9 Problems for Epiphany 9 Problems for Epiphany 9.2 Problems for section 5

9.1 Problems for section 4

Problem 51.

1.

Compute $\exp(X)$ for $X$ equal to $\begin{pmatrix}t&0\\ 0&s\end{pmatrix}$ , $\begin{pmatrix}0&t\\ -t&0\end{pmatrix}$ , and $\begin{pmatrix}0&t\\ t&0\end{pmatrix}$ (where $s,t\in\mathbb{R}$ ).
2.

Let $E_{a,b}$ be the elementary $n\times n$ matrix with $1$ in the $(a,b)$ -entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a\neq b$ and $a=b$ .

Solution

Problem 52.

Show that

\exp(tX)\exp(tY)=\exp\left(t(X+Y)+\frac{t^{2}}{2}[X,Y]+O(t^{3})\right)

as $t\to 0$ , where

[X,Y]=XY-YX.

Solution

Problem 53.

Let $\mathfrak{n}$ be the $\mathbb{C}$ -vector space of strictly upper triangular matrices ( $0$ ’s on the diagonal) and let $N=\{g\in\operatorname{GL}_{n}(\mathbb{C})\,:\,g=I+X,X\in\mathfrak{n}\}$ .

In this problem we will see that the restriction of the exponential to $\mathfrak{n}$ is a diffeomorphism onto $N$ .

1.

Let $X\in\mathfrak{n}$ . Show that $X^{n}=0$ .
2.

Show that $\exp(X)\in N$ for $X\in\mathfrak{n}$ .
3.

Show that, for $g\in N$ , the logarithm $\log(g)=\sum_{k=1}^{\infty}(-1)^{k+1}\tfrac{(g-I)^{k}}{k}$ is in fact a finite sum (and hence converges).
4.

Show that $\exp|_{\mathfrak{n}}$ and $\log|_{N}$ are inverses of each other. Hint: this boils down to an identity of formal power series, which you can actually deduce from the corresponding fact over $\mathbb{R}$ .

Solution

Problem 54.

1.

Using the previous question, fill in the gaps of the proof from the notes that

$\exp:\mathfrak{gl}_{n,\mathbb{C}}\rightarrow\operatorname{GL}_{n}(\mathbb{C})$

is surjective.
2.

(+) Is the exponential map $\exp:\mathfrak{sl}_{2,\mathbb{C}}\rightarrow\operatorname{SL}_{2}(\mathbb{C})$ surjective? What about $\exp:\mathfrak{gl}_{2,\mathbb{R}}\rightarrow\operatorname{GL}^{+}_{2}(\mathbb{% R})$ ?

Solution

Problem 55.

Let $v\in\mathbb{R}^{3}$ be a unit vector and let $f:\mathbb{R}\to SO(3)$ be the map with $f(\theta)$ being rotation by $\theta$ about the axis $v$ (the angle is measured anticlockwise as you look along the vector from the origin).

Show that $f$ is a one-parameter subgroup and find its infinitesimal generator in terms of $v$ .

Solution

Problem 56.

Prove that the Lie algebra of $\mathrm{U}(n)$ is

\mathfrak{u}_{n}=\{X\in\mathfrak{gl}_{n,\mathbb{C}}:X+X^{\dagger}=0\}

and find its (real) dimension. Is it a complex vector space?

Solution

Problem 57.

Let $I_{p,q}=\begin{pmatrix}I_{p}&\\ &-I_{q}\end{pmatrix}$ , where $I_{k}$ denotes the identity matrix of size $k$ . Let $n=p+q$ . Let

\operatorname{O}(p,q)=\{g\in\operatorname{GL}_{n}(\mathbb{R})\,:\,gI_{p,q}g^{T% }=I_{p,q}\}

be the orthogonal group of signature $(p,q)$ . Let $\mathrm{SO}(p,q)=\operatorname{O}(p,q)\cap\operatorname{SL}_{n}(\mathbb{R})$ . We let $\mathfrak{o}_{p,q}$ and $\mathfrak{so}_{p,q}$ be their Lie algebras.

Show that the Lie algebra $\mathfrak{o}_{p,q}$ is given by

\mathfrak{o}(p,q)=\{X\in M_{n}(\mathbb{R})\,:\,XI_{p,q}+I_{p,q}X^{T}=0\}

and that $\mathfrak{so}_{p,q}=\mathfrak{o}_{p,q}$ .

Solution

Problem 58.

1.

Show that the Lie algebras $\mathfrak{so}_{3}$ and $\mathfrak{su}_{2}$ are isomorphic. (Later on, we will see a conceptual reason for this).

Hint: it is enough to find a basis for $\mathfrak{so}_{3}$ and a basis for $\mathfrak{su}_{2}$ which satisfy the ‘same’ Lie bracket relations. Try using the basis of $\mathfrak{so}_{3}$ consisting of infinitesimal generators for rotations around the axes, and a basis for $\mathfrak{su}_{2}$ related to the quaternions.
2.

Show that the Lie algebras $\mathfrak{so}_{2,1}$ and $\mathfrak{sl}_{2,\mathbb{R}}$ are isomorphic.
3.

(+) Show that the Lie algebras $\mathfrak{so}_{3,1}$ and $\mathfrak{sl}_{2,\mathbb{C}}$ are isomorphic (as real Lie algebras).

Solution

Problem 59.

Show that:

1.

If $X\in\mathfrak{sp}_{2n}$ , then $\operatorname{tr}(X)=0$ .
2.

(+) Show that, if $g\in\operatorname{Sp}(2n)$ , then $\det(g)=1$ .

Solution

Problem 60.

Prove that if $G$ is a Lie group and $G^{0}$ is the connected component of the identity, then the subgroup $G^{0}$ is normal.

Solution

Problem 61.

1.

Give a direct proof that $\mathrm{SO}(3)$ is connected, by constructing a path from an arbitrary element of $\mathrm{SO}(3)$ to the identity. Hint: every element of $\mathrm{SO}(3)$ is rotation by some angle about some axis.
2.

Prove by induction on $n$ that $SO(n)$ is connected for all $n\geq 1$ .

Solution

Problem 62.

Show that a general element of $\mathrm{SU}(2)$ may be written

\begin{pmatrix}a&-\overline{b}\\ b&\overline{a}\end{pmatrix}

for $a,b\in\mathbb{C}$ with $|a|^{2}+|b|^{2}=1$ .

Deduce that $\mathrm{SU}(2)$ is diffeomorphic to the three-sphere $S^{3}=\{v\in\mathbb{R}^{4}:|v|=1\}$ .

In other words, write down a smooth bijection $\mathrm{SU}(2)\rightarrow S^{3}$ with smooth inverse. Don’t worry about checking that the maps are smooth, just write them down. The result of this problem implies that $\mathrm{SU}(2)$ is simply-connected.

Solution

Problem 63.

Show that, if $G$ is a connected (linear) Lie group with Lie algebra $\mathfrak{g}$ , then $G$ is abelian if and only if $\mathfrak{g}$ is (see Definition 4.29). Hint: for the converse, consider the adjoint map $G\to GL(\mathfrak{g})$ .

What goes wrong if $G$ is not connected?

Solution: see Proposition 5.14.

Problem 64.

If $\mathfrak{g}$ is a Lie algebra, let $\mathfrak{z}$ be its centre:

\mathfrak{z}=\{Z\in\mathfrak{g}:[X,Z]=0\,:\,\text{for all $X\in\mathfrak{g}$}.\}.

Suppose that $G$ is a connected Lie group with centre $Z$ and Lie algebra $\mathfrak{g}$ with centre $\mathfrak{z}$ .

Prove that $\mathfrak{z}$ is the Lie algebra of $Z$ .

Solution

Problem 65.

Solve the exercises in section 4.7