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.
Let be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra. Let be the associated root system. We then say that an element isintegral[5]if
is an integer for each root . Next, we choose a set of positive roots and we say that an element isdominantif for all . An element dominant integral if it is both dominant and integral. Finally, if and are in , we say that ishigher[6] than if is expressible as a linear combination of positive roots with non-negative real coefficients.
Aweight of a representation of is then called a highest weightif is higher than every other weight of.
Let be a connected compact Lie group with Lie algebra and let be the complexification of . Let be a maximal torusin 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[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 that do not come from representations of . On the other hand, if 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."
The theory of Verma modules contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representations. This approach is essentially due to H. Weyl and works quite well for classical groups.
Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN978-3319134666