> < ^ Date: Fri, 24 Aug 2001 14:16:11 +0200 (MET DST)
> < ^ From: Willem de Graaf <degraaf@math.uu.nl >
> < ^ Subject: Re: About HighestWeightModule

Dear Philippe Gaillard,

Thank you for your message. You wrote:

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.

I do not entirely understand your question. However, here is an example:

gap> K:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> L:= DirectSumOfAlgebras( K, K );;
gap> W1:= HighestWeightModule( L, [1,0] );
<2-dimensional left-module over <Lie algebra of dimension 6 over Rationals>>
gap> W2:= HighestWeightModule( L, [0,1] );
<2-dimensional left-module over <Lie algebra of dimension 6 over Rationals>>
gap> T:= TensorProductOfAlgebraModules( W1, W2 );
<4-dimensional left-module over <Lie algebra of dimension 6 over Rationals>>
gap> U:= HighestWeightModule( L, [1,1] );
<4-dimensional left-module over <Lie algebra of dimension 6 over Rationals>>

Here the L-modules T, U are isomorphic. But T has been constructed as
tensor product of W1, W2 and U directly as a highest weight module.

The function HighestWeightModule constructs an "abstract" module, i.e.,
a vector space together with the action of the Lie algebra. If you are
interested, I can send you some details about the algorithm.

I hope this helps you; if you have further questions, please ask.

Best wishes,

Willem de Graaf

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