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I am not sure of the difference between weights and roots. Am I correct in thinking that the weights are the eigenvalues of the action of the maximal torus on a given representation, and the roots are the weights that occur when the maximal torus acts on the Lie algebra itself under the adjoint representation?

If you have time, what would this look like for $\mathfrak{sl}_3\Bbb C$?

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Yes, that's right! – user29743 Apr 27 '13 at 17:13
up vote 6 down vote accepted

Certain representations of a (say, reductive) Lie algebra will decompose into eigenspaces for the action of a Cartan subalgebra $\mathfrak{h}$ (the Lie algebra analogue of a maximal torus). Thus, $\mathfrak{h}$ acts on these subspaces, called "weight spaces" via a linear form $\mathfrak{h} \to \mathbb{C}$ (if you're working over the complex numbers), that is, an element of the dual space $\mathfrak{h}^*$. These elements of $\mathfrak{h}^*$ (that correspond to nonzero eigenspaces) are called the "weights" of the representation. Then you are exactly right when you say that the "roots" of a Lie algebra are the weights of its action on itself via the adjoint representation. So roots are special types of weights.

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