-M(A2)*M(A1);
>  Print(" check ",B1=2*B2,"\n");   #   <---  extra factor of 2

In a structure constants Lie algebra the multiplication is
simply the Lie bracket, i.e., A1*A2 = [ A1, A2 ]. So
A1*A2-A2*A1 = [A1,A2]-[A2,A1] = 2*[A1,A2]. For example:

gap> L:= SimpleLieAlgebra( "A", 2, Rationals );
<Lie algebra of dimension 8 over Rationals>
gap> g:= Basis(L);
CanonicalBasis( <Lie algebra of dimension 8 over Rationals> )
gap> g[1]*g[2];
(-1)*v.3
gap> g[2]*g[1];
v.3

That is where the factor 2 comes from.

All the best,

Willem de Graaf