Abstract
This chapter discusses the representations of semisimple Lie algebras. E. Cartan studied representations of simple Lie algebras separately, but most of his work can be generalized to semisimple Lie algebras. V is generated by weight vectors. It is known that an irreducible representation of a semisimple Lie algebra is uniquely determined by the highest weight of it. Therefore, to find all irreducible representations of the Lie algebra, it is necessary to determine the elements in h*0 that can be highest weights of irreducible representations. The two irreducible representations of a semisimple Lie algebra are equivalent iff they have the same diagram. It is also known that that an irreducible representation with highest weight zero is the zero representation.
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