Abstract
In this chapter, we will study the finite-dimensional irreducible representations of a complex semisimple Lie algebra g. These will be classified by means of a “theorem of the highest weight.” The theorem states that every irreducible representation has a (unique) highest weight, that two irreducible representations with the same highest weight are equivalent, and that the elements that actually arise as highest weights of irreducible representations are precisely the “dominant integral” elements.
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