Representations of sl2

Example

Let $\mathfrak{g} = \mathfrak{sl}_2(\C)$, the special linear Lie algebra of complex $2 \times 2$ matrices of trace zero, with basis $$e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$ so that $[h, e] = 2e$, $[h, f] = -2f$ and $[e, f] = h$. For each integer $n \ge 0$, let $V_n$ have basis $v_0, \dots, v_n$, and put $v_{-1} = v_{n+1} = 0$. Define $$h v_k = (n - 2k) v_k, \qquad f v_k = (k + 1) v_{k+1}, \qquad e v_k = (n - k + 1) v_{k-1}.$$ This is a representation of dimension $n + 1$. Check of $[e, f] = h$ on $v_k$: $(ef - fe) v_k = \big( (k + 1)(n - k) - k(n - k + 1) \big) v_k = (n - 2k) v_k$. Check of $[h, e] = 2e$ on $v_k$: $(he - eh) v_k = (n - k + 1)\big( (n - 2k + 2) - (n - 2k) \big) v_{k-1} = 2 e v_k$; the relation $[h, f] = -2f$ is checked the same way.

The classification: each $V_n$ is irreducible with highest weight $n$, and each finite-dimensional irreducible $\mathfrak{g}$-module is isomorphic to exactly one $V_n$. By Weyl's theorem, every finite-dimensional $\mathfrak{g}$-module is a direct sum of copies of the $V_n$.