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Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow!

"There is a bijection between the complex representations of a real Lie algebra and the complex representations of its complexification."

I think this is true (and very useful) because the complex representation of the complexification has the extra requirement of being complex linear, which fixes any extra freedom you'd get from complexifying.

Morally though, this seems weird - particularly as a complex Lie algebra may have non-isomorphic real forms. Does anyone have any comments on this which might make it seem more intuitive?

Many thanks in advance.

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Well, I have an obvious comment that since you are only taking complex representations they don't see the difference in the real forms (which after all comes precisely from not being able to use complex numbers to change the signature of the Killing form). Not sure if you want something more than this. – Marek Sep 25 '13 at 16:51

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