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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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