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- In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (over ); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight. There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group K and the finite-dimensional representations of the complex semisimple Lie algebra that is the complexification of the Lie algebra of K (this fact is essentially a special case of the Lie group–Lie algebra correspondence). Also, finite-dimensional representations of a connected compact Lie group can be studied through finite-dimensional representations of the universal cover of such a group. Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of representations of connected compact Lie groups. The theory is a basis for the later works of Harish-Chandra that concern (infinite-dimensional) representation theory of real reductive groups. (en)
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- In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (over ); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight. (en)
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- Representation theory of semisimple Lie algebras (en)
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