${\mathfrak{sl}_2}$ is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over ${\mathbb{C}}$ of trace zero, with the Lie bracket defined by:

$\displaystyle [A,B] = AB - BA.$

The representation theory of ${\mathfrak{sl}_2}$ is important for several reasons.

1. It’s elegant.
2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
3. Knowing the theory for ${\mathfrak{sl}_2}$ is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, ${\mathfrak{sl}_2}$ is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.

Irreducibility

The notion of irreducibility parallels that in the theory of group representations:

Definition 1 Let ${L}$ be a Lie algebra. A ${L}$-representation ${V}$ is irreducible if it contains no nontrivial subrepresentations, i.e. there is no ${W \neq 0, V}$, ${W \subset V}$, such that “multiplication” by ${L}$ maps ${W}$ into itself.

Since we are working in an “artinian” category, we have, as a general fact:

Example 1 Every representation ${V}$ contains an irreducible representation. We prove this by induction on ${\dim V}$; if ${\dim V=1}$ this is trivial, since ${V}$ itself is irreducible. Suppose we know it for representations of smaller dimension than ${V}$. If ${V}$ contains no subrepresentations, then we’re done. Otherwise, it contains ${W}$ such that ${\dim W < \dim V}$, and ${\dim W}$ contains an irreducible subrepresentation.

There is in fact an analog of Maschke’s theorem:

Theorem 2 (Weyl’s Theorem) Any (finite-dimensional) representation ${V}$ of ${\mathfrak{sl}_2}$ decomposes uniquely (up to isomorphism) as a direct sum of irreducible representations.

Here this result is more difficult than Maschke’s theorem, but it can be proved more generally for semisimple Lie algebras, using either algebraic methods or the theory of compact groups. I shall therefore omit the proof, at least for now.

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A Presentation of ${\mathfrak{sl}_2}$

Now we want to classify the irreducible ${\mathfrak{sl}_2}$-modules. First, we need information on ${\mathfrak{sl}_2}$ itself:

Choose the matrices (I’m following Fulton and Harris’s conventions)

$\displaystyle H = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \ X = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \ \text{and} \ Y = \begin{bmatrix} 0 & 0 \\ 1 & 0. \end{bmatrix} \ \ \ \ \ (1)$

There are three key relations:

$\displaystyle [H,X] = 2X, \quad [H,Y] = -2Y, \quad [X,Y] = H. \ \ \ \ \ (2)$

The first two should be thought of in the following sense: There is a linear map ${\mathfrak{sl}_2 \rightarrow \mathfrak{sl}_2}$ sending ${A \rightarrow [H,A]}$. Then ${X}$ is an eigenvector with eigenvalue 2, and ${Y}$ is an eigenvector with eigenvalue -2. ${H}$ is an eigenvector with value ${0}$. This is the way these techniques generalize to semisimple Lie algebras, except that ${H}$ is replaced with a “Cartan subalgebra.”

There is another theorem I will quote without proof:

Theorem 3 In any (finite-dimensional) representation ${V}$ of ${\mathfrak{sl}_2}$, the map ${V \rightarrow V, \ v \rightarrow Hv}$, is diagonalizable.

The result follows from general facts about the Jordan-Chevalley decomposition in semisimple Lie algebras. I’d rather prove it in the abstract setting though.

We’re just about ready to get to the classification theorem itself. But I’ll leave that for another post.