Domenico Fiorenza representations for sl(2,C)

Representations of $\mathfrak{sl}_2(\mathbb{C})$

The Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ can be introduced as the tangent space to the identity for $SL_2(\mathbb{C})$. By an easy calculation it results that $\mathfrak{sl}_2(\mathbb{C})$ is three dimensional. Choose a basis $h,e,f$ as follows:

with commutation rules:

First we want to identify a typical irreducible representation for $\mathfrak{sl}_2(\mathbb{C})$. We resume its properties in a theorem.

Theorem. For each positive integer $m$ there exists up to isomorphism a unique irreducible complex linear representation $\pi$ of $\mathfrak{sl}_2(\mathbb{C})$ on a complex vector space $V$ of dimension $m$. Moreover, there are an integer $k$ and a basis $\{ v_{-k},v_{-k+2},\dots,v_{k-2},v_k\}$ of $V$ such that: 1. $h\cdot v_{k-2j} = (k-2j) v_i$, 1. $e\cdot v_{k-2j} = j(k-j+1)v_{k-(2j-1)}$, with $e\cdot v_k=0$, 1. $f\cdot v_{k-2j}=v_{k-2(j+1)}$, with $f\cdot v_{-k}=0$.

where $g\cdot v$ for $v \in V$ and $g \in \mathfrak{sl}_2(\mathbb{C})$ is the action of $\mathfrak{sl}_2(\mathbb{C})$ on $V$.

Proof. Let $V$ be a complex linear irreducible representation of $\mathfrak{sl}_2(\mathbb{C})$. We show that $V$ satisfy 1. to 3. Let $v \ne 0$ be an eigenvector for $h\cdot$, say $h\cdot v = \lambda v$. Then $e \cdot v, e \cdot e \cdot v, \dots$ are also eigen vectors because:

Since $\lambda, \lambda +2, \dots$ are distinct, these eigenvectors are indipendent. By finite dimensionality of $V$ we can find $v_0 \in V$ with

a. $v_0 \ne 0$ a. $h \cdot v_0 = \lambda v_0$ a. $e \cdot v_0 = 0$

Define $v_i = f^i \cdot v_0$. Then $h \cdot v_i = (\lambda - 2i) v_i$, by the same argument as above and so there is a minimum integer $k$ with $f^{k+1} \cdot v_0 = v_0$. Then $v_0 \dots v_k$ are indipendent and

a. $e \cdot v_0 = 0$ a. $h \cdot v_i = (\lambda -2i) v_i$ a. $f \cdot v_i = v_{i+1}$ with $v_{n+1}= 0$.

Now we show that $V = span \{ v_0, \dots, v_n \}$. It is sufficient to show that $span \{ v_0, \dots, v_n \}$ is stable under $e \cdot$. This results from:

with $v_{-1} = 0$. We prove the calculation above by induction. Assume case for i. Then for i+1:

For ending the proof of uniqueness, we show that $\lambda = k$. It results:

due to the cyclic property of the trace. Note that here we intend $h,e$ and $f$, being elements of $End(V)$. Then in the basis $\{ v_0, \dots v_n \}$ the trace is explicitely:

Then $\lambda=k$.

Remark that by 1. in the statement of the theorem the smallest of the eigenvalues is the negative of the greatest.

The proof of the existence is easily achieved by defining a $\pi : \mathfrak{sl}_2(\mathbb{C}) \to End(W)$ on the generators of $\mathfrak{sl}_2(\mathbb{C})$ such that properties 1. 2. 3. are satisfied, and then by extending through linearity to whole $\mathfrak{sl}_2(\mathbb{C})$. This is a representation for $\mathfrak{sl}_2(\mathbb{C})$. To see irreducibility, let $U$ be a nonzero invariant subspace. Since $U$ is invariant under $h \cdot$, $U$ is spanned by a subset of the basis $v_i$. Applying on such a $v_i$ the action of $e \cdot$ several times, we can see that $v_0$ is in $U$. Repeated application of $f \cdot$ then shows that $U = V$. Hence the representation is irreducible. This completes the proof.

Since we built directly a representation that is irreducible in a very concrete way, we want a result that assure us that we can decompose a representation in irreducible ones, that we report without proof.

Theorem. Let $\phi$ be a complex linear representation of $\mathfrak{sl}_2(\mathbb{C})$ on a finite-dimensional vector space $V$. Then V is completely reducible in the sense that there exist invariant subspaces $U_1,..U_r$ of $V$ such that $V = U_1 \oplus .. U_r$ and such that the restriction of the representation to each $U_i$ is irreducible

The importance of studying $\mathfrak{sl}_2(\mathbb{C})$ is that a semi-simple Lie algebra $g$ contains several copies of $\mathfrak{sl}_2(\mathbb{C})$. It follows that for a given $V$ representation of $g$, there are several representantions of $\mathfrak{sl}_2(\mathbb{C})$ embedded in $V$. This is shown shortly in Lie groups and algebras representations.