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Matrix Groups An Introduction to Lie Group Theory

Andrew Baker

Chapter 3

Tangent Spaces and Lie Algebras - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $\mathbf{k}=\mathbb{R}$ or $\mathbb{C}$.
a) Consider the 2 -dimensional $\mathbf{k}$-Lie algebras
$$
\mathbf{a}=\mathbf{k}^2, \quad \mathbf{b}=\left\{\left[\begin{array}{ll}
u & v \\
0 & 0
\end{array}\right]: u, v \in \mathbf{k}\right\},
$$
with the obvious brackets which make $a$ abelian and $b \leqslant M_2(\mathbf{k})$. Show that any 2-dimensional $k$-Lie algebra $g$ is isomorphic to $a$ if it is abelian and $b$ otherwise.
b) Find a matrix group $G \leqslant \mathrm{GL}_2(\mathbf{k})$ whose Lie algebra is $\mathbf{b}$.

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Problem 2

Let $G$ be a matrix group and $U \in G$.
a) Show that each of the functions
$$
\begin{array}{ll}
\mathrm{L}_U: G \longrightarrow G ; & \mathrm{L}_U(A)=U A, \\
\mathrm{R}_U: G \longrightarrow G ; & \mathrm{R}_U(A)=A U \\
\mathrm{C}_U: G \longrightarrow G ; & \mathrm{C}_U(A)=U A U^{-1},
\end{array}
$$
is a differentiable map and determine its derivative at $I$.
b) Using (a), show that there are $\mathbb{R}$-linear isomorphisms
$$
\lambda_U: \mathrm{T}_I G \longrightarrow \mathrm{T}_U G, \quad \rho_U: \mathrm{T}_I G \longrightarrow \mathrm{T}_U G, \quad \chi_U: \mathrm{T}_I G \longrightarrow \mathrm{T}_I G,
$$
such that for all $U, V \in G$,
$$
\lambda_{U V}=\lambda_U \circ \lambda_V, \quad \rho_{U V}=\rho_V \circ \rho_U, \quad \chi_{U V}=\chi_U \circ \chi_V .
$$

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Problem 3

For each of the following matrix groups $G$, find the Lie algebra g.
$$
\begin{array}{ll}
G_1=\left\{A \in \mathrm{GL}_2(\mathbb{R}): A^T Q_1 A=Q_1\right\}, \quad Q_1=\left[\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}\right] ; \\
G_2=\left\{A \in \mathrm{GL}_2(\mathbb{R}): A^T Q_2 A=Q_2\right\}, \quad Q_2=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] ; \\
G_3=\left\{A \in \mathrm{GL}_3(\mathbb{R}): A^T Q_3 A=Q_3\right\}, \quad Q_3=\left[\begin{array}{rcr}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1
\end{array}\right] ; \\
G_4=\operatorname{Aff}_n(\mathbf{k}) \quad(n=1,2, \ldots) ; \\
G_5=\operatorname{Symp}_{2 m}(\mathbb{R}) \quad(m=1,2, \ldots) .
\end{array}
$$

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Problem 4

Prove Proposition 3.9.

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Problem 5

Let $G$ be a matrix group with Lie algebra $g$ and let $X, Y \in g$. Show that $[X, Y]=0$ if and only if $\exp (s X) \exp (t Y)=\exp (t Y) \exp (s X)$ for all $s, t \in \mathbb{R}$.

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08:25

Problem 6

Consider the set of all $n \times n$ real special orthogonal matrices $\mathrm{SO}(n)$ and its subset
$$
U=\{A \in \operatorname{SO}(n): \operatorname{det}(I+A) \neq 0\} \subseteq \operatorname{SO}(n) .
$$

Define the function
$$
\Phi: U \longrightarrow \mathrm{M}_n(\mathbb{R}) ; \quad \Phi(A)=(I-A)(I+A)^{-1} .
$$
[ $\Phi$ is known as the real Cayley transform.] metric matrices. Hence we might as well write $\Phi: U \longrightarrow{\operatorname{Sk}-S_n}_n(\mathbb{R})$.
c) Use (b) to determine the dimension of $\mathrm{SO}(n)$.

Ely Crowder
Ely Crowder
Numerade Educator

Problem 7

Consider the set of all $n \times n$ unitary matrices $\mathrm{U}(n)$ and its subset
$$
V=\{A \in \mathrm{U}(n): \operatorname{det}(I+A) \neq 0\} \subseteq \mathrm{U}(n) .
$$

Define the function
$$
\Theta: V \longrightarrow \mathrm{M}_n(\mathbb{C}) ; \quad \Theta(A)=(I-A)(I+A)^{-1} .
$$
[ $\Theta$ is known as the complex Cayley transform.]
a) Show that $\operatorname{im} \Theta=\mathrm{Sk}_{\mathrm{H}} \mathrm{Herm}_n(\mathbf{C})$, the set of all $n \times n$ skew hermi-
c) Use (b) to determine the dimension of $\mathrm{U}(n)$.

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Problem 8

For $n \geqslant 1$, prove the following:
a) $\mathrm{O}(n)$ is the semi-direct product $\{1,-1\} \times \operatorname{SO}(n)$;
b) $\mathrm{U}(n)$ is the semi-direct product $\mathrm{T} \times \mathrm{SU}(n)$, where
$$
\mathbb{T}=\{z \in \mathbb{C}:|z|=1\}
$$
is the unit circle;
c) $\mathrm{GL}_n(\mathbb{R})$ is the semi-direct product $\mathbb{R}^{\times} \times \mathrm{SL}_n(\mathbb{R})$;
d) $\mathrm{GL}_n(\mathbb{C})$ is the semi-direct product $\mathbb{C}^{\times} \times \mathrm{SL}_n(\mathbf{C})$.

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Problem 9

Verify the formula of Lemma 3.29.

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