Chapter 2

Lie Groups

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Problem 2.1

A (global) group of $n \times n$ matrices is compact if it is a bounded, closed subset of the set of all $n \times n$ matrices. A set $U$ of $n \times n$ matrices is bounded if there exists a constant $K > 0$ such that $\left|A_{ik}\right| \leq K$ for $1 \leq i, k \leq n$ and all $A \in U$ . The set $U$ is closed provided every Cauchy sequence in $U$ converges to a matrix in $U$ . A sequence of $n \times n$ matrices $\left\{ A^{(p)}\right\}$ is a Cauchy sequence if each of the sequences of matrix elements $\left\{ A_{ik}^{(p)}\right\}$ , $1 \leq i, k \leq n$ is Cauchy. The orthogonal group $O(3, \mathbb {R})$ consists of all $3 \times 3$ matrices $A$ over $\mathbb {R}$ with $AA^{T} = I_{3}$ . Show that the orthogonal group $O(3, \mathbb {R})$ is compact.

Problem 2.2

(i)(ii)(iii)(iv)

The Lie group $SU(1,1)$ consists of the set of all $2 \times 2$ pseudo-unitary matrices (of determinant $1$ ) preserving the quadratic form

\left|z_{1}\right|^{2} - \left|z_{2}\right|^{2}, \quad (z_{1}, z_{2} \in \mathbb {C}) .

(i)

Show that the $2 \times 2$ matrix

U(\tau , \alpha , \beta ) = \begin{pmatrix} \cosh (\tau /2) e^{-i\alpha } & \sinh (\tau /2) e^{-i\beta } \\ \sinh (\tau /2) e^{i\beta } & \cosh (\tau /2) e^{i\alpha } \end{pmatrix} \tag {1}

preserves the quadratic form $\left|z_{1}\right|^{2} - \left|z_{2}\right|^{2}$ $(z_{1}, z_{2} \in \mathbb {C})$ , where $\tau , \alpha , \beta \in \mathbb {R}$ .

(ii)

Show that $\operatorname {det}\left(U(\tau , \alpha , \beta )\right) = 1$ .

(iii)

Give the inverse of $U(\tau , \alpha , \beta )$ .

(iv)

Show that $U_{1}(\tau_{1}, \alpha_{1}, \beta_{1}) U_{2}(\tau_{2}, \alpha_{2}, \beta_{2})$ with $\alpha_{1} + \alpha_{2} = \beta_{1} - \beta_{2}$ is again a matrix of the form (1).

Problem 2.3

The underlying field is the real numbers $\mathbb {R}$ . The elements of the Lie group $SO(n)$ satisfy $A^{T}A = I_{n}$ and $\operatorname {det}\left(A\right) = 1$ . Given any rotation matrix, $R \in SO(n)$ . If $R$ does not admit $-1$ as an eigenvalue, then there is a unique skew symmetric matrix, $S$ , ( $S^{T} = -S$ ) so that

R = (I_{n} - S)(I_{n} + S)^{-1} .

The matrix $R$ is called the Cayley transform of $S$ . Let $n = 2$ and

S = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

Find $R$ .

Problem 2.4

(i)(ii)

The group commutator $[A, B]$ of two elements $A$ and $B$ of a group $G$ is defined by

[A, B] := A^{-1}B^{-1}AB .

The commutator is a kind of measure as to how near $A$ and $B$ come to commuting, $[A, B]$ being the identity element of $G$ if and only if $AB = BA$ .

(i)

Consider a matrix group with matrix multiplication as composition. Calculate $\operatorname {det}\left([A,B]\right)$ .

(ii)

The Lie group $U(2)$ consists of all $2 \times 2$ unitary matrices $U$ , i.e. $UU^{*} = I_{2}$ . Now

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad B = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

are elements of $U(2)$ . Calculate $[A, B]$ .

Problem 2.5

The rotation matrix

A(\phi ) = \begin{pmatrix} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{pmatrix}

is an element of $SO(2)$ , i.e. $A(\phi )A^{T}(\phi ) = I_{2}$ and $\operatorname {det}\left(A(\phi )\right) = 1$ . Is the matrix

B(\phi ) = \begin{pmatrix} \cos \phi & 0 & 0 & \sin \phi \\ 0 & \cos \phi & \sin \phi & 0 \\ 0 & -\sin \phi & \cos \phi & 0 \\ -\sin \phi & 0 & 0 & \cos \phi \end{pmatrix}

an element of $SO(4)$ ?

Problem 2.6

Let $\epsilon \in \mathbb {R}$ . The matrix

B(\epsilon ) = \begin{pmatrix} 1 & 0 & \epsilon \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

is an element of the Lie group $SL(3, \mathbb {R})$ . Find a matrix $A$ such that $\mathrm{exp}\left(\epsilon A\right) = B(\epsilon )$ .

Problem 2.7

(i)(ii)(iii)

Consider the $2 \times 2$ matrix

g = \begin{pmatrix} a+ib & c+id \\ -c+id & a-ib \end{pmatrix}

where $a, b, c, d \in \mathbb {R}$ and $a^{2}+b^{2}+c^{2}+d^{2} = 1$ . Thus $g \in SU(2)$ . Let $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ be the Pauli spin matrices. We define $\tau_{j} = \frac{1}{2}\sigma_{j}$ for $j = 1, 2, 3$ .

(i)

Calculate

g^{-1}\tau _{1}g, \qquad g^{-1}\tau _{2}g, \qquad g^{-1}\tau _{3}g .

(ii)

Find the $3 \times 3$ matrix $G$ from the relation

(g^{-1}\tau _{1}g, g^{-1}\tau _{2}g, g^{-1}\tau _{3}g) = (\tau _{1}, \tau _{2}, \tau _{3})G

where $(\tau_{1}, \tau_{2}, \tau_{3})G$ is defined as

(\tau _{1}, \tau _{2}, \tau _{3})G = (\tau _{1}G_{11}+\tau _{2}G_{21}+\tau _{3}G_{31}, \ \tau _{1}G_{12}+\tau _{2}G_{22}+\tau _{3}G_{32}, \ \tau _{1}G_{13}+\tau _{2}G_{23}+\tau _{3}G_{33}) .

(iii)

Let

M = \begin{pmatrix} 0 & -b & c \\ b & 0 & -d \\ -c & d & 0 \end{pmatrix} .

Show that $G$ can be written as

G = I_{3} + 2aM + 2M^{2} . \tag {1}

Problem 2.8

(i)(ii)(iii)

(i)

Consider the Lie group $GL(2, \mathbb {R})$ . Let

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL(2, \mathbb {R}) .

Thus $\operatorname {det}\left(A\right) \neq 0$ . Show that the so called Mobius transformation

z \to \frac{az+b}{cz+d}

of the extended complex plane $\mathbb {C} \cup \left\{ \infty \right\}$ forms a group under function composition.

(ii)

Show that this group is isomorphic to the quotient of $GL(2, \mathbb {R})$ by its center.

(iii)

Show that the Mobius group is isomorphic to $SL(2, \mathbb {R})$ .

Problem 2.9

(i)(ii)(iii)

(i)

Let $A \in SL(2, \mathbb {R})$ , i.e. $\operatorname {det}\left(A\right) = 1$ . Show that

A^{2} - \tr (A)A + I_{2} = 0_{2} .

(ii)

Let $A \in SL(2, \mathbb {R})$ . Show that $\tr (A) = \tr (A^{-1})$ .

(iii)

$A, B \in SL(2, \mathbb {R})$ . Show that $\tr (AB) = \tr A \, \tr B - \tr (AB^{-1})$ .

Problem 2.10

(i)(ii)

(i)

Let $A \in SL(2, \mathbb {C})$ . Show that

A^{2} = \tr (A)A - I_{2}, \qquad A + A^{-1} = \tr (A)I_{2} .

(ii)

Show that

A^{n} = U_{n-1}(x)A - U_{n-2}I_{2}

where $x := \frac{1}{2}\tr A$ and $U_{n}(x)$ are Chebyshev’s polynomial of the second kind defined by

\begin{align} U_{-1}(x) & := 0, \qquad U_{0}(x) := 1 \\ U_{n+1}(x) & := 2xU_{n}(x) - U_{n-1}(x), \qquad n = 1, 2, \ldots . \end{align}

Apply the Cayley-Hamilton theorem. The Cayley-Hamilton theorem states that every $n \times n$ matrix over a commutative ring (such as $\mathbb {R}$ , $\mathbb {C}$ , $\operatorname {char}\mathbb {F} = 2$ ) satisfies its own characteristic equation.

Problem 2.11

Consider the Lie group $SL(2, \mathbb {R})$ and its Lie algebra $s\ell (2, \mathbb {R})$ . The Lie algebra $s\ell (2, \mathbb {R})$ consists of all $2 \times 2$ matrices over $\mathbb {R}$ with trace equal to $0$ . Let

A = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \in SL(2, \mathbb {R})

with eigenvalues $-1$ (twice). Note that the matrix $A$ is not normal, i.e. $AA^{T} \neq A^{T}A$ . Show that $A \notin \mathrm{exp}\left(s\ell (2,\mathbb {R})\right)$ . This shows that the map $\exp : s\ell (2, \mathbb {R}) \to SL(2, \mathbb {R})$ is not surjective although the Lie group $SL(2, \mathbb {R})$ is connected.

Problem 2.12

Given the invertible $4 \times 4$ matrix

G = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Find two non-diagonal matrices $L$ over $\mathbb {R}$ such that $L^{T}GL = G$ . Such transformations are known as Lorentz transformations.

Problem 2.13

The group $Sp(2n, \mathbb {R})$ consists of all real $2n \times 2n$ matrices $S$ which obey the condition

S^{T}JS = J

where $J$ is the $2n \times 2n$ skew-symmetric matrix

J := \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix}

with $I_{n}$ the $n \times n$ identity matrix and $0_{n}$ the $n \times n$ zero matrix. Let $V$ be a $2n \times 2n$ real symmetric positive definite matrix. Show that there exists an $S \in Sp(2n, \mathbb {R})$ such that

S^{T}VS = D^{2} > 0, \qquad D^{2} = \operatorname {diag}(\kappa _{1}, \kappa _{2}, \ldots , \kappa _{n}, \kappa _{1}, \kappa _{2}, \ldots , \kappa _{n}) .

Problem 2.14

Consider the trace-less matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad C = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .

Let $t \in \mathbb {R}$ . Show that the matrices

e^{tA}, \quad e^{tB}, \quad e^{tC}

are elements of the Lie group $SL(2, \mathbb {R})$ .

Problem 2.15

(i)(ii)

The group of complex rotations $O(n, \mathbb {C})$ is defined as the group of all $n \times n$ complex matrices $O$ , such that $OO^{T} = O^{T}O = I_{n}$ , where $T$ means transpose. These transformations preserve the real scalar product

\overrightarrow {x} \cdot \overrightarrow {y} = \sum _{j=1}^{n} x_{j}y_{j}

so that $(O\overrightarrow {x}) \cdot O\overrightarrow {y} = \overrightarrow {x} \cdot \overrightarrow {y}$ , where $\overrightarrow {x}$ and $\overrightarrow {y}$ are complex vectors in general, i.e. $x_{j}, y_{j} \in \mathbb {C}$ .

(i)

Show that the $2 \times 2$ matrix ( $\alpha \in \mathbb {R}$ )

O = \begin{pmatrix} \cosh \alpha & i\sinh \alpha \\ -i\sinh \alpha & \cosh \alpha \end{pmatrix}

is an element of $O(2, \mathbb {C})$ .

(ii)

Find the partial derivatives under complex orthogonal transformations $O \in O(n, \mathbb {C})$

w_{j}(\overrightarrow {x}) := (O\overrightarrow {x})_{j} = \sum _{k=1}^{n} O_{jk}x_{k}, \qquad j = 1, 2, \ldots , n

i.e. $\partial /\partial w_{j}$ with $j = 1, 2, \ldots , n$ .

Problem 2.16

(i)(ii)(iii)

(i)

The non-compact Lie group $SL(2, \mathbb {C})$ consists of all $2 \times 2$ matrices over $\mathbb {C}$ with determinant equal to $1$ . Give a $2 \times 2$ matrix $A$ which is an element of $SL(2, \mathbb {C})$ but not an element of the compact Lie group $SU(2)$ .

(ii)

The maximal compact Lie subgroup of $SL(3, \mathbb {R})$ is the Lie group $SO(3)$ . Give an element which is in $SL(3, \mathbb {R})$ but not in $SO(3)$ .

(iii)

Consider the Lie group $SL(2, \mathbb {C})$ . Give a compact and non-compact subgroup of $SL(2, \mathbb {C})$ .

Problem 2.17

Consider the Lie group $SL(2, \mathbb {R})$ , i.e. the set of all real $2 \times 2$ matrices with determinant equal to $1$ . A dynamical system in $SL(2, \mathbb {R})$ can be defined by

M_{k+2} = M_{k}M_{k+1}, \qquad k = 0, 1, 2, \ldots \tag {1}

with the initial matrices $M_{0}, M_{1} \in SL(2, \mathbb {R})$ . Let $F_{k} := \tr M_{k}$ . Show that

F_{k+3} = F_{k+2}F_{k+1} - F_{k}, \qquad k = 0, 1, 2, \ldots . \tag {2}

Problem 2.18

The Lie group $SO(m,n)$ consists of all real matrices $S$ that satisfy

S^{T}gS = g

where $\operatorname {det}\left(S\right) = 1$ and $g = \operatorname {diag}(+1,+1,\ldots ,+1,-1,\ldots ,-1)$ with $n$ $+1$ 's and $m$ $-1$ 's. Let $V$ be a real symmetric positive definite matrix of dimension $N$ . Show that for any choice of partition $N = m+n$ , there exists an $S \in SO(m,n)$ such that

S^{T}VS = D^{2} = \text{diagonal (and} > 0) .

Problem 2.19

The compact Lie group $SO(3, \mathbb {R})$ has a realization as the group of all $3 \times 3$ real matrices $A$ such that $A^{T}A = I_{3}$ and $\det A = 1$ . This is the natural realization of $SO(3, \mathbb {R})$ as the group of all rotations in the vector space $\mathbb {R}^{3}$ which leave the origin fixed. One convenient parameterization of $SO(3, \mathbb {R})$ is in terms of the Euler angles. A rotation through angle $\phi$ about the $z$ -axis is given by

R_{z}(\phi ) = \begin{pmatrix} \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{pmatrix} \in SO(3, \mathbb {R})

and rotations through angle $\phi$ about the $x$ and $y$ axis are given by

R_{x}(\phi ) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \phi & -\sin \phi \\ 0 & \sin \phi & \cos \phi \end{pmatrix} \in SO(3, \mathbb {R})

R_{y}(\phi ) = \begin{pmatrix} \cos \phi & 0 & \sin \phi \\ 0 & 1 & 0 \\ -\sin \phi & 0 & \cos \phi \end{pmatrix} \in SO(3, \mathbb {R})

respectively. Differentiate each of these matrices with respect to $\phi$ and set $\phi = 0$ to find a basis $L_{x}$ , $L_{y}$ , $L_{z}$ of the simple Lie algebra $so(3)$ . Calculate the commutators. Find $\mathrm{exp}\left(\phi L_{x}\right)$ , $\mathrm{exp}\left(\phi L_{y}\right)$ , $\mathrm{exp}\left(\phi L_{z}\right)$ .

Problem 2.20

(i)(ii)

Consider the symplectic group, $Sp(2, \mathbb {R})$ , consisting of $2 \times 2$ real matrices $M$ satisfying

MNM^{T} = N, \qquad N = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

These matrices can also be characterized by the condition, $\operatorname {det}\left(M\right) = 1$ , i.e. $Sp(2, \mathbb {R})$ is identical with the group $SL(2, \mathbb {R})$ of all $2 \times 2$ real matrices of determinant $1$ .

(i)

Show that the matrix

A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}

is an element of $Sp(2, \mathbb {R})$ . Find $\operatorname {det}\left(A\right)$ .

(ii)

An element $M \in Sp(2, \mathbb {R})$ has the decomposition

M = \begin{pmatrix} 1 & 0 \\ -v & 1 \end{pmatrix} \begin{pmatrix} u^{-1/2} & 0 \\ 0 & u^{1/2} \end{pmatrix} \begin{pmatrix} \cos \theta & -\sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}

where $v \in \mathbb {R}$ , $u > 0$ , $0 < \theta \leq 2\pi$ . Find the conditions on $v, u, \theta$ .

Problem 2.21

We know that all $n \times n$ unitary matrices form the Lie group $U(n)$ under matrix multiplication. Let $U$ be an $n \times n$ unitary matrix. Is $\mathrm{exp}\left(U\right)$ unitary? Is $\mathrm{exp}\left(i(U+U^{*})\right)$ unitary?

Problem 2.22

Consider the Lie group $U(n)$ . The eigenvalues of a unitary matrix take the form $e^{i\alpha }$ , where $\alpha \in \mathbb {R}$ . Consider the unitary matrix

U = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} .

Find the eigenvalues of $U$ and the group generated by the eigenvalues under multiplication.

Problem 2.23

(i)(ii)(iii)

(i)

Let $K$ be a skew-hermitian $n \times n$ matrix, i.e. $K^{*} = -K$ , and $\epsilon \in \mathbb {R}$ . Then

V(\epsilon ) := \mathrm{exp}\left(\epsilon K\right)

is a unitary matrix. The unitary matrices form a Lie group under matrix multiplication. Given $V(\epsilon )$ reconstruct the skew-hermitian matrix $K$ .

(ii)

Apply the reconstruction technique to the unitary matrix

V(\epsilon ) = \begin{pmatrix} \cos (\epsilon ) & i\sin (\epsilon ) \\ i\sin (\epsilon ) & \cos (\epsilon ) \end{pmatrix} .

(iii)

Apply the reconstruction technique to the unitary matrix

V(\epsilon ) = \begin{pmatrix} 0 & e^{i\epsilon } \\ e^{i\epsilon } & 0 \end{pmatrix} .

Problem 2.24

(i)(ii)

(i)

Consider the semisimple Lie group $SL(2, \mathbb {R})$ . Let $A \in SL(2, \mathbb {R})$ . Then $A^{-1} \in SL(2, \mathbb {R})$ . Show that $A$ and $A^{-1}$ have the same eigenvalues.

(ii)

Is this still true for $A \in SL(3, \mathbb {R})$ ?

Problem 2.25

(i)(ii)

Consider the $2 \times 2$ matrices

A_{j} = \begin{pmatrix} 1 & 1/j \\ 0 & 1 \end{pmatrix}, \qquad j = 1, 2, \ldots , n

and the $2 \times 2$ matrices

B_{j} = \begin{pmatrix} 1 & 0 \\ 1/j & 1 \end{pmatrix}, \qquad j = 1, 2, \ldots , n .

Obviously, these matrices are elements of the Lie group $SL(2, \mathbb {R})$ .

(i)

Calculate the inverse matrices $A_{j}^{-1}$ , $B_{j}^{-1}$ .

(ii)

Calculate the matrix products

A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}, \qquad A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1} .

Problem 2.26

Let $X, Y$ be real Banach spaces. Let $L(X,Y)$ be the space of bounded linear operators. If the operator $f$ is differentiable at $u \in X$ then for every $v \in X$

Df(u)(v) = \left. \frac{d}{d\epsilon } f(u+\epsilon v) \right|_{\epsilon =0} .

This is called the directional derivative (Gateaux derivative) of $f$ at $u$ in the direction of $v$ . In the following we have $X = Y$ . Consider the Banach space of all real $n \times n$ matrices with the norm induced by the scalar product

\left\langle A, B \right\rangle := \tr (AB^{*}) .

Let $f$ be the map $A \mapsto A^{-1}$ in the Lie group $GL(n, \mathbb {R})$ . Show that

Df(A)(B) = -A^{-1}BA^{-1} .

Problem 2.27

Show that $SU(1,1,\mathbb {C})$ is isomorphic to $SL(2, \mathbb {R})$ .

Problem 2.28

Show that $SU(2)$ is the double covering group of $SO(3)$ .

Problem 2.29

Let $t \in \mathbb {R}$ . Consider the matrices

A_{1}(t) = \begin{pmatrix} e^{it} & 0 \\ 0 & 1 \end{pmatrix}, \qquad A_{2}(t) = \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \qquad A_{3}(t) = \begin{pmatrix} 1 & it \\ 0 & 1 \end{pmatrix}

which are elements in the Lie groups $GL(2, \mathbb {R})$ . Find matrices $X_{1}, X_{2}, X_{3}$ (infinitesimal generators) such that

A_{1}(t) = e^{tX_{1}}, \qquad A_{2}(t) = e^{tX_{2}}, \qquad A_{3}(t) = e^{tX_{3}} .

Problem 2.30

(i)(ii)

The following three matrices are elements of the Lie group $SL(2, \mathbb {R})$ with $t \in \mathbb {R}$

A_{1}(t) = \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \qquad A_{2}(t) = \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}, \qquad A_{3}(t) = \begin{pmatrix} e^{t} & 0 \\ 0 & e^{-t} \end{pmatrix} .

(i)

Find $2 \times 2$ matrices $X_{1}, X_{2}, X_{3}$ such that

A_{1}(t) = e^{tX_{1}}, \qquad A_{2}(t) = e^{tX_{2}}, \qquad A_{3}(t) = e^{tX_{3}} .

(ii)

Calculate the commutators $[X_{1}, X_{2}]$ , $[X_{3}, X_{2}]$ , $[X_{3}, X_{1}]$ .

Problem 2.31

One-parameter subgroups of the Lie group $SL(2, \mathbb {R})$ are given by

\begin{align} A_{1}(t) & = \begin{pmatrix} e^{t/2} & 0 \\ 0 & e^{-t/2} \end{pmatrix} \\ A_{2}(\tau ) & = \begin{pmatrix} \cosh (\tau /2) & \sinh (\tau /2) \\ \sinh (\tau /2) & \cosh (\tau /2) \end{pmatrix} \\ A_{3}(\phi ) & = \begin{pmatrix} \cos (\phi /2) & \sin (\phi /2) \\ -\sin (\phi /2) & \cos (\phi /2) \end{pmatrix} . \end{align}

Find the infinitesimal generators.

Problem 2.32

Consider the subgroups of the Lie group $SU(1,1)$

\begin{pmatrix} e^{i\theta /2} & 0 \\ 0 & e^{-i\theta /2} \end{pmatrix}, \qquad \begin{pmatrix} \cosh (t/2) & i\sinh (t/2) \\ -i\sinh (t/2) & \cosh (t/2) \end{pmatrix}, \qquad \begin{pmatrix} 1+\zeta i/2 & \zeta /2 \\ \zeta /2 & 1-\zeta i/2 \end{pmatrix}

where $\theta , t, \zeta \in \mathbb {R}$ . Find the infinitesimal generators.

Problem 2.33

Consider the subgroups of the Lie group $SU(2)$

\begin{pmatrix} e^{i\theta /2} & 0 \\ 0 & e^{-i\theta /2} \end{pmatrix}, \qquad \begin{pmatrix} \cos (t/2) & -\sin (t/2) \\ \sin (t/2) & \cos (t/2) \end{pmatrix}, \qquad \begin{pmatrix} \cos (t/2) & i\sin (t/2) \\ i\sin (t/2) & \cos (t/2) \end{pmatrix}

where $t \in \mathbb {R}$ . Find the infinitesimal generators.

Problem 2.34

Consider $2 \times 2$ real matrices $A$ satisfying the following two conditions. First $A$ is a rotation matrix, i.e. its rows (and therefore its columns) are pairwise orthonormal. Second we impose $A^{2} = -I_{2}$ . Find all these matrices.

Problem 2.35

Let $\alpha \in \mathbb {R}$ . The $4 \times 4$ matrices

A(\alpha ) = \begin{pmatrix} \cosh \alpha & 0 & 0 & \sinh \alpha \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\sinh \alpha & 0 & 0 & \cosh \alpha \end{pmatrix}

form a Lie group. Find the infinitesimal generator.

Problem 2.36

Let $\alpha , \beta \in \mathbb {R}$ . Let $T$ be the $2 \times 2$ unitary diagonal matrix

T(\alpha , \beta ) = \begin{pmatrix} e^{i\alpha } & 0 \\ 0 & e^{i\beta } \end{pmatrix} .

Find the condition on a unitary $2 \times 2$ matrix $U$ such that

UT(\alpha , \beta )U^{-1} = T(\alpha ', \beta ') \equiv \begin{pmatrix} e^{i\alpha '} & 0 \\ 0 & e^{i\beta '} \end{pmatrix} .

Problem 2.37

Consider the compact Lie group $SO(2, \mathbb {R})$ . Let

A(\phi ) = \begin{pmatrix} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{pmatrix}

be an element of $SO(2, \mathbb {R})$ . Find a $2 \times 2$ matrix $B$ such that $B^{-1}A(\phi )B$ is diagonal. To find $B$ calculate the eigenvalues and normalized eigenvectors of $A(\phi )$ . Is $B$ an element of the Lie group $SO(2, \mathbb {R})$ ?

Problem 2.38

Consider the Lie group $SO(1,1,\mathbb {R})$ . Let

A(\phi ) = \begin{pmatrix} \cosh \phi & \sinh \phi \\ \sinh \phi & \cosh \phi \end{pmatrix}

be an element of $SO(1,1,\mathbb {R})$ . Find a $2 \times 2$ matrix $B$ such that $B^{-1}A(\phi )B$ is diagonal. To find $B$ calculate the eigenvalues and normalized eigenvectors of $A(\phi )$ . Is $B$ an element of the Lie group $SO(1,1,\mathbb {R})$ ?

Problem 2.39

Consider the Lie group $GL(2, \mathbb {R})$ . Show that the Lie group $SL(2, \mathbb {R})$ is a normal subgroup of $GL(2, \mathbb {R})$ . The group elements of $SL(2, \mathbb {R})$ have determinant $1$ . Apply the theorem:

A subgroup $H$ of the group $G$ is normal in $G$ if and only if $gHg^{-1} \subseteq H$ for all $g$ in $G$ .

Problem 2.40

Consider the Lie group $SL(2, \mathbb {R})$ . Show that each element of $SL(2, \mathbb {R})$ can be generated by the two matrices

\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}

where $t \in \mathbb {R}$ .

Problem 2.41

(i)(ii)

(i)

Find an invertible $2 \times 2$ matrix $X$ , i.e. $X \in GL(2, \mathbb {C})$ , such that

X \begin{pmatrix} a & b \\ c & d \end{pmatrix} X^{-1} = \begin{pmatrix} a & -b \\ -c & d \end{pmatrix} .

(ii)

Find an invertible $2 \times 2$ matrix $Y$ , i.e. $Y \in GL(2, \mathbb {C})$ , such that

Y \begin{pmatrix} a & b \\ c & d \end{pmatrix} Y^{-1} = \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} .

Problem 2.42

(i)(ii)

The Pauli spin matrices are elements of the Lie group $U(2)$ . Additionally they are hermitian.

(i)

Let $z \in \mathbb {C}$ . Find

\mathrm{exp}\left(z\sigma _{x}\right), \qquad \mathrm{exp}\left(z\sigma _{y}\right), \qquad \mathrm{exp}\left(z\sigma _{z}\right) .

(ii)

Find

\mathrm{exp}\left(-z\sigma _{x}\right)\sigma _{y}\mathrm{exp}\left(z\sigma _{x}\right), \qquad \mathrm{exp}\left(-z\sigma _{y}\right)\sigma _{z}\mathrm{exp}\left(z\sigma _{y}\right), \qquad \mathrm{exp}\left(-z\sigma _{z}\right)\sigma _{x}\mathrm{exp}\left(z\sigma _{z}\right) .

Problem 2.43

The Pauli spin matrices are elements of the Lie group $U(2)$ . Additionally they are hermitian. Let $\otimes$ be the Kronecker product. Then $\sigma_{x} \otimes \sigma_{x}$ , $\sigma_{y} \otimes \sigma_{y}$ , $\sigma_{z} \otimes \sigma_{z}$ are elements of the Lie group $U(4)$ . The matrices are also hermitian. Let $z \in \mathbb {C}$ . Find

\mathrm{exp}\left(z\sigma _{x} \otimes \sigma _{x}\right), \qquad \mathrm{exp}\left(z\sigma _{y} \otimes \sigma _{y}\right), \qquad \mathrm{exp}\left(z\sigma _{z} \otimes \sigma _{z}\right) .

Consider also the exponential functions of the terms $(\sigma_{x} \oplus \sigma_{x})$ , $(\sigma_{y} \oplus \sigma_{y})$ , $(\sigma_{z} \oplus \sigma_{z})$ , where $\oplus$ is the direct sum. These terms are also elements of the Lie group $U(4)$ .

Problem 2.44

Consider the Lie group $U(n)$ of the unitary $n \times n$ matrices. Let $\mathbb {T}^{n}$ be the $n$ -torus of diagonal unitary $n \times n$ matrices. Obviously $\mathbb {T}^{n}$ is a subgroup of $U(n)$ . Describe the homogeneous space $U(n)/\mathbb {T}^{n}$ .

Problem 2.45

(i)(ii)

(i)

Let $G$ be a semisimple Lie group and $H$ a maximal compact Lie subgroup of $G$ . Give an example for $G$ and $H$ .

(ii)

Find the maximal compact subgroup $K$ of $SL(2, \mathbb {C})$ .

Problem 2.46

(i)(ii)

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . If $AB = BA$ , then $\mathrm{exp}\left(A+B\right) = \mathrm{exp}\left(A\right)\mathrm{exp}\left(B\right)$ . Let $t \in \mathbb {R}$ . Consider the $2 \times 2$ matrices

C = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad D = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} .

Then $\mathrm{exp}\left(tC\right)$ is an element of $SL(2, \mathbb {R})$ and $\mathrm{exp}\left(tD\right)$ is a diagonal matrix with determinant $e^{t}$ , i.e. $\mathrm{exp}\left(tD\right)$ is an element of $GL(2, \mathbb {R})$ .

(i)

Show that $CD \neq DC$ .

(ii)

Calculate

e^{tC}e^{tD} \qquad \text{and} \qquad e^{t(C+D)} .

Discuss.

Problem 2.47

Given two $2 \times 2$ matrices

A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \qquad B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}

in the Lie group $GL(2, \mathbb {C})$ , i.e. $\operatorname {det}\left(A\right) \neq 0$ , $\operatorname {det}\left(B\right) \neq 0$ . The Kronecker product of $A$ and $B$ is given by the $4 \times 4$ matrix

A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix} .

The action of an $n \times n$ matrix $M = (m_{jk})$ on a polynomial $p(\overrightarrow {x}) \in \mathbf{R}_{n} = \mathbb {C}^{x_{1}, x_{2}, \ldots , x_{n}}$ may be defined by setting

T_{M}p(\overrightarrow {x}) := p(\overrightarrow {x}M)

where $\overrightarrow {x}M$ is the multiplication of the row $n$ -vector $\overrightarrow {x}$ by the $n \times n$ matrix $M$ . Thus $\overrightarrow {x}M$ is a row vector again. We denote by

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})}

the ring of polynomials in $\mathbf{R}_{4}$ that are invariant under the action of $A \otimes B$ for all pairs $A, B \in GL(2, \mathbb {C})$ , i.e.

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})} := \left\{ p \in \mathbf{R}_{4} : T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})\right\} .

The action preserves degree and homogeneity. Consider

p(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{1}

and

A = B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Show that $T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})$ .

Problem 2.48

Let $D \geq 4$ . Show that $SO(3) \otimes SO(D-3)$ is a subgroup of $SO(D)$ .

Problem 2.49

Consider the compact Lie groups $H = SU(2)$ and $G = SO(3)$ . Show that we have a homeomorphism $\psi : SU(2) \to SO(3)$ .

Problem 2.50

The circle

S^{1} := \left\{ (x_{1}, x_{2}) : x_{1}^{2}+x_{2}^{2} = 1\right\}

is a $C^{\infty }$ -manifold which is a one-dimensional submanifold of the vector space $\mathbb {R}^{2}$ . An atlas is given by

\begin{align} U_{1} & := \left\{ (x_{1}, x_{2}) \in S^{1} : x_{1} > 0\right\} & \phi _{1}(x_{1}, x_{2}) & = x_{2} \\ U_{2} & := \left\{ (x_{1}, x_{2}) \in S^{1} : x_{1} < 0\right\} & \phi _{2}(x_{1}, x_{2}) & = x_{2} \\ U_{3} & := \left\{ (x_{1}, x_{2}) \in S^{1} : x_{2} > 0\right\} & \phi _{3}(x_{1}, x_{2}) & = x_{1} \\ U_{4} & := \left\{ (x_{1}, x_{2}) \in S^{1} : x_{2} < 0\right\} & \phi _{4}(x_{1}, x_{2}) & = x_{1} . \end{align}

Show that the overlap functions are differentiable.

Problem 2.51

Consider the Lie group $GL(n, \mathbb {R})$ . The connected component of $GL(n, \mathbb {R})$ is defined by

GL^{+}(n, \mathbb {R}) := \left\{ A \in GL(n, \mathbb {R}) : \operatorname {det}\left(A\right) > 0\right\}

and is itself a differentiable manifold of dimension $n^{2}$ . Show that $GL^{+}(n, \mathbb {R})$ is a subgroup of $GL(n, \mathbb {R})$ .

Problem 2.52

We define

O(n, \mathbb {R}) := \left\{ A \in GL(n, \mathbb {R}) : AA^{T} = I_{n}\right\}

where $T$ stands for transpose. This is the real orthogonal group. Show that $O(n, \mathbb {R})$ is a subgroup of $GL(n, \mathbb {R})$ . Give the condition $AA^{T} = I_{n}$ .

Problem 2.53

Consider the Lie group $O(n, \mathbb {R})$ . Show that the $n$ -sphere

S^{n} := \left\{ (x_{1}, \ldots , x_{n}, x_{n+1}) : x_{1}^{2}+\cdots +x_{n+1}^{2} = 1\right\}

is diffeomorphic to $O(n+1, \mathbb {R})/O(n, \mathbb {R})$ .

Problem 2.54

Consider the linear action of $SU(n+1)$ on the vector space $\mathbb {C}^{n+1}$ . This maps complex lines into complex lines and therefore passes to an action on the complex projective space $\mathbb {C} P^{n}$ which is defined to be the vector space of all such lines in $\mathbb {C}^{n+1}$ . Let $n = 1$ , i.e. we consider $SU(2)$ . Find an element $A$ in $SU(2)$ such that

A\frac{1}{\sqrt{2}} \begin{pmatrix} i \\ -i \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} .

Problem 2.55

Consider the $2 \times 2$ matrix

S = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

(parity reversal) where $S \in O(2)$ but $S \notin SO(2)$ . Show that every $A \in O(2)$ , $A \notin SO(2)$ can be written as $A = \tilde{A}S$ with $\tilde{A} \in SO(2)$ .

Problem 2.56

Let

A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \in GL(2, \mathbb {R})

i.e. $\operatorname {det}\left(A\right) = a_{11}a_{22}-a_{12}a_{21} \neq 0$ . Find the conditions on the entries $a_{jk}$ such that

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} A \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = A .

Do these matrices form a subgroup of $GL(2, \mathbb {R})$ under matrix multiplication?

Problem 2.57

The Lie groups $SO(3)$ and $SU(2)$ have isomorphic Lie algebras $so(3)$ and $su(2)$ , respectively. A basis for the Lie algebra $su(2)$ is given by

\tau _{1} = \frac{1}{2} \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}, \qquad \tau _{2} = \frac{1}{2} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad \tau _{3} = \frac{1}{2} \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}

with $[\tau_{1}, \tau_{2}] = \tau_{3}$ , $[\tau_{2}, \tau_{3}] = \tau_{1}$ , $[\tau_{3}, \tau_{1}] = \tau_{2}$ . Construct a homomorphism

\varphi : SU(2) \to SO(3) .

Problem 2.58

(i)(ii)

Let $S \in SL(2, \mathbb {C})$ and

E = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

Obviously, $E \in SL(2, \mathbb {C})$ .

(i)

Find $SES^{T}$ , where $T$ denotes the transpose.

(ii)

Let $A, B \in SL(2, \mathbb {C})$ . Find $(A \otimes B)(E \otimes E)(A \otimes B)^{T}$ , where $\otimes$ denotes the Kronecker product.

Problem 2.59

A quandle, $X$ , is a set with a binary operation $(a,b) \mapsto a*b$ such that

For any $a \in X$ , $a*a = a$ .
For any $a, b \in X$ , there is a unique $c \in X$ such that $a = c*b$ .
For any $a, b, c \in X$ , we have $(a*b)*c = (a*c)*(b*c)$ . Examples of quandles are: (1) A group $X = G$ with conjugation as the quandle operation $a*b := bab^{-1}$ . We denote by $\operatorname {Conj}(G)$ the quandle defined for a group $G$ by $a*b = bab^{-1}$ . Any subset that is closed under such conjugation is also a quandle. (2) Any $\Lambda (= \mathbb {Z}^{t,t^{-1}})$ -module $M$ is a quandle with $a*b := ta+(1-t)b$ , where $a, b \in M$ , that is called an Alexander quandle. (3) Let $n$ be a positive integer, and for elements $i, j \in \left\{ 0,1,\ldots ,n-1\right\}$ , define

i*j := 2j-i \pmod{n} .

Then $*$ defines a quandle structure called the dihedral quandle.

Let $A, B \in SL(2, \mathbb {R})$ with

A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} .

Calculate $A*B$ and $B*A$ . Is $A*B = B*A$ ?

Problem 2.60

Let $GL(1, \mathbb {C})$ be the multiplicative group of nonzero complex numbers also denoted by $\mathbb {C}^{*}$ . Show that $GL(1, \mathbb {C})$ is a Lie group.

Problem 2.61

$GL(n, \mathbb {R})$ is the set of nondegenerate (determinant nonzero) $n \times n$ matrices with matrix multiplication for its group operation. Show that $GL(n, \mathbb {R})$ is a Lie group.

Problem 2.62

Let $V \in M(2n, \mathbb {R})$ , $V^{T} = V$ and $V > 0$ . Show that there exists $S \in Sp(2n, \mathbb {R})$ and $D \in M(n, \mathbb {R})$ diagonal and positive definite such that

V = S^{T} \begin{pmatrix} D & 0_{n} \\ 0_{n} & D \end{pmatrix} S \tag {1}

where $0_{n}$ is the $n \times n$ zero matrix. The matrices $S$ and $D$ are unique up to a permutation of the elements of $D$ .

Problem 2.63

The $n$ -sphere defined by

S^{n} := \left\{ \overrightarrow {x} = (x_{1}, \ldots , x_{n+1}) \in \mathbb {R}^{n+1} : x_{1}^{2}+\cdots +x_{n+1}^{2} = 1\right\}

is a manifold of dimension $n$ . Find two different atlases.

Problem 2.64

Consider the manifold $\mathbb {R}^{2}$ . Consider the analytic function $\overrightarrow {f} : \mathbb {R}^{2} \to \mathbb {R}^{2}$

\overrightarrow {f}(\overrightarrow {x}) = \begin{pmatrix} e^{x_{1}}\cos x_{2} \\ e^{x_{1}}\sin x_{2} \end{pmatrix} .

Show that $\overrightarrow {f}$ satisfies the conditions of the inverse function theorem on $\mathbb {R}^{2}$ , but the function is not $1-1$ on $\mathbb {R}^{2}$ .

Problem 2.65

Let $\alpha , \beta , \gamma \in \mathbb {R}$ . Consider the invertible $3 \times 3$ matrices

S_{1}(\alpha ) = \begin{pmatrix} \cosh \alpha & 0 & \sinh \alpha \\ 0 & 1 & 0 \\ \sinh \alpha & 0 & \cosh \alpha \end{pmatrix}, \qquad S_{2}(\beta ) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cosh \beta & \sinh \beta \\ 0 & \sinh \beta & \cosh \beta \end{pmatrix}

R(\gamma ) = \begin{pmatrix} \cos \gamma & -\sin \gamma & 0 \\ \sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{pmatrix} .

Find matrices $K_{1}, K_{2}, L$ such that

S_{1} = \mathrm{exp}\left(-i\alpha K_{1}\right), \qquad S_{2} = \mathrm{exp}\left(-i\beta K_{2}\right), \qquad R = \mathrm{exp}\left(-i\gamma L\right) .

Problem 2.66

Consider the rotation matrix around the $z$ -axis

A(\phi ) = \begin{pmatrix} \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{pmatrix} .

Let

G = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} .

Find $GA(\phi )G^{-1}$ . Discuss.

Problem 2.67

(i)(ii)

The Lie group $U(2,2)$ consists of all linear transformations of $\mathbb {C}^{4}$ which preserves a non-degenerate hermitian form with two positive and two negative eigenvalues. For a suitable choice of basis in $\mathbb {C}^{4}$ we can characterize a $g \in U(2,2)$ by

g^{*}Jg = J, \qquad J = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} \equiv \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} .

If we write the $4 \times 4$ matrix $g \in U(2,2)$ in a $2 \times 2$ block matrix form we have

g = \begin{pmatrix} A & B \\ C & D \end{pmatrix}

where

B^{*}D+D^{*}B = 0_{2}, \quad A^{*}C+C^{*}A = 0_{2}, \quad B^{*}D+D^{*}B = 0_{2}, \quad A^{*}D+C^{*}B = I_{2} .

(i)

U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

an element of $U(2,2)$ ?

(ii)

\begin{pmatrix} \cos (\alpha /2) & i\sin (\alpha /2) \\ i\sin (\alpha /2) & \cos (\alpha /2) \end{pmatrix} \otimes \begin{pmatrix} \cos (\beta /2) & i\sin (\beta /2) \\ i\sin (\beta /2) & \cos (\beta /2) \end{pmatrix}

an element of $U(2,2)$ ?

Problem 2.68

(i)(ii)(iii)

Let $0 < \tau < \infty$ and $S$ be positive real number. Consider the vector

\overrightarrow {S}_{\tau } := (\tau /(1+\tau ), \, i\tau /(1+\tau ), \, 1/(1+\tau ))S .

For $\tau = 0$ we have $\overrightarrow {S}_{0} = (0,0,S)$ and for $\tau \to \infty$ we have $\overrightarrow {S}_{\infty } = (S, iS, 0)$ . Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. We define the $2 \times 2$ matrix

\hat{S}_{\tau } := S_{1\tau }\sigma _{1} + S_{2\tau }\sigma _{2} + S_{3\tau }\sigma _{3} .

(i)

Calculate the matrix $(S_{1\tau }\sigma_{1})^{2} + (S_{2\tau }\sigma_{2})^{2} + (S_{3\tau }\sigma_{3})^{2}$ .

(ii)

Let $\sigma_{+} := \sigma_{1}+i\sigma_{2}$ . Let $\phi := \phi_{1}+i\phi_{2}$ , where $\phi_{1}, \phi_{2}$ are real. Calculate

H_{\tau }(\phi ) := \mathrm{exp}\left(-i\frac{\phi }{2}(\sigma _{3}+\tau \sigma _{+})\frac{1}{1+\tau }\right) .

(iii)

Calculate $H_{\tau }(\phi )\hat{S}_{\tau }H_{\tau }^{-1}(\phi )$ . Discuss.

Problem 2.69

(i)(ii)

Consider the Lie group $SL(2, \mathbb {R})$ . Consider the Jacobi elliptic functions $\operatorname {sn}(x,k)$ , $\operatorname {cn}(x,k)$ , $\operatorname {dn}(x,k)$ , where $x \in \mathbb {R}$ and $k \in [0,1]$ .

(i)

Show that the matrix ( $\epsilon \in \mathbb {R}$ )

\begin{pmatrix} \operatorname {cn}(\epsilon ,k) & \operatorname {sn}(\epsilon ,k) \\ -\operatorname {sn}(\epsilon ,k) & \operatorname {cn}(\epsilon ,k) \end{pmatrix}

is an element of $SL(2, \mathbb {R})$ . Find the inverse of the matrix.

(ii)

Study the limiting cases with $k=0$ and $k=1$ . Are the matrix elements bounded?

Problem 2.70

(i)(ii)

(i)

Show that the matrix ( $\epsilon \in \mathbb {R}$ )

\begin{pmatrix} \operatorname {dn}(\epsilon ,k) & k\operatorname {sn}(\epsilon ,k) \\ -k\operatorname {sn}(\epsilon ,k) & \operatorname {dn}(\epsilon ,k) \end{pmatrix}

is an element of $SL(2, \mathbb {R})$ . Find the inverse of the matrix.

(ii)

Study the limiting cases with $k=0$ and $k=1$ . Are the matrix elements bounded?

Problem 2.71

(i)(ii)

Consider the Lie group $SL(2, \mathbb {R})$ and the matrix ( $x \in \mathbb {R}$ )

M(x) = \begin{pmatrix} \sqrt{\pi }\operatorname {Ai}(x) & \sqrt{\pi }\operatorname {Bi}(x) \\ \sqrt{\pi }\operatorname {Ai}'(x) & \sqrt{\pi }\operatorname {Bi}'(x) \end{pmatrix}

where $\operatorname {Ai}$ and $\operatorname {Bi}$ are Airy functions which are solutions of the linear second order differential equation $d^{2}u/dx^{2} = xu$ and $'$ denotes the derivative. The Airy functions are defined by the improper integrals

\operatorname {Ai}(x) = \frac{1}{\pi }\int _{0}^{\infty } \cos \left( \frac{1}{3}t^{3}+xt \right) dt

\operatorname {Bi}(x) = \frac{1}{\pi }\int _{0}^{\infty } \left( \mathrm{exp}\left(-\frac{1}{3}t^{3}+xt\right) + \sin \left( \frac{1}{3}t^{3}+xt \right) \right) dt .

The matrix $M(x)$ is an element of $SL(2, \mathbb {R})$ since

\operatorname {Ai}(x)\operatorname {Bi}'(x) - \operatorname {Ai}'(x)\operatorname {Bi}(x) = \frac{1}{\pi } .

(i)

Find the inverse of $M(x)$ .

(ii)

Find $M(0)$ .

Find $M(x)$ for $x \to \infty$ using the asymptotic expansion

\operatorname {Ai}(x) \sim \frac{e^{-2x^{3/2}/3}}{2\sqrt{\pi }x^{1/4}}, \qquad \operatorname {Bi}(x) \sim \frac{e^{2x^{3/2}/3}}{\sqrt{\pi }x^{1/4}} .

Problem 2.72

Consider the group $G$ of $n \times n$ real matrices under matrix addition. Show that this is an $n^{2}$ -dimensional Lie group.

Problem 2.73

Let $M(n, \mathbb {R})$ be the vector space of all $n \times n$ matrices over the real number $\mathbb {R}$ . Show that the real general linear group

GL(n, \mathbb {R}) := \left\{ g \in M(n, \mathbb {R}) : \det (g) \neq 0\right\}

which is the group of all non-singular real $n \times n$ matrices is a Lie group.

Problem 2.74

Consider the group of $n \times n$ upper triangular real matrices under matrix multiplication with nonzero determinants. Show that this is an $n(n+1)/2$ -dimensional Lie group. Describe its group manifold.

Problem 2.75

(i)(ii)(iii)

Let

S^{n} := \left\{ (x_{1}, x_{2}, \ldots , x_{n+1}) \in \mathbb {R}^{n+1} : x_{1}^{2}+x_{2}^{2}+\cdots +x_{n+1}^{2} = 1\right\} .

(i)

Show that $S^{3}$ can be considered as a subset of $\mathbb {C}^{2}$ ( $\mathbb {C}^{2} \cong \mathbb {R}^{4}$ )

S^{3} = \left\{ (z_{1}, z_{2}) \in \mathbb {C}^{2} : \left|z_{1}\right|^{2}+\left|z_{2}\right|^{2} = 1\right\} .

(ii)

The Hopf map $\pi : S^{3} \to S^{2}$ is defined by

\pi (z_{1}, z_{2}) := (\bar{z}_{1}z_{2}+\bar{z}_{2}z_{1}, \ -i\bar{z}_{1}z_{2}+i\bar{z}_{2}z_{1}, \ \left|z_{1}\right|^{2}-\left|z_{2}\right|^{2}) .

Find the parameterization of $S^{3}$ , i.e. find $z_{1}(\theta ,\phi )$ , $z_{2}(\theta ,\phi )$ and thus show that indeed $\pi$ maps $S^{3}$ onto $S^{2}$ .

(iii)

Show that $\pi (z_{1}, z_{2}) = \pi (z_{1}', z_{2}')$ if and only if $z_{j}' = e^{i\alpha }z_{j}$ ( $j=1,2$ ) and $\alpha \in \mathbb {R}$ .

Problem 2.76

(i)(ii)

Consider the compact differentiable manifold

S^{2} := \left\{ (x_{1}, x_{2}, x_{3}) : x_{1}^{2}+x_{2}^{2}+x_{3}^{2} = 1\right\} .

An element $\eta \in S^{2}$ can be written as

\eta = (\cos \phi \sin \theta , \sin \phi \sin \theta , \cos \theta )

where $\phi \in [0, 2\pi )$ and $\theta \in [0, \pi ]$ . The stereographic projection is a map

\Pi : S^{2} \setminus \left\{ (0,0,-1)\right\} \to \mathbb {R}^{2}

given by

x_{1}(\theta ,\phi ) = \frac{2\sin (\theta )\cos (\phi )}{1+\cos (\theta )}, \qquad x_{2}(\theta ,\phi ) = \frac{2\sin (\theta )\sin (\phi )}{1+\cos (\theta )} .

(i)

Let $\theta = 0$ and $\phi$ arbitrary. Find $x_{1}, x_{2}$ . Give a geometric interpretation.

(ii)

Find the inverse of the map, i.e., find

\Pi ^{-1} : \mathbb {R}^{2} \to S^{2} \setminus \left\{ (0,0,-1)\right\} .

Problem 2.77

One natural parameterization of the Lie algebra $su(1,1)$ as a sub Lie algebra of $sp(2,2)$ is given by the $4 \times 4$ matrices

R_{1} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \qquad R_{2} = \begin{pmatrix} i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & -i \end{pmatrix}, \qquad R_{3} = \begin{pmatrix} 0 & 0 & 0 & i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}

with $[R_{1}, R_{2}] = -2R_{3}$ , $[R_{3}, R_{1}] = 2R_{2}$ , $[R_{2}, R_{3}] = -2R_{1}$ . Find $g_{2}g_{1}g_{2}$ , where $g_{1} = \mathrm{exp}\left(\alpha R_{1}\right)$ and $g_{2} = \mathrm{exp}\left(\beta R_{2}/2\right)$ .

Problem 2.78

Find the factor group $GL(n, \mathbb {R})/SL(n, \mathbb {R})$ .

Problem 2.79

The $n \times n$ unitary matrices form a Lie group. Do the $n \times n$ unitary matrices satisfying the additional condition $U = U^{T}$ form a sub Lie group?

Problem 2.80

The $n \times n$ unitary matrices form a Lie group $U(n)$ under matrix multiplication. Let $0_{n}$ be the $n \times n$ zero matrix. Do the $2n \times 2n$ matrices

\begin{pmatrix} 0_{n} & U \\ U & 0_{n} \end{pmatrix}

form a group under matrix multiplication?

Problem 2.81

Let $x$ be a fixed element of a group $G$ . The centralizer of $x$ in $G$ , $C(x)$ , is the set of all elements in $G$ that commute with $x$ , i.e.

C(x) := \left\{ g \in G : gx = xg\right\} .

Consider the Lie group $SL(2, \mathbb {R})$ and

A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

Then $A \in SL(2, \mathbb {R})$ since $\det (A) = 1$ . Find the centralizer $C(A)$ of $A$ .

Problem 2.82

The kernel of a homomorphism $\phi$ from a group $G$ with identity $e$ is the set

\left\{ g \in G : \phi (g) = e\right\} .

The kernel of $\phi$ is denoted by $\operatorname {Ker}\phi$ . Let $\mathbb {R}^{*}$ be the abelian group of nonzero real numbers under multiplication. Let $g \in GL(2, \mathbb {R})$ . The mapping (determinant) $g \to \det (g)$ is a homomorphism from $GL(2, \mathbb {R})$ to $\mathbb {R}^{*}$ . Find the kernel.

Problem 2.83

(i)(ii)

(i)

Consider the compact Lie group $SO(3, \mathbb {R})$ . Let $A \in SO(3, \mathbb {R})$ . Find the Frobenius norm

\left\| A\right\| = \tr (AA^{T}) .

(ii)

Find the norm

\left\| A\right\| = \sup _{\left\| \overrightarrow {x}\right\| =1} \left\| A\overrightarrow {x}\right\| .

Problem 2.84

(i)(ii)(iii)

The vector space of all $n \times n$ matrices over $\mathbb {C}$ form a Hilbert space with the scalar product

\left\langle A, B \right\rangle := \tr (AB^{*}) .

This implies a norm $\left\| A\right\|^{2} = \tr (AA^{*})$ .

(i)

Let $U$ be a unitary matrix. Find $\left\langle U, U \right\rangle$ .

(ii)

Consider the $2 \times 2$ unitary matrices

U_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad U_{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} .

Find the norm $\left\| U_{1}-U_{2}\right\|$ .

(iii)

Let $V$ be a $2 \times 2$ unitary matrix. Let $\phi \in \mathbb {R}$ . Find the norm $\left\| V-e^{i\phi }V\right\|$ .

Problem 2.85

Consider the Lie group $U(n)$ of all $n \times n$ unitary matrices. Let $K$ be an $n \times n$ skew-hermitian matrix with eigenvalues $\mu_{1}, \ldots , \mu_{n}$ (counted according to multiplicity) and the corresponding normalized eigenvectors $\overrightarrow {u}_{1}, \ldots , \overrightarrow {u}_{n}$ , where $\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{k} = 0$ for $k \neq j$ . Then $K$ can be written as

K = \sum _{j=1}^{n} \mu _{j}\overrightarrow {u}_{j}\overrightarrow {u}_{j}^{*}

and $\overrightarrow {u}_{j}\overrightarrow {u}_{j}^{*}\overrightarrow {u}_{k}\overrightarrow {u}_{k}^{*} = 0$ for $k \neq j$ and $j,k = 1,2,\ldots ,n$ . Note that the matrices $\overrightarrow {u}_{j}\overrightarrow {u}_{j}^{*}$ are projection matrices and

\sum _{j=1}^{n} \overrightarrow {u}_{j}\overrightarrow {u}_{j}^{*} = I_{n} .

(i)

Calculate $\mathrm{exp}\left(K\right)$ .

(ii)

Every $n \times n$ unitary matrix can be written as $U = \mathrm{exp}\left(K\right)$ , where $K$ is a skew-hermitian matrix. Find $U$ from a given $K$ .

(iii)

Use the result from (ii) to find for a given $U$ a possible $K$ .

(iv)

Apply the result from (ii) and (iii) to the unitary $2 \times 2$ matrix

U(\theta ) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} .

(v)

Apply the result from (ii) and (iii) to the $2 \times 2$ unitary matrix

V(\theta , \phi ) = \begin{pmatrix} \cos \theta & -e^{i\phi }\sin \theta \\ e^{-i\phi }\sin \theta & \cos \theta \end{pmatrix} .

(vi)

Every hermitian matrix $H$ can be written as $H = iK$ , where $K$ is a skew-hermitian matrix. Find $H$ for the examples given above.

Problem 2.86

(i)(ii)

An action or realization of a group $G$ on a set $M$ is defined as map $\varphi_{g}$

\varphi _{g} : \overrightarrow {x} \mapsto \overrightarrow {x}' = \varphi _{g}(\overrightarrow {x})

where $\overrightarrow {x}, \overrightarrow {x}' \in M$ and $g \in G$ . An element $\overrightarrow {x}$ of the manifold $M$ is a fixed point if

\varphi _{g}(\overrightarrow {x}) = \overrightarrow {x} \qquad \text{for all} \qquad g \in G .

(i)

Find the fixed points for the Lie group $SO(3)$ on $M = \mathbb {R}^{3}$ .

(ii)

Find the fixed points for $SO(3)$ on $M = S^{2}$ .

Problem 2.87

Consider the subgroups of the Lie group $SU(1,1)$

\begin{pmatrix} e^{i\theta /2} & 0 \\ 0 & e^{-i\theta /2} \end{pmatrix}, \qquad \begin{pmatrix} \cosh (t/2) & i\sinh (t/2) \\ -i\sinh (t/2) & \cosh (t/2) \end{pmatrix}, \qquad \begin{pmatrix} 1+\zeta i/2 & \zeta /2 \\ \zeta /2 & 1-\zeta i/2 \end{pmatrix}

where $\theta , t, \zeta \in \mathbb {R}$ . The Lie group $SU(1,1)$ acts as a linear group of transformations of linear vector space $\mathbb {R}^{2}$ . Find the infinitesimal generators of the group as vector fields.

Problem 2.88

(i)(ii)

(i)

The representation of the Lie group $SO(2)$ by $2 \times 2$ matrices is

\begin{pmatrix} \cos (\phi ) & -\sin (\phi ) \\ \sin (\phi ) & \cos (\phi ) \end{pmatrix}, \qquad \phi \in \mathbb {R} .

Show that the quadratic form (Euclid's metric) $s^{2} = x^{2}+y^{2}$ is an invariant of the group action of $SO(2)$ on vectors $(x,y)$ on the plane.

(ii)

The representation of the Lie group $SH(2)$ group by $2 \times 2$ matrices is

\begin{pmatrix} \cosh (t) & \sinh (t) \\ \sinh (t) & \cosh (t) \end{pmatrix}, \qquad t \in \mathbb {R} .

Show that the quadratic form (Minkowski's metric) $s^{2} = x^{2}-y^{2}$ is an invariant of the group action of $SH(2)$ on vectors $(x,y)$ on the plane.

Problem 2.89

Let $z \in \mathbb {C}$ . Consider the complex form $SO(2, \mathbb {C})$ of the $SO(2)$ group

\begin{pmatrix} \cos (z) & -\sin (z) \\ \sin (z) & \cos (z) \end{pmatrix} .

Show that $SO(2, \mathbb {R})$ is an abelian subgroup of $SO(2, \mathbb {C})$ . Show that $SH(2)$ is isomorphic to a subgroup of $SO(2, \mathbb {C})$ .

Problem 2.90

Let $R = \mathbb {C}^{x,y}$ be the polynomial ring in two variables $x,y$ over the field $\mathbb {C}$ . Let $q$ be a positive integer and $R_{q}$ be the sub vector space of polynomials which are homogeneous of degree $q$ . Then $R_{q}$ has the basis

x^{q}, \ x^{q-1}y, \ x^{q-2}y^{2}, \ \ldots , \ y^{q}

and dimension $q+1$ . An element of $SL(2, \mathbb {C})$

\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad ad-bc = 1

operates on $R$ by the transformation $x \to ax+cy$ , $y \to bx+dy$ and $R_{q}$ is the representation space for $SL(2, \mathbb {C})$ for each $q$ under this operation. We could also write in matrix form

\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ax+cy & bx+dy \end{pmatrix} .

Let $p_{j} := x^{j}y^{q-j}$ , $j = 0, 1, \ldots , q$ . Show that the images of $p_{j}$ under the transformation by

\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \qquad \text{and} \qquad \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}

are given by

\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \cdot p_{j} = \sum _{k=0}^{q-j} \binom {q-j}{k} t^{k}p_{j+k}

\begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} \cdot p_{j} = \sum _{k=0}^{j} \binom {j}{k} t^{k}p_{j-k} .

Problem 2.91

(i)(ii)(iii)

(i)

Let $t \in \mathbb {R}$ . Show that the matrix

g_{t} = \begin{pmatrix} \cosh t & i\sinh t \\ -i\sinh t & \cosh t \end{pmatrix}

has an inverse and find $g_{t}^{-1}$ . $g_{t}$ is an element of $SO(1,1)$ .

(ii)

The symmetric space $SL(2, \mathbb {C})/SU(2)$ can be identified with the space $H^{3}$ of positive $2 \times 2$ matrices with determinant $1$

A = \begin{pmatrix} a+b & x+iy \\ x-iy & a-b \end{pmatrix}, \qquad a,b,c,d \in \mathbb {R}

i.e. $\det A = a^{2}-b^{2}-x^{2}-y^{2} = 1$ . The group action is defined by

g_{t} \cdot A := g_{t}Ag_{t}^{*} .

Find $g_{t} \cdot A$ .

(iii)

Show that on the hyperboloid $a^{2} = 1+b^{2}+x^{2}+y^{2}$ , the matrix $g_{t}$ only changes the coordinates $y$ and $a$ .

Problem 2.92

Find an abelian Lie subgroup of the Lie group $U(n)$ .

Problem 2.93

(i)(ii)

We consider the Lie groups $U(2,2)$ , $SU(2,2)$ and the corresponding Lie algebras $u(2,2)$ , $su(2,2)$ , respectively. We define the matrix

L := \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Then

\begin{align} U(2,2) & := \left\{ G \in GL(4, \mathbb {C}) : G^{*}LG = L\right\} \\ SU(2,2) & := \left\{ G \in U(2,2) : \det G = 1\right\} \\ u(2,2) & := \left\{ X \in M_{4}(\mathbb {C}) : XL+LX = 0_{4}\right\} \\ su(2,2) & := \left\{ X \in u(2,2) : \tr (X) = 0\right\} \end{align}

where $M_{4}(\mathbb {C})$ is the vector space of the $4 \times 4$ matrices over the complex numbers.

(i)

Show that

X = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}

is an element of $u(2,2)$ and $su(2,2)$ .

(ii)

Let $t \in \mathbb {R}$ . Calculate $\mathrm{exp}\left(tX\right)$ to find an element of $SU(2,2)$ .

Problem 2.94

Show that the Lie group $U(n)$ is connected.

Problem 2.95

(i)(ii)(iii)

Consider the Lie group $T_{3}$ consisting of all $4 \times 4$ matrices

g(z_{1}, z_{2}, z_{3}) = \begin{pmatrix} 1 & 0 & 0 & z_{3} \\ 0 & e^{-z_{3}} & 0 & z_{2} \\ 0 & 0 & e^{z_{3}} & z_{1} \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad z_{1}, z_{2}, z_{3} \in \mathbb {C} .

(i)

Find the inverse $g^{-1}(z_{1}, z_{2}, z_{3})$ .

(ii)

Find the product $g(z_{1}, z_{2}, z_{3})g(w_{1}, w_{2}, w_{3})$ .

(iii)

Find a basis for the Lie algebra $t_{3}$ of the Lie group $T_{3}$ and the commutators.

Problem 2.96

Consider the vector space $V_{2}^{n}$ of homogeneous polynomials of two complex variables. An element of this vector space can be written as

p(z_{1}, z_{2}) = c_{0}z_{1}^{n} + c_{1}z_{1}^{n-1}z_{2} + \cdots + c_{n}z_{2}^{n} .

The compact Lie group $SU(2)$ acts on this vector space through the action of $U \in SU(2)$ as a linear transform on the vector $\overrightarrow {z} = (z_{1}, z_{2})^{T}$ by

\pi (U)f(\overrightarrow {z}) = f(U^{-1}\overrightarrow {z}) .

Show that this a group homomorphism.

Problem 2.97

(i)(ii)

Consider all $2 \times 2$ matrices with $UU^{*} = I_{2}$ , $\det U = 1$ i.e., $U \in SU(2)$ . Then $U$ can be written as

U = \begin{pmatrix} a & b \\ -b^{*} & a^{*} \end{pmatrix}, \qquad a, b \in \mathbb {C}

with the constraint $aa^{*}+bb^{*} = 1$ . Let

\begin{pmatrix} z_{1}' \\ z_{2}' \end{pmatrix} = \begin{pmatrix} a & b \\ -b^{*} & a^{*} \end{pmatrix} \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix} .

(i)

Show that

(z_{1}')(z_{1}')^{*} + (z_{2}')(z_{2}')^{*} = z_{1}z_{1}^{*} + z_{2}z_{2}^{*} .

(ii)

Consider

\begin{pmatrix} z_{1}' \\ z_{2}' \end{pmatrix} = \begin{pmatrix} a & b \\ -b^{*} & a^{*} \end{pmatrix} \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix} .

Show that $dz_{1}' \wedge dz_{2}' = dz_{1} \wedge dz_{2}$ , where $\wedge$ is the exterior product and $d$ the exterior derivative. Note that $d$ is linear.

Problem 2.98

(i)(ii)(iii)

(i)

Consider the $2 \times 2$ matrix

A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} .

Let $t \in \mathbb {R}$ . Find $\mathrm{exp}\left(tA\right)$ .

(ii)

Consider the vector field

V = x_{2}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{2}}

defined on $\mathbb {R}^{2}$ . Let $t \in \mathbb {R}$ . Calculate

\mathrm{exp}\left(tV\right) \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} \equiv \begin{pmatrix} \mathrm{exp}\left(tV\right)x_{1} \\ \mathrm{exp}\left(tV\right)x_{2} \end{pmatrix}, \qquad \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} \in \mathbb {R}^{2} .

(iii)

Use the results from either (i) or (ii) to solve the initial value problem of the system of linear first order differential equations

\frac{dx_{1}}{dt} = x_{2}, \qquad \frac{dx_{2}}{dt} = -x_{1} .

Problem 2.99

Consider the vector fields

x\frac{d}{dx}, \qquad \frac{d}{dx} .

The manifold is $M = \mathbb {R}$ , i.e. $x \in \mathbb {R}$ . Calculate

\mathrm{exp}\left(tx\frac{d}{dx}\right) \mathrm{exp}\left(t\frac{d}{dx}\right) x, \qquad \mathrm{exp}\left(t\frac{d}{dx}\right) \mathrm{exp}\left(tx\frac{d}{dx}\right) x .

Problem 2.100

Consider the nonlinear ordinary differential equation

\frac{dx}{dt} = \cos ^{2}(x)

with the initial condition $x(t=0) = x_{0} = 0$ . Find the solution of the initial value problem using the exponential map (Lie series)

x(t) = \left. \mathrm{exp}\left(tV\right)x \right|_{x \to x_{0}}

where $V$ is the corresponding vector field

V = \cos ^{2}(x)\frac{d}{dx} .

Section 2.1Programming Problems[2/100%]

Problem 2.101

Consider the $2 \times 2$ matrix

A(\alpha _{0}, \alpha _{1}, \alpha _{2}) = \begin{pmatrix} (\cos \alpha _{0}\cos \alpha _{1}+i\sin \alpha _{0}\sin \alpha _{1})e^{i\alpha _{2}} & -\cos \alpha _{0}\sin \alpha _{1}+i\sin \alpha _{0}\cos \alpha _{1} \\ \cos \alpha _{0}\sin \alpha _{1}+i\sin \alpha _{0}\cos \alpha _{1} & (\cos \alpha _{0}\cos \alpha _{1}-i\sin \alpha _{0}\sin \alpha _{1})e^{-i\alpha _{2}} \end{pmatrix}

where $-\pi \leq \alpha_{0} < \pi$ , $-\pi \leq \alpha_{1} < \pi$ , $0 \leq \alpha_{2} \leq \pi$ . Write a Computer algebra program in SymbolicC++ that calculates

\left. \frac{\partial }{\partial \alpha _{j}} A(\alpha _{0}, \alpha _{1}, \alpha _{2}) \right|_{\alpha _{0}=0, \alpha _{1}=0, \alpha _{2}=0}

and thus finds the generators

X_{0} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \qquad X_{1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad X_{2} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} .

Thus

A(\alpha _{0}, \alpha _{1}, \alpha _{2}) = \exp (\alpha _{0}X_{0}+\alpha _{1}X_{1}+\alpha _{2}X_{2}) .

Problem 2.102

Let $\alpha \in \mathbb {R}$ . Given the matrix

A(\alpha ) = \begin{pmatrix} \cosh \alpha & 0 & 0 & \sinh \alpha \\ 0 & \cos \alpha & \sin \alpha & 0 \\ 0 & -\sin \alpha & \cos \alpha & 0 \\ \sinh \alpha & 0 & 0 & \cosh \alpha \end{pmatrix} .

Write a SymbolicC++ program that calculates the vector

A(\alpha ) \begin{pmatrix} x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}

and shows that $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$ is invariant.

Section 2.2Supplementary Problems[8/0%]

Problem 2.103

Show that the compact Lie groups $SO(3)$ and $SU(2)$ have isomorphic Lie algebras. A basis for $so(3)$ is given by the skew-symmetric matrices

T_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad T_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \qquad T_{3} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} .

A basis for $su(2)$ is given by skew-hermitian matrices

\tau _{1} = \frac{1}{2} \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}, \qquad \tau _{2} = \frac{1}{2} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad \tau _{3} = \frac{1}{2} \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix} .

Thus $\tau_{1} = -i\sigma_{1}/2$ , $\tau_{2} = -i\sigma_{2}/2$ , $\tau_{3} = -i\sigma_{3}/2$ , where $\sigma_{1}, \sigma_{2}, \sigma_{3}$ denote the Pauli spin matrices.

Problem 2.104

Let $n \geq 1$ . We define the $(2n+1)$ -sphere

S^{2n+1} := \left\{ \overrightarrow {z} \in \mathbb {C}^{n+1} : \sum _{j=1}^{n+1} \left|z_{j}\right|^{2} = 1\right\} .

Show that

S^{2n+1} \cong U(n+1)/U(n) \cong SU(n+1)/SU(n) .

For $n=1$ one has $S^{3} \cong U(2)/U(1) \cong SU(2)$ .

Problem 2.105

(i)(ii)

(i)

Consider the Pauli spin matrices

\sigma _{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma _{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .

Let $\alpha_{1}, \alpha_{2}, \alpha_{3} \in \mathbb {R}$ . Calculate

e^{i\alpha _{1}\sigma _{1}+i\alpha _{2}\sigma _{2}+i\alpha _{3}\sigma _{3}} \qquad \text{and} \qquad e^{i\alpha _{1}\sigma _{1}}e^{i\alpha _{2}\sigma _{2}}e^{i\alpha _{3}\sigma _{3}} .

Discuss.

(ii)

A basis for $so(3)$ is given by the skew-symmetric matrices

T_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad T_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \qquad T_{3} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} .

Let $\alpha_{1}, \alpha_{2}, \alpha_{3} \in \mathbb {R}$ . Calculate

e^{\alpha _{1}T_{1}+\alpha _{2}T_{2}+\alpha _{3}T_{3}} \qquad \text{and} \qquad e^{i\alpha _{1}T_{1}}e^{i\alpha _{2}T_{2}}e^{i\alpha _{3}T_{3}} .

Discuss.

Problem 2.106

Show that any element $U$ of $SU(n)$ can be diagonalized by an $n \times n$ unitary matrix and the diagonal elements are given by $\exp (i\phi_{kk})$ .

Problem 2.107

The Pauli matrices $\sigma_{1}, \sigma_{2}, \sigma_{3}$ and the $2 \times 2$ identity matrix $I_{2} = \sigma_{0}$ are elements of the Lie group $U(2)$ . What is the condition on $c_{j} \in \mathbb {C}$ ( $j=0,1,2,3$ ) such that

\sum _{j=0}^{3} c_{j}\sigma _{j} \otimes \sigma _{j}

is an invertible matrix, i.e. an element of $GL(2, \mathbb {C})$ ?

Problem 2.108

The matrix

U = \begin{pmatrix} 0 & 0 & i \\ 0 & i & 0 \\ i & 0 & 0 \end{pmatrix}

is an element of the Lie group $U(3)$ . Find a $3 \times 3$ unitary matrix $V$ such that $V^{-1}UV$ is a diagonal matrix.

Problem 2.109

(i)(ii)

(i)

Show that the group $SO^{+}(3,1)$ is isomorphic to $SL(2, \mathbb {C})/\left\{ \pm \operatorname {id}\right\}$ the group of $2 \times 2$ complex matrices with determinant $1$ . The quotient means that two matrices differing by an overall sign are identified.

(ii)

Show that $SL(2, \mathbb {C})/\left\{ \operatorname {id}\right\}$ is isomorphic to the group of complex Mobius transformations in the extended complex plane $\mathbb {C} \cup \left\{ \infty \right\}$ .

Problem 2.110

Let $G$ denote the five-dimensional manifold $\mathbb {C} \times \mathbb {C} \times \mathbb {R}$ with multiplication defined as follows

(c_{1}, c_{2}, r)(c_{1}', c_{2}', r') := (c_{1}+e^{2\pi i r}c_{1}', \ c_{2}+e^{2\pi i h r}c_{2}', \ r+r')

where $h$ is a fixed irrational number and $c_{1}, c_{1}', c_{2}, c_{2}' \in \mathbb {C}$ , $r, r' \in \mathbb {R}$ . Show that $G$ is a Lie group.