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Sorry for what is probably a stupid question, but our lecturer did not define weights (just told us to find the "standard definition") and I have some questions to do concerning them.

Wikipedia (https://en.wikipedia.org/wiki/Weight_%28representation_theory%29) tells me it's a linear map from the Lie algebra to the field such that the commutator vanishes over the map. Wolfram tells me something to do with Cartan subalgebras (http://mathworld.wolfram.com/LieAlgebraWeight.html). Some other lecture notes seem to define it kind of similar to one of these two.

Which is the definition I should be looking at? This is a 4th year introduction class to Lie Algebras so I'm assuming it would be the 1st definition (as the second looks too complicated for this level of maths).

Thanks

D. P
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A weight on a Lie algebra is a 1-dimensional representation of the Lie algebra; equivalently, it's a linear functional which vanishes on commutators.

But the most common application of this definition to Lie theory is the special case where we take the Lie algebra to be the Cartan subalgebra of a semisimple Lie algebra. When people say things like "the weights of a representation of a semisimple Lie algebra," they're referring to the decomposition of that representation into 1-dimensional representations of the Cartan subalgebra.

Qiaochu Yuan
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    @D. P: no problem. In general I would advise you to ignore Wolfram. There are a lot of mistakes and idiosyncratic definitions on it, and no procedure (that I know of) for correcting them, unlike with Wikipedia. – Qiaochu Yuan Feb 21 '16 at 00:52
  • One additional question, how would we define the dimension of a representation. I've been operating under the assumption it is the dimension of the image of the representation, and only just realised I didn't actually know if that is the definition or not. – D. P Feb 21 '16 at 21:44
  • @D.P.: it has nothing to do with the image; it's the dimension of the underlying vector space. – Qiaochu Yuan Feb 21 '16 at 22:22
  • Okay, on reflection that makes a lot more sense, thanks. – D. P Feb 22 '16 at 15:09