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Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra .[1][2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group .[3] The theorem states that there is a bijection

from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of or . The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If is simply connected, this distinction disappears.

The theorem was originally proved by Élie Cartan in his 1913 paper.[4] The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.

  1. ^ Dixmier 1996, Theorem 7.2.6.
  2. ^ Hall 2015 Theorems 9.4 and 9.5
  3. ^ Hall 2015 Theorem 12.6
  4. ^ Knapp, A. W. (2003). "Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann". The American Mathematical Monthly. 110 (5): 446–455. doi:10.2307/3647845. JSTOR 3647845.

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Satz vom höchsten Gewicht German Teorema del peso máximo Spanish Теорема найбільшої ваги Ukrainian

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