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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?

edited Feb 1 2010 at 7:04

Yemon Choi
8,455●1●19●45

asked Jan 29 2010 at 9:30

Q.Q.J.
1,349●1●10●20

	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 2010 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 2010 at 11:56

Pavel Etingof · Accepted Answer · 2010-02-01 16:24:03Z UTC

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

answered Feb 1 2010 at 16:24

Pavel Etingof
2,846●12●14

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 2010 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 2010 at 12:47

Ah, that is very nice. Thanks very much. – Q.Q.J. Feb 2 2010 at 12:57

Name of upper triangular/lower triangular Lie Algebra decomposition

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