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

${\displaystyle \lambda \mapsto [V^{\lambda }]}$

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

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

${\displaystyle 2{\frac {\langle \lambda ,\alpha \rangle }{\langle \alpha ,\alpha \rangle }}}$

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

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

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

• If ${\displaystyle V}$ is a finite-dimensional irreducible representation of ${\displaystyle {\mathfrak {g}}}$, then ${\displaystyle V}$ 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 ${\displaystyle \lambda }$, there exists a finite-dimensional irreducible representation with highest weight ${\displaystyle \lambda }$.

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.

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

${\displaystyle \langle \lambda ,H\rangle }$

is an integer whenever

${\displaystyle e^{2\pi H}=I}$

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

The theorem of the highest weight for representations of ${\displaystyle K}$^[9] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."

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.

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

• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 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.