6 SL₂ 6.3 Decomposing representations 6.5 SO(3) (non-examinable)

6.4 Real forms and complete decomposability

We use our understanding of the representation theory of $\mathfrak{sl}_{2,\mathbb{C}}$ into an understanding of the representation theory of $\operatorname{SL}_{2}(\mathbb{R})$ and $\mathrm{SU}(2)$ , and prove complete reducibility.

Definition 6.23.

A real form of a complex Lie algebra $\mathfrak{g}$ is a real Lie algebra $\mathfrak{h}\subset\mathfrak{g}$ such that every element $Z$ of $\mathfrak{g}$ can be written uniquely as $X+iY$ for $X,Y\in\mathfrak{h}$ .

Remark 6.24.

In lectures, we didn’t give this definition, just doing the special case of $\mathfrak{su}_{n}\subset\mathfrak{sl}_{n,\mathbb{C}}$ .

For dimension reasons, necessary and sufficient conditions are that $\mathfrak{h}\cap i\mathfrak{h}=0$ and $\dim_{\mathbb{R}}\mathfrak{h}=\dim_{\mathbb{C}}\mathfrak{g}$ .

Example 6.25.

The Lie algebra $\mathfrak{sl}_{n,\mathbb{R}}$ is a real form of $\mathfrak{sl}_{n,\mathbb{C}}$ .

Example 6.26.

The Lie algebra $\mathfrak{su}_{n}$ is a real form of $\mathfrak{sl}_{n,\mathbb{C}}$ . Indeed, $\dim_{\mathbb{R}}\mathfrak{su}_{n}=n^{2}-1=\dim_{\mathbb{C}}\mathfrak{sl}_{n,%\mathbb{C}}$ and if $X,iX\in\mathfrak{su}_{n}$ then

iX=-(iX)^{\dagger}=iX^{\dagger}=-iX

so $X=0$ .

Explicitly, we may write $A\in\mathfrak{sl}_{n,\mathbb{C}}$ as $X+iY$ with

X=\frac{1}{2}(A-A^{\dagger})

and

Y=\frac{-i}{2}(A+A^{\dagger})

in $\mathfrak{su}_{n}$ .

Example 6.27.

Recall that

\mathfrak{so}_{n}=\{A\in{}_{n,\mathbb{R}}:A+A^{T}=0\}.

Let

\mathfrak{so}_{n,\mathbb{C}}=\{A\in{}_{n,\mathbb{C}}:A+A^{T}=0\}.

Since the defining equation $A+A^{T}=0$ can be checked separately on the real and imaginary parts of $A$ ,

\mathfrak{so}_{n,\mathbb{C}}=\{B+iC:B,C\in\mathfrak{so}_{n}\}

and so $\mathfrak{so}_{n}$ is a real form of $\mathfrak{so}_{n,\mathbb{C}}$ .

Proposition 6.28.

Let $\mathfrak{h}$ be a real form of $\mathfrak{g}$ .

There is a one-to-one correspondence between representations of $\mathfrak{h}$ and complex-linear representations of $\mathfrak{g}$ under which irreducible representations correspond to irreducible representations.

Proof.

Given a $\mathbb{C}$ -linear representation of $\mathfrak{g}$ , it is a representation of $\mathfrak{h}$ by restriction. Conversely, if $(\rho,V)$ is a representation of $\mathfrak{h}$ , then it extends to a unique $\mathbb{C}$ -linear representation of $\mathfrak{g}$ given by the formula (forced by $\mathbb{C}$ -linearity)

\rho(X+iY)v=\rho(X)v+i\rho(Y)v.

It is easy to see that this preserves the Lie bracket. The proposition follows (the final statement is left as an exercise). ∎

As a corollary we immediately obtain

Theorem 6.29.

The representation theories of $\mathfrak{sl}_{n,\mathbb{R}}$ and $\mathfrak{su}_{n}$ are ‘the same’ as the complex-linear representation theory of $\mathfrak{sl}_{n,\mathbb{C}}$ . All finite-dimensional irreducible representations of $\mathfrak{sl}_{2,R}$ , $\operatorname{SL}_{2}(\mathbb{R})$ , $\mathfrak{su}_{2}$ , or $\mathrm{SU}(2)$ are of the form $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ .

Proof.

The claims about Lie algebras follow from the above discussion. Every (finite-dimensional) irreducible representation of $\mathfrak{sl}_{2,\mathbb{R}}$ is of the form $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , and these clearly exponentiate to representations of $\operatorname{SL}_{2}(\mathbb{R})$ , despite this not being a simply connected group! Similarly for $\mathrm{SU}(2)$ (which is simply connected). Since $\operatorname{SL}_{2}(\mathbb{R})$ and $\mathrm{SU}(2)$ are connected, every representation of them is determined by its derivative, so we have a complete list of the irreducible representations. ∎

Theorem 6.30.

Every finite-dimensional complex-linear representation of $\mathfrak{sl}_{n,\mathbb{C}}$ is completely reducible.

Proof.

Let $V$ be a finite-dimensional complex-linear representation of $\mathfrak{sl}_{n,\mathbb{C}}$ and let $W\subset V$ be a subrepresentation. Then $W$ is an $\mathfrak{su}_{n}$ -subrepresentation. As $\mathrm{SU}(n)$ is simply-connected, $V$ and $W$ exponentiate to representations of $\mathrm{SU}(n)$ . Since $\mathrm{SU}(n)$ is compact, by Maschke’s theorem there is a complementary $\mathrm{SU}(n)$ -subrepresentation $W^{\prime}$ with

V=W\oplus W^{\prime}.

Then $W^{\prime}$ is a $\mathfrak{su}_{n}$ -subrepresentation, and so a $\mathbb{C}$ -linear $\mathfrak{sl}_{n,\mathbb{C}}$ -subrepresentation, so that

V=W\oplus W^{\prime}

as representations of $\mathfrak{sl}_{n,\mathbb{C}}$ . Complete reducibility follows. ∎

The argument in this theorem is called Weyl’s unitary trick. For a similar application of this idea, see the proof of Proposition 6.2.