> < ^ Date: Mon, 27 Aug 2001 13:33:10 +0200 (MET DST)
< ^ From: Philippe Gaillard <pgaillar@maths.univ-rennes1.fr >
< ^ Subject: Re: About HighestWeightModule

Dear GAP-forum, and Willem de Graaf,

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.

To be more precise, my first, and false, attempt was working with
TensorProductOfAlgebraModules( W1, W1 ), with your notations, and this
didn't give me the results I expected. After that, I worked with
HighestWeightModule( L, [1,1] ) and I obtained interesting results.
I noticed after I had the same results with
TensorProductOfAlgebraModules( W1, W2 ) but I wanted to be sure this was
what I was looking for. Thanks to the mail of Mr Draisma, whom I thanked by
private mail, and yours, I now know I can go on to work with what you call U.
Thank you for your help,

Philippe Gaillard

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