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Finite-dimensional simple modules are classified by dominant highest weights

Statement

Every finite-dimensional simple $g$ -module is a cyclic highest-weight module whose highest weight is dominant integral, and for each dominant integral weight

$λ = \sum_{i = 1}^{r} m_{i} ω_{i} (m_{i} \in N_{0}),$

there exists, up to isomorphism, a unique finite-dimensional simple module $L (λ)$ with highest weight $λ$ .

Facts & Assumptions

Given: A complex semisimple Lie algebra $g$ with chosen simple roots $α_{1}, \dots, α_{r}$ and fundamental weights $ω_{1}, \dots, ω_{r}$ .

[F1]

For each weight $λ$ , there is a unique irreducible highest-weight module $L (λ)$ , and every irreducible highest-weight module of weight $λ$ is isomorphic to $L (λ)$ .

[F2]

If $λ$ is dominant integral, then $L (λ)$ is finite-dimensional.

Proof

technique · direct

1.1

In any finite-dimensional module, one can choose a weight maximal with respect to the positive-root order. Its weight space contains a nonzero vector killed by $n^{+}$ , so every finite-dimensional simple module is a cyclic highest-weight module in the sense of Highest-weight vectors and cyclic highest-weight modules.

givenchoose

2.1

Restricting that module to each simple-root ${s l}_{2}$ -subalgebra shows that the highest weight pairs nonnegatively and integrally with every simple coroot. By Fundamental weights for a chosen simple root system, the highest weight is therefore a dominant integral combination of the fundamental weights.

step 1.1

3.1

Let $V$ be a finite-dimensional simple module. By step 1.1 it is an irreducible highest-weight module of some weight $λ$ , and step 2.1 shows that $λ$ is dominant integral. By [F1], $V ≅ L (λ)$ . Conversely, if $λ$ is dominant integral, then [F2] gives a finite-dimensional module $L (λ)$ , and [F1] makes it the unique simple module with highest weight $λ$ . This is exactly the claimed classification.

F1F2step 1.1step 2.1∎

Depends on

Used by

Weyl-invariant polynomials on the Cartan extend to invariant polynomials on g Lemma

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)