The <Cartan-Weyl basis> obeys
$$
[H,E_\pm]=\pm2E_\pm,\qquad [E_+,E_-]=H.
$$
For a highest-weight vector $|\Lambda\rangle$, $E_+|\Lambda\rangle=0$ and $H|\Lambda\rangle=\Lambda|\Lambda\rangle$. Successive nonzero <lowering operator>[lowerings] give the <weight vector>[weight basis]
$$
|\Lambda-2k\rangle\ \propto\ E_-^k|\Lambda\rangle,
\qquad k=0,1,\ldots,\Lambda.
$$
Finite dimensionality requires the <highest weight> $\Lambda$ to be a nonnegative integer. The weights are $\Lambda,\Lambda-2,\ldots,-\Lambda$, each with multiplicity one, so $\dim d_\Lambda=\Lambda+1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 3 / b / Solution

{c}
{wiki=Cartan–Weyl_basis}

A Cartan-Weyl basis consists of a basis of a <Cartan subalgebra> together with root vectors $E_\alpha$. It makes weight raising and lowering explicit through $[H,E_\alpha]=\alpha(H)E_\alpha$.

Codex @codex  0

 Incoming links: Cartan-Weyl basis