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 [math]\displaystyle{ \mathfrak g }[/math].^[1][2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group [math]\displaystyle{ K }[/math].^[3] The theorem states that there is a bijection

[math]\displaystyle{ \lambda \mapsto [V^\lambda] }[/math]

from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of [math]\displaystyle{ \mathfrak g }[/math] or [math]\displaystyle{ K }[/math]. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If [math]\displaystyle{ K }[/math] 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.

Statement

Lie algebra case

Let [math]\displaystyle{ \mathfrak{g} }[/math] be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra [math]\displaystyle{ \mathfrak{h} }[/math]. Let [math]\displaystyle{ R }[/math] be the associated root system. We then say that an element [math]\displaystyle{ \lambda\in\mathfrak h^* }[/math] is integral^[5] if

[math]\displaystyle{ 2\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle} }[/math]

is an integer for each root [math]\displaystyle{ \alpha }[/math]. Next, we choose a set [math]\displaystyle{ R^+ }[/math] of positive roots and we say that an element [math]\displaystyle{ \lambda\in\mathfrak h^* }[/math] is dominant if [math]\displaystyle{ \langle\lambda,\alpha\rangle\geq 0 }[/math] for all [math]\displaystyle{ \alpha\in R^+ }[/math]. An element [math]\displaystyle{ \lambda\in\mathfrak h^* }[/math] dominant integral if it is both dominant and integral. Finally, if [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ \mu }[/math] are in [math]\displaystyle{ \mathfrak h^* }[/math], we say that [math]\displaystyle{ \lambda }[/math] is higher^[6] than [math]\displaystyle{ \mu }[/math] if [math]\displaystyle{ \lambda-\mu }[/math] is expressible as a linear combination of positive roots with non-negative real coefficients.

A weight [math]\displaystyle{ \lambda }[/math] of a representation [math]\displaystyle{ V }[/math] of [math]\displaystyle{ \mathfrak g }[/math] is then called a highest weight if [math]\displaystyle{ \lambda }[/math] is higher than every other weight [math]\displaystyle{ \mu }[/math] of [math]\displaystyle{ V }[/math].

The theorem of the highest weight then states:^[2]

• If [math]\displaystyle{ V }[/math] is a finite-dimensional irreducible representation of [math]\displaystyle{ \mathfrak{g} }[/math], then [math]\displaystyle{ V }[/math] has a unique highest weight, and this highest weight is dominant integral.
• If two finite-dimensional irreducible representations have the same highest weight, they are isomorphic.
• For each dominant integral element [math]\displaystyle{ \lambda }[/math], there exists a finite-dimensional irreducible representation with highest weight [math]\displaystyle{ \lambda }[/math].

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 [math]\displaystyle{ K }[/math] be a connected compact Lie group with Lie algebra [math]\displaystyle{ \mathfrak k }[/math] and let [math]\displaystyle{ \mathfrak g:=\mathfrak k+i\mathfrak k }[/math] be the complexification of [math]\displaystyle{ \mathfrak g }[/math]. Let [math]\displaystyle{ T }[/math] be a maximal torus in [math]\displaystyle{ K }[/math] with Lie algebra [math]\displaystyle{ \mathfrak t }[/math]. Then [math]\displaystyle{ \mathfrak h:=\mathfrak t+i\mathfrak t }[/math] is a Cartan subalgebra of [math]\displaystyle{ \mathfrak g }[/math], and we may form the associated root system [math]\displaystyle{ R }[/math]. 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 [math]\displaystyle{ \lambda\in\mathfrak h }[/math] is analytically integral^[7] if

[math]\displaystyle{ \langle\lambda,H\rangle }[/math]

is an integer whenever

[math]\displaystyle{ e^{2\pi H}=I }[/math]

where [math]\displaystyle{ I }[/math] is the identity element of [math]\displaystyle{ K }[/math]. 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 [math]\displaystyle{ K }[/math] is not simply connected, there may be representations of [math]\displaystyle{ \mathfrak g }[/math] that do not come from representations of [math]\displaystyle{ K }[/math]. On the other hand, if [math]\displaystyle{ K }[/math] is simply connected, the notions of "integral" and "analytically integral" coincide.^[3]

The theorem of the highest weight for representations of [math]\displaystyle{ K }[/math]^[9] 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:

• Hermann Weyl's original proof from the compact group point of view,^[10] based on the Weyl character formula and the Peter–Weyl theorem.
• 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.

See also

• Classifying finite-dimensional representations of Lie algebras
• Representation theory of a connected compact Lie group
• Weights in the representation theory of semisimple Lie algebras

Notes

References

• Dixmier, Jacques (1996), Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, https://books.google.com/books?isbn=0821805606 
• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9. 
• Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666 
• Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7, https://archive.org/details/introductiontoli00jame .

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