Todd Trimble
Notes on SU(2) reps

Contents

Definition of spin representations

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

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

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

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

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

Definition

The spin kk representation of SU(2)SU(2), where kk is a half-integer 0,12,1,32,0, \frac1{2}, 1, \frac{3}{2}, \ldots, is the representation V 2kV_{2k}. The spin kk representation is irreducible.

We note in passing that exp(V)= n0V n/n!\exp(V) = \sum_{n \geq 0} V^{\otimes n}/n!, where the homogeneous component V n/n!V^{\otimes n}/n! is the n thn^{th} symmetric power of VV. Hence the spin representation V nV_n is the n thn^{th} symmetric power of the tautological representation V 1V_1.

Lie algebra representations

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

z 1+z 2j(z 1+z 2j)q 1z_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 3SU(2) = S^3 is 3-dimensional. Moreover, it is connected and simply connected, so the continuous linear representations SU(2)GL(W)SU(2) \to GL(W) on real vector spaces WW will be in natural bijection with Lie algebra representations su(2)gl(W)su(2) \to gl(W).

Elements of SU(2)SU(2), i.e., 2×22 \times 2 unitary matrices of determinant 11, can be represented as e iAe^{i A} where AA is a 2×22 \times 2 Hermitian matrix of trace 0. (Similarly, any unit quaternion qq is of the form cos(θ)+sin(θ)v=e θv\cos(\theta) + \sin(\theta)v = e^{\theta v} for some purely quaternionic unit vector vv, seen as generating a subalgebra of the quaternion algebra isomorphic to \mathbb{C}.) Thus we identify su(2)su(2) with the Lie algebra of Hermitian matrices of trace 00. A conventional basis is given by

H=(1 0 0 1),Q=(0 1 1 0),P=(0 i i 0)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 2\mathbb{C}^2 given by these matrices can be written as

H(a,b)=(a,b),Q(a,b)=(b,a),P(a,b)=(ib,ia).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)SU(2)-action on [x,y]\mathbb{C}[x, y] by algebra automorphisms, we see the su(2)su(2)-action on [x,y]\mathbb{C}[x, y] is by derivations. From the previous display, where HH takes ax+byaxbya x + by \mapsto a x - b y etc., it is easy to read off the derivation representation:

H=xxyy,Q=yx+xy,P=iyx+ixyH = 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)su(2) on the spin representation V nV_{n} (which in our representation is generated by monomials x jy kx^j y^k of total degree n=j+kn = j + k). We have

H(x jy k)=(jk)x jy kH(x^j y^k) = (j - k)x^j y^k
\,
Q(x jy k)=jx j1y k+1+kx j+1y k1Q(x^j y^k) = j x^{j-1} y^{k+1} + k x^{j+1} y^{k-1}
\,
P(x jy k)=i(jx j1y k+1+kx j+1y k1)P(x^j y^k) = i (-j x^{j-1} y^{k+1} + k x^{j+1} y^{k-1})

Ladder operators

Since we have been dealing with complex representations of su(2)su(2), we may as well complexify su(2)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)su(2) is the complex Lie algebra sl 2()sl_2(\mathbb{C}), and su(2)su(2) is known as the real compact form of sl 2()sl_2(\mathbb{C}).

Representations of sl 2()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=Q+iP2,A *=QiP2A = \frac{Q + i P}{2}, \qquad A^\ast = \frac{Q - i P}{2}

In matrix form,

A=(0 0 1 0),A *=(0 1 0 0)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()sl_2(\mathbb{C}):

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

In terms of derivations, we have

A=yx,A *=xyA = y \frac{\partial}{\partial x}, \qquad A^\ast = x\frac{\partial}{\partial y}

and the action of sl 2()sl_2(\mathbb{C}) on the space V nV_n of homogeneous degree nn polynomials is therefore given by

A(x jy k)=jx j1y k+1A(x^j y^k) = j x^{j-1} y^{k+1}
\,
A *(x jy k)=kx j+1y k1.A^\ast (x^j y^k) = k x^{j+1} y^{k-1}.

Thus AA lowers the exponent on xx, and A *A^\ast raises it.

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

Pretty much the same idea shows that any irreducible representation of sl 2()sl_2(\mathbb{C}) is one of the spin nn representations. The point is that any irreducible representation of su(2)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 vv, or alternatively, of a lowest weight vector. The structure of the submodule generated by v,Av,A 2v,,A nvv, A v, A^2 v, \ldots, A^n v, where A nvA^n v is a lowest weight vector, must be that of V nV_n, just by using the Lie algebra relations between AA, A *A^\ast, and HH. (See Fulton and Harris for the more detailed proof.)

Tensor products

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

g(vw)=gvgw.g (v \otimes w) = g v \otimes g w.

At the “infinitesimal” or Lie algebra level, the corresponding action of the enveloping algebra Usl 2()U sl_2(\mathbb{C}) on a tensor product is given in terms of generators xsl 2()x \in sl_2(\mathbb{C}) by

x(vw)=(xv)w+v(xw)x \cdot (v \otimes w) = (x \cdot v) \otimes w + v \otimes (x \cdot w)

(corresponding to the fact that the standard coalgebra comultiplication δ:U𝔤U𝔤U𝔤\delta: U\mathfrak{g} \to U\mathfrak{g} \otimes U\mathfrak{g} is determined by its effect on generators xx, which is δ(x)=x1+1x\delta(x) = x \otimes 1 + 1 \otimes x).

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

V nV m.V_n \otimes V_m.

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

χ VW(γ)= γ=α+βχ V(α)χ W(β)\chi_{V \otimes W}(\gamma) = \sum_{\gamma = \alpha + \beta} \chi_V(\alpha) \chi_W(\beta)

or in other words by a convolution product:

χ VW=χ V*χ W.\chi_{V \otimes W} = \chi_V \ast \chi_W.

In particular, if V,WV, W are irreducible, then χ V,χ W\chi_V, \chi_W are “uniform distributions” of finite support centered at 00, 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 mnm \leq n, then

V mV n=V n+mV n+m2V n+m4V nmV_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