For semisimple Lie groups it's just a lookup (I define "small" as dim(R)<5 and
put the dimensions of the Clebsch-Gordon series of RxR in parentheses):
A1(1*1=1),A1(2*2=1+3),A1(3*3=1+3+5),A1(4*4=1+3+5+7),SO2(2*2=1+1+2),A2(3*3=3+6),B2(4*4=1+5+10),
and the products SO2#SO2,SO2#A1 and A1#A1. Hope I forgot none.
But Lie groups can be non-this, non-that and non-bugsme. (In the "magic" series
D(2,1,$\alpha$) and OSP(2,1) pop up, but these are super algebras and I have
no idea what "dimension" means for a super algebra since they seem to have
more than one.) Can you amend my list with, say, non-solvable Lie groups?
Especially interesting would be if RxR contains more than one copy of the
1 irrep...if that's possible at all! (SO2 doesn't count since the other 1 is no 1.
Dim 1, but no unity irrep. Or whatever. Don't ask, I don't understand it :-)
1 Answer
These dimension formula only really make sense for reductive Lie groups or algebras. This means that every finite dimenional representation is completely reducible. If you want to do this for Lie groups or algebras which are not reductive then you need to make it clear what it is you are asking for. Reductive includes semisimple and the standard example of a Lie group which is reductive but not semisimple is $GL(n)$.
For super vector spaces the super dimension is given by taking the dimension of the even space and subtracting the dimension of the odd space.
However I believe that the only super algebra which is reductive in the above sense is $Osp(2,1)$.
If V is an irreducible representation then $V\otimes V$ contains at most one copy of the trivial representation. This is an application of Schur's lemma.
