The Lumenary · First-year graduate

Lie groups and Lie algebras

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You will learn to

Define Lie groups and Lie algebras, and compute the Lie algebras of the classical groups $GL(n)$ , $SL(n)$ , $O(n)$ , $SO(n)$ , $U(n)$ and $SU(n)$
Use the exponential map and one-parameter subgroups, prove that homomorphisms are determined by their differentials on connected groups, and state the Lie correspondence between connected Lie subgroups and Lie subalgebras
Prove that the exponential map of $SL(2, \R)$ is not surjective, and explain why $SU(2)$ and $SO(3)$ have isomorphic Lie algebras but are not isomorphic
Prove complete reducibility for compact groups from the existence of Haar measure
Classify the finite-dimensional irreducible representations of $\mathfrak{sl}(2, \C)$ , and describe the root system of $\mathfrak{sl}(3, \C)$

1.
What is the dimension of $SU(3)$ as a real Lie group?
2.
Which of these matrices belong to the Lie algebra $\mathfrak{su}(2)$ ? (Choose all that apply.)
- A. $\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$
- B. $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
- C. $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
- D. $\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}$
3.
Which element of $SL(2, \R)$ is not of the form $e^X$ with $X \in \mathfrak{sl}(2, \R)$ ?
- A. $\begin{pmatrix} -2 & 0 \\ 0 & -1/2 \end{pmatrix}$
- B. $-I$
- C. $\begin{pmatrix} 2 & 0 \\ 0 & 1/2 \end{pmatrix}$
- D. $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
4.
$SU(2)$ and $SO(3)$ have isomorphic Lie algebras. Which property distinguishes the two groups?
- A. $SU(2)$ is simply connected, while $\pi_1(SO(3)) \cong \Z/2$
- B.Their dimensions
- C.Compactness
- D.Connectedness
5.
In the classification of irreducible $\mathfrak{sl}(2, \C)$ -representations, why must the string $v, fv, f^2v, \dots$ eventually reach $0$ ?
- A.Its non-zero members are $h$ -eigenvectors with distinct eigenvalues $\mu, \mu - 2, \mu - 4, \dots$ , hence linearly independent, and $V$ is finite-dimensional
- B.Because $f$ is nilpotent as a matrix in $\mathfrak{sl}(2)$
- C.Because $\mu$ is a natural number
- D.Because $V$ is irreducible
6.
In the representation $V_2 \otimes V_2$ of $\mathfrak{sl}(2, \C)$ , what is the dimension of the eigenspace of $h$ for the eigenvalue (weight) $0$ ?
7.
Decompose $V_3 \otimes V_1$ into irreducible representations $V_n$ . List the values of $n$ that occur, separated by commas.
8.
For real $n \times n$ matrices, with exp the matrix exponential, which of these Lean statements is false in general (so it could not be proved)?
- A.∀ X Y, exp (X + Y) = exp X * exp Y
- B.∀ X (s t : ℝ), exp ((s + t) • X) = exp (s • X) * exp (t • X)
- C.∀ A, Aᵀ = -A → (exp A)ᵀ * exp A = 1
- D.∀ X, exp (-X) * exp X = 1
9.
Write exp_sl2(a, b, c) returning the matrix exponential of $X = \begin{pmatrix} a & b \\ c & -a \end{pmatrix}$ as a nested list, using $X^2 = \delta I$ with $\delta = a^2 + bc$ : $e^X = C(\delta) I + S(\delta) X$ , where $C = \cosh\sqrt\delta$ , $S = \sinh\sqrt\delta/\sqrt\delta$ if $\delta > 0$ ; $C = \cos\sqrt{-\delta}$ , $S = \sin\sqrt{-\delta}/\sqrt{-\delta}$ if $\delta < 0$ ; and $C = S = 1$ if $\delta = 0$ .
Computational — Practice

Answer key — Lie groups and Lie algebras

1.
A skew-Hermitian $3 \times 3$ matrix has $3$ imaginary diagonal entries ( $3$ real parameters) and $3$ complex entries above the diagonal ( $6$ real parameters): $\dim\mathfrak u(3) = 9$ . Trace zero removes one: $\dim SU(3) = \dim\mathfrak{su}(3) = 8$ — the eight gluons of quantum chromodynamics, and the octet of the eightfold way.
2.
(a) and (b) are skew-Hermitian with trace zero. (c) is Hermitian. (d) is skew-Hermitian but has trace $2i$ , so it lies in $\mathfrak u(2)$ and not in $\mathfrak{su}(2)$ ; its exponentials $e^{it}I$ are the centre of $U(2)$ , outside $SU(2)$ unless $e^{2it} = 1$ .
3.
By Theorem 6, $\operatorname{tr} e^X \in \{2\cosh\mu\} \cup \{2\} \cup \{2\cos\theta\}$ , always $\ge -2$ , with equality only for $e^X = -I$ . The matrix $\operatorname{diag}(-2, -\tfrac12)$ has trace $-5/2$ , so it is not an exponential, although it is the product $e^{\pi(f-e)}e^{(\log 2)h}$ of two. The others are exponentials as shown in the feedback.
4.
The Lie algebra determines the simply connected group $SU(2)$ ; $SO(3)$ is its quotient by the central subgroup $\{\pm I\}$ . So $SU(2)$ is simply connected with centre $\{\pm I\}$ , while $SO(3)$ has $\pi_1 = \Z/2$ (the loop of rotations by $0$ to $2\pi$ about a fixed axis) and trivial centre. Dimension, compactness and connectedness agree.
5.
From $hf = f(h - 2)$ we get $h f^kv = (\mu - 2k)f^kv$ , so each non-zero $f^kv$ is an eigenvector of $h$ with its own eigenvalue. Eigenvectors for distinct eigenvalues are linearly independent, so only finitely many can be non-zero: some $f^{n+1}v = 0$ with $f^nv \ne 0$ . Then $0 = ef^{n+1}v = (n+1)(\mu - n)f^nv$ gives $\mu = n$ .
6.
The weights of $V_2 \otimes V_2$ are the sums $a + b$ with $a, b \in \{2, 0, -2\}$ ; the sum $0$ arises from $(2, -2)$ , $(0, 0)$ and $(-2, 2)$ , so the zero weight space is $3$ -dimensional. This matches Clebsch–Gordan, $V_2 \otimes V_2 \cong V_4 \oplus V_2 \oplus V_0$ : each summand contributes one weight- $0$ vector.
7.
The weights are $4, 2, 2, 0, 0, -2, -2, -4$ . The highest, $4$ , gives $V_4$ (weights $4, 2, 0, -2, -4$ ); what remains, $2, 0, -2$ , is $V_2$ . So $V_3 \otimes V_1 \cong V_4 \oplus V_2$ , with dimensions $5 + 3 = 8$ , as Clebsch–Gordan predicts ( $V_{m+n} \oplus \dots \oplus V_{|m-n|}$ ). In physics: spin $3/2$ combined with spin $1/2$ gives spin $2$ or spin $1$ .
8.
$e^{X+Y} = e^Xe^Y$ holds when $XY = YX$ and fails otherwise; for $X = \begin{pmatrix}0&1\\0&0\end{pmatrix}$ , $Y = \begin{pmatrix}0&0\\1&0\end{pmatrix}$ , $e^Xe^Y = \begin{pmatrix}2&1\\1&1\end{pmatrix}$ while $e^{X+Y} = \begin{pmatrix}\cosh 1&\sinh 1\\\sinh 1&\cosh 1\end{pmatrix}$ . Statements (b), (c), (d) involve only commuting pairs and are proved in the formal section or follow from the same lemma.
9.
```
import math

def exp_sl2(a, b, c):
    delta = a * a + b * c
    if delta > 0:
        r = math.sqrt(delta)
        C, S = math.cosh(r), math.sinh(r) / r
    elif delta < 0:
        r = math.sqrt(-delta)
        C, S = math.cos(r), math.sin(r) / r
    else:
        C, S = 1.0, 1.0
    return [[C + S * a, S * b], [S * c, C - S * a]]
```
Since $\delta = -\det X$ , this is the formula from the proof of Theorem 6, and the trace of the result is $2C \ge -2$ in every case. The tests also check $\det e^X = 1 = e^{\operatorname{tr} X}$ . (In floating point, $\delta$ very close to $0$ should really use Taylor expansions of $C$ and $S$ ; the exact formula is what matters for the theorem.)