Notes on SU(2) reps

The unitary irreducible representations of $SU(2)$ (which is isomorphic to the group of unit quaternions $S^3$) couldn’t be simpler to describe. Start with the tautological representation

$SU(2) \stackrel{id}{\hookrightarrow} U(\mathbb{C}^2) \hookrightarrow Vect(\mathbb{C}^2, \mathbb{C}^2).$

If $\exp \colon Vect \to CAlg$ is the free commutative algebra functor, aka the polynomial algebra functor, then by functoriality of $\exp$ we get

$SU(2) \to Vect(\mathbb{C}^2, \mathbb{C}^2) \to CAlg(\exp(\mathbb{C}^2), \exp(\mathbb{C}^2))$

so that $SU(2)$ acts by graded algebra automorphisms on $\mathbb{C}[x, y]$. Thus $SU(2)$ acts linearly on each degree $n$ homogeneous component; this component has complex dimension $n+1$. Let $V_n$ denote this component.

The **spin $k$** representation of $SU(2)$, where $k$ is a *half-integer* $0, \frac1{2}, 1, \frac{3}{2}, \ldots$, is the representation $V_{2k}$. The spin $k$ representation is irreducible.

We note in passing that $\exp(V) = \sum_{n \geq 0} V^{\otimes n}/n!$, where the homogeneous component $V^{\otimes n}/n!$ is the $n^{th}$ *symmetric power* of $V$. Hence the spin representation $V_n$ is the $n^{th}$ symmetric power of the tautological representation $V_1$.

It is worth going into more concrete detail. The tautological $\mathbb{C}$-linear action of $SU(2) = S^3$ on $\mathbb{C}^2$ can be described quaternionically as assigning to a unit quaternion $q$ the linear map

$z_1 + z_2 j \mapsto (z_1 + z_2 j)q^{-1}$

(this is a left $\mathbb{C}$-module homomorphism). As a real Lie group, $SU(2) = S^3$ is 3-dimensional. Moreover, it is connected and simply connected, so the continuous linear representations $SU(2) \to GL(W)$ on real vector spaces $W$ will be in natural bijection with Lie algebra representations $su(2) \to gl(W)$.

Elements of $SU(2)$, i.e., $2 \times 2$ unitary matrices of determinant $1$, can be represented as $e^{i A}$ where $A$ is a $2 \times 2$ Hermitian matrix of trace 0. (Similarly, any unit quaternion $q$ is of the form $\cos(\theta) + \sin(\theta)v = e^{\theta v}$ for some purely quaternionic unit vector $v$, seen as generating a subalgebra of the quaternion algebra isomorphic to $\mathbb{C}$.) Thus we identify $su(2)$ with the Lie algebra of Hermitian matrices of trace $0$. A conventional basis is given by

$H = \left(
\begin{aligned}
1 & 0\\
0 & -1
\end{aligned}
\right),
\qquad
Q = \left(
\begin{aligned}
0 & 1\\
1 & 0
\end{aligned}
\right),
\qquad
P = \left(
\begin{aligned}
0 & i\\
-i & 0
\end{aligned}
\right)$

known as **Pauli matrices**. The tautological Lie algebra representation on $\mathbb{C}^2$ given by these matrices can be written as

$H(a, b) = (a, -b), \qquad Q(a, b) = (b, a), \qquad P(a, b) = (-i b, i a).$

By differentiating at the identity the $SU(2)$-action on $\mathbb{C}[x, y]$ by algebra automorphisms, we see the $su(2)$-action on $\mathbb{C}[x, y]$ is by derivations. From the previous display, where $H$ takes $a x + by \mapsto a x - b y$ etc., it is easy to read off the derivation representation:

$H = x\frac{\partial}{\partial x} - y\frac{\partial}{\partial y}, \qquad
Q = y \frac{\partial}{\partial x} + x\frac{\partial}{\partial y},
\qquad
P = -i y\frac{\partial}{\partial x} + i x\frac{\partial}{\partial y}$

This gives us the Lie algebra action of $su(2)$ on the spin representation $V_{n}$ (which in our representation is generated by monomials $x^j y^k$ of total degree $n = j + k$). We have

$H(x^j y^k) = (j - k)x^j y^k$

$\,$

$Q(x^j y^k) = j x^{j-1} y^{k+1} + k x^{j+1} y^{k-1}$

$\,$

$P(x^j y^k) = i (-j x^{j-1} y^{k+1} + k x^{j+1} y^{k-1})$

Since we have been dealing with complex representations of $su(2)$, we may as well complexify $su(2)$, and this will allow us a neater presentation in terms of raising and lowering (aka creation and annihilation) operators. The complexification of $su(2)$ is the complex Lie algebra $sl_2(\mathbb{C})$, and $su(2)$ is known as the real compact form of $sl_2(\mathbb{C})$.

Representations of $sl_2(\mathbb{C})$ are of course very classical, and we relate them to our development above as follows. Define lowering and raising operators by

$A = \frac{Q + i P}{2}, \qquad A^\ast = \frac{Q - i P}{2}$

In matrix form,

$A = \left(
\begin{aligned}
0 & 0\\
1 & 0
\end{aligned}
\right),
\qquad
A^\ast = \left(
\begin{aligned}
0 & 1\\
0 & 0
\end{aligned}
\right)$

and we have the classical Lie algebra relations for $sl_2(\mathbb{C})$:

$[A^\ast, A] = H, \qquad [H, A^\ast] = 2A^\ast, \qquad [H, A] = -2A.$

In terms of derivations, we have

$A = y \frac{\partial}{\partial x}, \qquad A^\ast = x\frac{\partial}{\partial y}$

and the action of $sl_2(\mathbb{C})$ on the space $V_n$ of homogeneous degree $n$ polynomials is therefore given by

$A(x^j y^k) = j x^{j-1} y^{k+1}$

$\,$

$A^\ast (x^j y^k) = k x^{j+1} y^{k-1}.$

Thus $A$ lowers the exponent on $x$, and $A^\ast$ raises it.

This representation makes it palpably obvious that the spin $n$ representation is irreducible: starting with any non-zero vector $v$ (i.e., a homogeneous polynomial of degree $n$), if we hit it enough times with a lowering operator $A$, we eventually obtain a lowest weight vector (a nonzero scalar multiple of $y^n$), and from there we can get a nonzero scalar multiple of any monomial $x^j y^k$ by applying the raising operator $A^\ast$ enough times. Hence the submodule generated by any nonzero $v$ is the whole module $V_n$.

Pretty much the same idea shows that any irreducible representation of $sl_2(\mathbb{C})$ is one of the spin $n$ representations. The point is that any irreducible representation of $su(2)$ is finite-dimensional (by a compactness argument), and then the yoga of raising and lowering operators allows us to deduce the existence, in a finite-dimensional representation, of a highest weight vector $v$, or alternatively, of a lowest weight vector. The structure of the submodule generated by $v, A v, A^2 v, \ldots, A^n v$, where $A^n v$ is a lowest weight vector, must be that of $V_n$, just by using the Lie algebra relations between $A$, $A^\ast$, and $H$. (See Fulton and Harris for the more detailed proof.)

The category of $SU(2)$-representations is a symmetric monoidal category, with the action on a tensor product being a diagonal action

$g (v \otimes w) = g v \otimes g w.$

At the “infinitesimal” or Lie algebra level, the corresponding action of the enveloping algebra $U sl_2(\mathbb{C})$ on a tensor product is given in terms of generators $x \in sl_2(\mathbb{C})$ by

$x \cdot (v \otimes w) = (x \cdot v) \otimes w + v \otimes (x \cdot w)$

(corresponding to the fact that the standard coalgebra comultiplication $\delta: U\mathfrak{g} \to U\mathfrak{g} \otimes U\mathfrak{g}$ is determined by its effect on generators $x$, which is $\delta(x) = x \otimes 1 + 1 \otimes x$).

In particular, if $H(v) = \alpha v$ and $H(w) = \beta w$, then $H(v \otimes w) = (\alpha + \beta)(v \otimes w)$. This makes it easy to work out the structure of a tensor product

$V_n \otimes V_m.$

Namely, if a representation $V$ is specified by a function $\chi_V: \mathbb{Z} \to \mathbb{N}$ where $\chi_V(\alpha)$ is defined to be the dimension of the $H$-eigenspace of $V$ with eigenvalue $\alpha$, then the tensor product of modules is specified by

$\chi_{V \otimes W}(\gamma) = \sum_{\gamma = \alpha + \beta} \chi_V(\alpha) \chi_W(\beta)$

or in other words by a convolution product:

$\chi_{V \otimes W} = \chi_V \ast \chi_W.$

In particular, if $V, W$ are irreducible, then $\chi_V, \chi_W$ are “uniform distributions” of finite support centered at $0$, and we just work out the convolution product of two such uniform distributions. This has the characteristic shape of an Egyptian or Mayan pyramid, which is a “stacked” sum of uniform distributions. This can be left as an exercise, but anyway here is the answer: if $m \leq n$, then

$V_m \otimes V_n = V_{n+m} \oplus V_{n+m-2} \oplus V_{n+m-4} \oplus \ldots \oplus V_{n-m}$

going from the bottom of the pyramid to the top. And that’s all for now, folks.

Revised on July 17, 2011 at 22:06:21
by
Todd Trimble