representation theory of $𝔰{𝔩}_2ℂ$

The special linear Lie algebra of $2\times 2$ matricies, denoted by $𝔰{𝔩}_2ℂ$, is defined to be the span (over $ℂ$) 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: $[E,F]=H,[H,E]=2E,[H,F]=-2F$.

The representation theory of $𝔰{𝔩}_2ℂ$ 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 $𝔰{𝔩}_2ℂ$ are in bijection with the non-negative integers ${\mathbb{Z}}_{\ge 0}$ as follows. Let $k\in {\mathbb{Z}}_{\ge 0}$, $V$ be a $ℂ$-vector space spanned by vectors $v_0,\mathrm{\dots },v_k$. The following action of $E,H,F$ on $V$ define the unique (up to isomorphism) irreducible representation of $𝔰{𝔩}_2ℂ$ of dimension $k+1$ (or of highest weight $k$):

The main points are that the one dimensional spaces $ℂ\cdot v_i$ are eigenspaces for $H$ with eigenvalue $k-2i$, the operator corresponding to $E$ kills $v_0$ and otherwise sends $ℂ\cdot v_i\to ℂ\cdot v_{i-1}$, while $F$ kills $v_k$ and otherwise sends $ℂ\cdot v_i\to ℂ\cdot v_{i+1}$. 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 $H$.

$𝔰{𝔩}_2ℂ$ is a simple Lie algebra, thus by Weyl’s Theorem all finite dimensional representations for $𝔰{𝔩}_2ℂ$ are completely reducible. So any finite dimensional representation of $𝔰{𝔩}_2ℂ$ splits into a direct sum of irreducible representations for various non-negative integers as described above.