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. There is a closely related theorem classifying the irreducible representations of a connected compact Lie group. 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. 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 be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra. Let be the associated root system. We then say that an element is integral if
is an integer for each root. Next, we choose a set of positive roots and we say that an element is dominant if for all. An element dominant integral if it is both dominant and integral. Finally, if and are in, we say that is higher than if is expressible as a linear combination of positive roots with non-negative real coefficients.
A weight of a representation of is then called a highest weight if is higher than every other weight of.
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 be a connected compact Lie group with Lie algebra and let be the complexification of. Let be a maximal torus in with Lie algebra. Then is a Cartan subalgebra of, and we may form the associated root system. 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 is analytically integral if
is an integer whenever
where is the identity element of. 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 is not simply connected, there may be representations of that do not come from representations of. On the other hand, if is simply connected, the notions of "integral" and "analytically integral" coincide.
The theorem of the highest weight for representations of 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: