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
The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element.
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.
be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra
be the associated root system.
is integral[5] if is an integer for each root
of positive roots and we say that an element
⟨ λ , α ⟩ ≥ 0
is dominant integral if it is both dominant and integral.
λ − μ
is expressible as a linear combination of positive roots with non-negative real coefficients.
is then called a highest weight if
The theorem of the highest weight then states:[2] The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.
be a connected compact Lie group with Lie algebra
, 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[7] if is an integer whenever where
is the identity element of
Every analytically integral element is integral in the Lie algebra sense,[8] 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
is simply connected, the notions of "integral" and "analytically integral" coincide.
[3] The theorem of the highest weight for representations of
[9] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."