Dear forum, and Philippe,

yes, if you enter:

sl2:=SimpleLieAlgebra("A",1,Rationals);
so4:=DirectSumOfAlgebras(sl2,sl2);
M:=HighestWeightModule(so2,[1,1]);

you get the 4-dimensional module over so(4,C) (i.e., the standard
one).

In general: if L1 and L2 are (semi-)simple Lie algebras, and V1 (resp. V2) is
an irreducible finite-dimensional L1- (resp. L2-) module of highest
weight w1 (resp w2), then V1 tensor V2 is an irreducible
finite-dimensional (L1 + L2)-module, and its highest weight is the
concatenation of w1 and w2. Every irreducible (L1+L2)-modules is
obtained in this way.

Best wishes,

Jan

answering:

Dear gap-forum,

Maybe it will seem absolutely obvious to you but I have a question about
the use of HighestWeightModule. If I call V the sl(2,C)-module with highest
weight 1 and if I see so(4,C) as T:=sl(2,C)+sl(2,C), I'm interested in
considering VxV or VxV* as a so(4,C)-module.
I former used HighestWeightModule by writing HighestWeightModule(T,[1,0])
in place of V but I noticed further some things which made me really
doubtfull about my use of HighestWeightModule. I tried after to use
HighestWeightModule(T,[1,1]) in place of my tensor product, but I'd like
to know if it's a good way, because I obtained more interesting results
for my goal (I spoke about it in a previous mail) by this way. The help
didn't give me an easy answer and I didn't understand enough how
HighestWeightModule build the module to know if I'm right.
Hoping it may interest some of you,
Best regards,

Philippe Gaillard