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 formula_1. There is a closely related theorem classifying the irreducible representations of a connected compact Lie group formula_2. 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 formula_1 or formula_2. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If formula_2 is simply connected, this distinction disappears.
The theorem was originally proved by Élie Cartan in his 1913 paper. 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.
Statement.
Lie algebra case.
Let formula_7 be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra formula_8. Let formula_9 be the associated root system. We then say that an element formula_10 is integral if 
is an integer for each root formula_12. Next, we choose a set formula_13 of positive roots and we say that an element formula_10 is dominant if formula_15 for all formula_16. An element formula_10 dominant integral if it is both dominant and integral. Finally, if formula_18 and formula_19 are in formula_20, we say that formula_18 is higher than formula_19 if formula_23 is expressible as a linear combination of positive roots with non-negative real coefficients. 
A weight formula_18 of a representation formula_25 of formula_1 is then called a highest weight if formula_18 is higher than every other weight formula_19 of formula_25.
The theorem of the highest weight then states:
The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.
The compact group case.
Let formula_2 be a connected compact Lie group with Lie algebra formula_36 and let formula_37 be the complexification of formula_1. Let formula_39 be a maximal torus in formula_2 with Lie algebra formula_41. Then formula_42 is a Cartan subalgebra of formula_1, and we may form the associated root system formula_9. The theory then proceeds in much the same way as in the Lie algebra case, with one crucial difference: the notion of integrality is different. Specifically, we say that an element formula_10 is analytically integral if 
is an integer whenever
where formula_48 is the identity element of formula_2. Every analytically integral element is integral in the Lie algebra sense, but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if formula_2 is not simply connected, there may be representations of formula_1 that do not come from representations of formula_2. On the other hand, if formula_2 is simply connected, the notions of "integral" and "analytically integral" coincide.
The theorem of the highest weight for representations of formula_2 is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."
Proofs.
There are at least four proofs: