[GAP Forum] semsimple algebras and subalgebras

[R.N. Tsai]

Dear forum,

I am using the sla package to do calcualtions with semisimple lie algebras
and their subalgebras. The calculations involve decomposing an irrep of the
main algebra into irreps of one of its subalgebras. The "Branching" functions
works great at providing this data. I'd like to extract some more information
on this decomposition for the adjoint irrep of the main algebra. I'd like to identify
the subalgebra irreps with concrete subspaces of the main algebra. Is there
a (hopefully simple) way to extract this information?

Here's an example for main algebra "G2" and its "A2" subalgebra :

LoadPackage("sla");
# get all semisimple subalgebras of "G2"

LS:=LieAlgebraAndSubalgebras("G2");
# alg is "G2"
alg:=LS.liealg;
# subalgebra number 5 is "A2"; 
sub:=LS.subalgs[5];
# rep with highest weight [0,1] is adjoint rep of "G2"
rep:=[0,1];
# get branching of G2 rep to A2 reps

bra:=Branching(alg,sub,rep);
# display results

for k in [1..Length(bra[1])] do 
 Print("subspcace ",k," ");
 Print("weight=",bra[1][k]," ");
 Print("multiplicity = ",bra[2][k]," ");
 Print("dimensions = ",Dimension(HighestWeightModule(sub,bra[1][k]))," ");
 Print("\n");
od;

in shorthand this says that G2 decomposes as 14 = 8 + 3 + 3' under A2.
Here 14=G2, 8=A2, 3 and 3' are 3-dimensional subspaces of G2 invraiant under A2.
I'd like for example to multiply elements of 3 with those of 3'...

As a side question, for G2 I happen to know that heighst weight rep [0,1] is the
adjoint rep; is there a general way to get the weight of the adjoint rep for an
arbitrary semisimple algebra?

Thanks,

R.N.