The special linear Lie algebra of $2 \times 2$ matricies, denoted by $\mathfrak{sl}_2 \mathbb{C}$ , is defined to be the span (over $\mathbb{C}$ ) of the matricies

with Lie bracket given by the commutator of matricies: $[X,Y] := X \cdot Y - Y \cdot X$ . The matricies $E, F, H$ satisfy the commutation relations: $[E, F] = H, [H, E] = 2 E, [H, F] = -2 F$ .

The representation theory of $\mathfrak{sl}_2 \mathbb{C}$ is a very important tool for understanding the structure theory and representation theory of other Lie algebras (semi-simple finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).

The finite dimensional, irreducible, representations of $\mathfrak{sl}_2 \mathbb{C}$ are in bijection with the non-negative integers $\mathbb{Z}_{\ge 0}$ as follows. Let $k \in \mathbb{Z}_{\ge 0}$ , $V$ be a $\mathbb{C}$ -vector space spanned by vectors $v_0, \ldots, v_k$ . The following action of $E, H, F$ on $V$ define the unique (up to isomorphism) irreducible representation of $\mathfrak{sl}_2 \mathbb{C}$ of dimension $k+1$ (or of highest weight $k$ ):

The main points are that the one dimensional spaces $\mathbb{C} \cdot v_i$ are eigenspaces for $H$ with eigenvalue $k - 2i$ , the operator corresponding to $E$ kills $v_0$ and otherwise sends $\mathbb{C} \cdot v_i \to \mathbb{C} \cdot v_{i-1}$ , while $F$ kills $v_k$ and otherwise sends $\mathbb{C} \cdot v_i \to \mathbb{C} \cdot v_{i+1}$ . The operator corresponding to $E$ is often called a raising operator since it raises the eigenvalue for $H$ , and that of $F$ is called a lowering operator since it lowers the eigenvalue for $H$ .

$\mathfrak{sl}_2 \mathbb{C}$ is a simple Lie algebra, thus by Weyl's Theorem all finite dimensional representations for $\mathfrak{sl}_2 \mathbb{C}$ are completely reducible. So any finite dimensional representation of $\mathfrak{sl}_2 \mathbb{C}$ splits into a direct sum of irreducible representations for various non-negative integers as described above.