Lie groups and Lie algebras: Fortnight 1 Questions

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 . Find the $0$ -eigenspace of $K_v$ and deduce that $v$ is fixed by $\exp \left(\theta K_v\right)$ for any $\theta \in 𝐑$ .

2. Exp and rotations

Depends on the previous question

1. Given vectors $v,w\in {𝐑}^3$ , 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 ${𝐑}^3$ (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 ${\mathrm{ad}}_X:𝔤𝔩(n,𝐑)\to 𝔤𝔩(n,𝐑)$ 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 $Z\in 𝔤𝔩(n,𝐑)$ 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 $M^{†}M=I$ . Here, $M^{†}$ is the conjugate-transpose of $M$ , for example Show that the Lie algebra $𝔲(n)$ of $U(n)$ is the space of $n$ -by-$n$ skew-Hermitian matrices $\{M:M^{†}=-M\}$ .

5. Example: surjective exp

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

We will show that $\exp :𝔤𝔩(2,𝐂)\to GL(2,𝐂)$ is surjective.

1. Calculate , and hence find a logarithm for the matrix for any $\lambda \ne 0\in 𝐂$ .

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

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 $SL(2,𝐂)$ is related)

Let $H\in 𝔤𝔩(n,𝐑)$ be a matrix.

1. Show that $det(I+tH)=1+t\mathrm{Tr}(H)+𝒪(t^2)$ where $\mathrm{Tr}(H)=H_{11}+H_{22}+\mathrm{\cdots }+H_{nn}$ is the trace of $H$ . Hint: Write it out in full, i.e.

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

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 $\varphi (t):=det(\exp (tH))$ then $\varphi$ satisfies the differential equation $$\frac{d\varphi }{dt}=\varphi (t)\mathrm{Tr}(H).$$

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 $𝔰𝔩(n,𝐑)$ of $SL(n,𝐑)=\{M\in GL(n,𝐑):det(M)=1\}$ is the space of matrices with trace zero.

9. Example: non-surjective exp

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

We will show that $\exp :𝔰𝔩(2,𝐑)\to SL(2,𝐑)$ is not surjective.

1. Given $B\in 𝔰𝔩(2,𝐑)$ , show that its Jordan normal form (when considered as a complex matrix) is one of

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 $SL(2,𝐑)$ whose eigenvalues satisfy none of these conditions, deduce that $\exp :𝔰𝔩(2,𝐑)\to SL(2,𝐑)$ 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 $M:V\to V$ and $N:V\to V$ 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 $N_j:M^{j}V\to M^{j}V$ 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 $X:V\to V$ be a linear map. Recall that the generalised $\lambda$ -eigenspace of a linear map $X:V\to V$ is the subspace $$V_{\lambda }:=\{v\in V:{(X-\lambda I)}^{k}v=0\text{ for some }k\}.$$ In this question, you may assume that $V={\oplus }_{\lambda }{V}_{\lambda }$ : that is, the generalised eigenspaces span $V$ and $V_{\lambda }\cap V_{\mu }=0$ unless $\lambda =\mu$ .

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

1. Show that $Y$ preserves the generalised eigenspaces of $X$ . Let $Y_{\lambda }:V_{\lambda }\to V_{\lambda }$ 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 $V={\oplus }_{\lambda ,\mu }{V}_{\lambda ,\mu }$ .

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

(Depends on an optional video about invertibility of $\exp :𝔤\to G$ )

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

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