# Lie groups and Lie algebras: Fortnight 1 Questions

Remember, you'll need to do at least 4, 4, or 5 questions to get a C, B, or A respectively. For a B or an A, this will need to include 2 (respectively 3) $\alpha$ questions.

## β questions

Answer as many as you want.

### 1. Exp and eigenvectors

(Watch the videos about matrix exponentiation and its properties first)

1. Suppose that $M$ is a matrix and $v$ is an eigenvector of $M$ with eigenvalue $\lambda$ . Show that $v$ is an eigenvector of $\exp(M)$ with eigenvalue $e^{\lambda}$ .

2. For a vector $v=(x,y,z)$ , define $K_{v}:=\begin{pmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{pmatrix}$ . Find the $0$ -eigenspace of $K_{v}$ and deduce that $v$ is fixed by $\exp\left(\theta K_{v}\right)$ for any $\theta\in\mathbf{R}$ .

### 2. Exp and rotations

Depends on the previous question

1. Given vectors $v,w\in\mathbf{R}^{3}$ , show that $K_{v}w=v\times w$ .

2. Given a unit vector $u$ , pick vectors $v$ and $w$ such that $u,v,w$ is a right-handed orthonormal basis of $\mathbf{R}^{3}$ (so $u\times v=w$ and $u\times w=-v$ ). Calculate $\exp(\theta K_{u})v$ and $\exp(\theta K_{u})w$ and deduce that $\exp(\theta K_{u})$ is a rotation. What are the axis and angle of rotation?

### 3. Jacobi identity

(Technically no prerequisites, but the video about abstract Lie algebras is related.)

1. Prove the Jacobi identity $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$ for matrices $X,Y,Z$ where $[A,B]:=AB-BA$ .

2. Given a matrix $X$ , define $\mathrm{ad}_{X}\colon\mathfrak{gl}(n,\mathbf{R})\to\mathfrak{gl}(n,\mathbf{R})$ by $\mathrm{ad}_{X}(Y):=[X,Y]$ . Prove that $[\mathrm{ad}_{X},\mathrm{ad}_{Y}]=\mathrm{ad}_{[X,Y]}$ . (Hint: evaluate both sides on some $Z\in\mathfrak{gl}(n,\mathbf{R})$ and use the Jacobi identity.)

### 4. Unitary group

(Watch the video about orthogonal matrices first)

The unitary group $U(n)$ is the group of complex $n$ -by-$n$ matrices $M$ such that $M^{\dagger}M=I$ . Here, $M^{\dagger}$ is the conjugate-transpose of $M$ , for example $\begin{pmatrix}a&b\\ c&d\end{pmatrix}^{\dagger}=\begin{pmatrix}\bar{a}&\bar{c}\\ \bar{b}&\bar{d}\end{pmatrix}.$ Show that the Lie algebra $\mathfrak{u}(n)$ of $U(n)$ is the space of $n$ -by-$n$ skew-Hermitian matrices $\{M\ :\ M^{\dagger}=-M\}$ .

### 5. Example: surjective exp

(You need to remember Jordan normal form and watch the first video about matrix exponentiation).

We will show that $\exp\colon\mathfrak{gl}(2,\mathbf{C})\to GL(2,\mathbf{C})$ is surjective.

1. Calculate $\exp\begin{pmatrix}a&b\\ 0&a\end{pmatrix}$ , and hence find a logarithm for the matrix $\begin{pmatrix}\lambda&1\\ 0&\lambda\end{pmatrix}$ for any $\lambda\neq 0\in\mathbf{C}$ .

2. If $M\in GL(2,\mathbf{C})$ , let $N=P^{-1}MP$ be its Jordan normal form. Prove that $N=\exp(X)$ for some $X\in\mathfrak{gl}(2,\mathbf{C})$ and deduce that $M=\exp(Y)$ for some $Y\in\mathfrak{gl}(2,\mathbf{C})$ .

## α questions

Answer as many as you want. You will need to do well on at least 2 to get a B- and at least 3 to get an A-.

### 6. BCH formula

(Watch the video about the BCH formula first. WARNING: This question gets messy before it gets better.)

Show that the third-order term in the Baker-Campbell-Hausdorff formula is $\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]].$

### 7. det(exp) = exp(tr), part 1

(Technically, this has no prerequisites, but the video about $SL(2,\mathbf{C})$ is related)

Let $H\in\mathfrak{gl}(n,\mathbf{R})$ be a matrix.

1. Show that $\det(I+tH)=1+t\mathrm{Tr}(H)+\mathcal{O}(t^{2})$ where $\mathrm{Tr}(H)=H_{11}+H_{22}+\cdots+H_{nn}$ is the trace of $H$ . Hint: Write it out in full, i.e. $\det(I+tH)=\det\begin{pmatrix}1+tH_{11}&tH_{12}&\cdots&tH_{1n}\\ tH_{21}&1+tH_{22}&&\vdots\\ \vdots&&\ddots&\vdots\\ tH_{n1}&tH_{n2}&\cdots&1+tH_{nn}\end{pmatrix}$

2. Deduce that if $M$ is invertible then $\det(M+tH)=\det(M)+t\det(M)\mathrm{Tr}(M^{-1}H)+\mathcal{O}(t^{2})$ .

3. If $A(t)$ is a path of matrices, use the previous parts to show that $\frac{d}{dt}\det(A(t))=\det(A(t))\mathrm{Tr}\left(A(t)^{-1}\frac{dA}{dt}(t)\right)$ (Hint: Use the Taylor expansion of $A(t)$ to find the Taylor expansion of $\det(A(t))$ and drop terms of higher order.)

Note: By a "path of matrices" I basically mean a matrix $A(t)$ whose entries depend on $t$ . You may assume that this dependence is sufficiently differentiable to run your argument.

### 8. det(exp) = exp(tr), part 2

(Depends on the previous question, the properties of the exponential map, and for the final part the definition of the Lie algebra of a matrix group)

1. Using the formula for $\frac{d}{dt}\det(A(t))$ from the previous question, show that if $\phi(t):=\det(\exp(tH))$ then $\phi$ satisfies the differential equation $\frac{d\phi}{dt}=\phi(t)\mathrm{Tr}(H).$

2. Assuming that differential equations have unique solutions (with specified initial conditions), prove that $\det(\exp(tH))=\exp(t\mathrm{Tr}(H))$ .

3. Using this formula, show that the Lie algebra $\mathfrak{sl}(n,\mathbf{R})$ of $SL(n,\mathbf{R})=\{M\in GL(n,\mathbf{R})\ :\ \det(M)=1\}$ is the space of matrices with trace zero.

### 9. Example: non-surjective exp

(You need to remember Jordan normal form. Recall that $\mathfrak{sl}(2,\mathbf{R})$ is the group of 2-by-2 tracefree matrices and $SL(2,\mathbf{R})$ is the group of 2-by-2 matrices with determinant 1. I also highly recommend trying the previous question first).

We will show that $\exp\colon\mathfrak{sl}(2,\mathbf{R})\to SL(2,\mathbf{R})$ is not surjective.

1. Given $B\in\mathfrak{sl}(2,\mathbf{R})$ , show that its Jordan normal form (when considered as a complex matrix) is one of $\begin{pmatrix}\lambda&0\\ 0&-\lambda\end{pmatrix},\lambda\in\mathbf{R}\mbox{ or }i\mathbf{R},\quad\begin% {pmatrix}0&1\\ 0&0\end{pmatrix}.$

2. Deduce that one of the following must be true about $\exp(B)$ :

1. both its eigenvalues are positive;

2. both its eigenvalues are unit complex numbers;

3. both its eigenvalues are equal to 1.

3. By exhibiting a matrix in $SL(2,\mathbf{R})$ whose eigenvalues satisfy none of these conditions, deduce that $\exp\colon\mathfrak{sl}(2,\mathbf{R})\to SL(2,\mathbf{R})$ is not surjective.

### 10. Simultaneous strict upper-triangularisability

(Just uses good ol'-fashioned linear algebra. It will become relevant if you choose to do a project about Lie's theorem or Engel's theorem on solvable (respectively nilpotent) Lie algebras.)

Recall that a linear map $N$ is called nilpotent if $N^{k}=0$ for some $k$ . In this question, you may use the fact that, if $N$ is a nilpotent linear map then there is some basis with respect to which its matrix is strictly upper-triangular.

Let $V$ be a vector space and suppose that $M\colon V\to V$ and $N\colon V\to V$ are linear maps which are both nilpotent and which commute with one another: $MN=NM$ .

1. Show that if $v\in M^{j}V$ then $Nv\in M^{j}V$ . We will write $N_{j}\colon M^{j}V\to M^{j}V$ for the restriction $N|_{M^{j}V}$ .

2. Suppose we have picked a basis of $M^{j}V$ such that the matrix of $N_{j}$ is strictly upper-triangular. Show that we can extend this to get a basis of $M^{j-1}V$ for which the matrix of $N_{j-1}$ is strictly upper-triangular. (Hint: Pick a complement $C$ for $M^{j}V\subset M^{j-1}V$ and consider the map $C\to C$ induced by $N$ ).

3. Use this to prove by induction that we can find a basis of $V$ making both $M$ and $N$ simultaneously upper-triangular.

### 11. Simultaneous upper triangularisability

(Uses Q.10).

Let $V$ be a complex vector space and $X\colon V\to V$ be a linear map. Recall that the generalised $\lambda$ -eigenspace of a linear map $X\colon V\to V$ is the subspace $V_{\lambda}:=\{v\in V\,:\,(X-\lambda I)^{k}v=0\mbox{ for some }k\}.$ In this question, you may assume that $V=\bigoplus_{\lambda}V_{\lambda}$ : that is, the generalised eigenspaces span $V$ and $V_{\lambda}\cap V_{\mu}=0$ unless $\lambda=\mu$ .

Now let $X\colon V\to V$ and $Y\colon V\to V$ be linear maps and suppose they commute with one another.

1. Show that $Y$ preserves the generalised eigenspaces of $X$ . Let $Y_{\lambda}\colon V_{\lambda}\to V_{\lambda}$ be the restriction $Y|_{V_{\lambda}}$ .

2. Let $V_{\lambda,\mu}$ be the $\mu$ -generalised eigenspace of $Y_{\lambda}$ . Show that it is preserved by $X$ .

3. Deduce that both $X$ and $Y$ are block-diagonal with respect to the splitting $V=\bigoplus_{\lambda,\mu}V_{\lambda,\mu}$ .

4. Show that we can pick a basis for $V_{\lambda,\mu}$ such that the corresponding blocks of $X$ and $Y$ are upper-triangular. (Hint: Consider $X-\lambda I$ and $Y-\mu I$ and use the previous question about strict upper triangularisability.)

### 12. Technicality about local coordinates

By mimicking the proof that $\exp$ is locally invertible, prove the following claim (which was used in this optional video to prove that $\exp\colon\mathfrak{g}\to G$ is locally invertible for a matrix group $G$ ):

If $\mathfrak{g}\subset\mathfrak{gl}(n,\mathbf{R})$ is a Lie subalgebra and $W\subset\mathfrak{gl}(n,\mathbf{R})$ is a vector space complement for $\mathfrak{g}$ , show that the map $F\colon\mathfrak{g}\oplus W\to GL(n,\mathbf{R})$ defined by $F(v,w)=\exp(v)\exp(w)$ is locally invertible.