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What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?

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What do you mean by upper triangular and lower triangular? Do you have a specific matrix realization of your (semisimple?) Lie algebra? –  Steven Sam Jan 29 '10 at 10:42
    
Good point, yes. Consider the fundamental representation of the semisimple Lie algebra and then conjugate by the general Linear group to obtain the form mentioned in the question for the representation matrices. –  Q.Q.J. Jan 29 '10 at 11:56

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up vote 4 down vote accepted

I think people use the term "triangular decomposition" or sometimes "polarization"

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Makes sense doesn't it? Is there a nice way to prove that such a transformation can always be done without having to give explicit conjugations case by case for A,B,C,D type? –  Q.Q.J. Feb 2 '10 at 8:27
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If you have any finite dimensional irreducible representation of your Lie algebra, then by Engel's theorem there is always an invariant vector v for the positive root generators, which is an eigenvector of the Cartan subalgebra. Applying the negative root generators f_i to this vector, one can get a basis of the representation in which positive root elements act as strictly upper triangular matrices, and negative root generators as strictly lower triangular matrices. –  Pavel Etingof Feb 2 '10 at 12:47
    
Ah, that is very nice. Thanks very much. –  Q.Q.J. Feb 2 '10 at 12:57

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