10 Problems for Epiphany

1. 1.

   Compute $\exp (X)$ for $X$ equal to , , and (where $s,t\in ℝ$).

2. 2.

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

Show that

as $t\to 0$, where

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

In this problem we will see that the restriction of the exponential to $𝔫$ is a homeomorphism onto $N$ (i.e. a continuous bijection with continuous inverse).

1. 1.

   Let $X\in 𝔫$. Show that $X^n=0$.

2. 2.

   Show that $\exp (X)\in N$ for $X\in 𝔫$.

3. 3.

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

4. 4.

   Show that ${\exp |}_{𝔫}$ 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 $ℝ$.

1. 1.

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

   $$\exp :𝔤{𝔩}_{n,ℂ}\to {\mathrm{GL}}_n(ℂ)$$

   is surjective.

2. 2.

   (+) Is the exponential map $\exp :𝔰{𝔩}_{2,ℂ}\to {\mathrm{SL}}_2(ℂ)$ surjective? What about $\exp :𝔤{𝔩}_{2,ℝ}\to {\mathrm{GL}}_2^{+}(ℝ)$?

Let $v\in {ℝ}^3$ be a unit vector and let $f:ℝ\to SO(3)$ 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 $v$.

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

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

Let , where $I_k$ denotes the identity matrix of size $k$. Let $n=p+q$. Let

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

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

and that $𝔰{𝔬}_{p,q}={𝔬}_{p,q}$.

1. 1.

   Show that the Lie algebras $𝔰{𝔬}_3$ and $𝔰{𝔲}_2$ are isomorphic. (Later on, we will see a conceptual reason for this).

   Hint: it is enough to find a basis for $𝔰{𝔬}_3$ and a basis for $𝔰{𝔲}_2$ which satisfy the ‘same’ Lie bracket relations. Try using the basis of $𝔰{𝔬}_3$ consisting of infinitesimal generators for rotations around the axes, and a basis for $𝔰{𝔲}_2$ related to the quaternions.

2. 2.

   Show that the Lie algebras $𝔰{𝔬}_{2,1}$ and $𝔰{𝔩}_{2,ℝ}$ are isomorphic. See Problem 55 for the definition of $𝔰{𝔬}_{2,1}$.

3. 3.

   (+) Show that the Lie algebras $𝔰{𝔬}_{3,1}$ and $𝔰{𝔩}_{2,ℂ}$ are isomorphic (as real Lie algebras). See Problem 55 for the definition of $𝔰{𝔬}_{3,1}$.

Show that:

1. 1.

   If $X\in 𝔰{𝔭}_{2n}$, then $\mathrm{tr}(X)=0$.

2. 2.

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

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.

1. 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. 2.

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

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

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

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

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

Show that, if $G$ is a connected (linear) Lie group with Lie algebra $𝔤$, then $G$ is abelian if and only if $𝔤$ is (see Definition 5.30). Hint: for the converse, consider the adjoint map $G\to GL(𝔤)$.

What goes wrong if $G$ is not connected?

Solution: see Proposition 6.14.

If $𝔤$ is a Lie algebra, let $𝔷$ be its centre:

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

Prove that $𝔷$ is the Lie algebra of $Z$.

You will always have $\mathrm{Lie}(Z)\subset 𝔷$ but the reverse inclusion requires that $G$ is connected.

Solve the exercises in section 5.8

1. 1.

   If $(\rho ,V)$ is an irreducible finite-dimensional complex representation of $𝔤$ and $𝔷$ is the centre of $𝔤$ (see problem 62), show that there is a linear map $\alpha :𝔷\to ℂ$ such that $\rho (Z)v=\alpha (Z)v$ for all $Z\in 𝔷$.

2. 2.

   For $𝔤={\mathrm{GL}}_{n,ℂ}$, find $𝔷$. Find $\alpha$ when $V={\mathrm{\Lambda }}^k{ℂ}^n$, where ${ℂ}^n$ is the standard representation and $1\le k\le n$. These representations are in fact irreducible, though we haven’t proved that yet; you can just directly show that $\alpha$ exists.

Prove that, for $X,Y\in 𝔤{𝔩}_n$,

Hence give a direct proof that $\exp ({\mathrm{ad}}_X)={\mathrm{Ad}}_{\exp (X)}$.

Consider $G=\mathrm{U}(1)$.

1. 1.

   For $\phi$ a continuous function on $G$, we define its integral

   $${\int }_G\phi (g)𝑑g=\frac{1}{2\pi }{\int }_0^{2\pi }\phi (e^{it})𝑑t.$$

   Note that ${\int }_{g}1𝑑g=1$. Show that

   $${\int }_G\phi (hg)𝑑g={\int }_G\phi (gh)𝑑g={\int }_G\phi (g)𝑑g$$

   for any $h\in G$.

2. 2.

   Let $(V,\rho )$ be a finite dimensional representation of $G$ and let $(,)$ be any Hermitian form on $V$. Define a new Hermitian form by

   $${(v,w)}_{\rho }={\int }_G(\rho (g)v,\rho (g)w)𝑑g.$$

   Show that ${(,)}_{\rho }$ is a $G$-invariant Hermitian form on $V$.

3. 3.

   Conclude that every finite-dimensional representation of $\mathrm{U}(1)$ is completely reducible. (This is analogous to Maschke’s theorem for finite groups.)

Consider the orthogonal group $\mathrm{O}(2)$.

1. 1.

   Show that $\mathrm{SO}(2)$ has index $2$ in $\mathrm{O}(2)$. Deduce that every element in $\mathrm{O}(2)$ can be uniquely written as $r_{\theta }$ or $r_{\theta }s$ with and $r_{\theta }$ the matrix for rotation by $\theta$. Show that

   $$s{r}_{\theta }=r_{-\theta }s.$$

2. 2.

   Mimic the method we used for dihedral groups to classify all irreducible finite-dimensional representations of $\mathrm{O}(2)$.

Let $V$ be the space of functions on ${ℂ}^2$ that are polynomials in the coordinates $x$ and $y$. Consider the (left) action of ${\mathrm{GL}}_2(ℂ)$ on $V$ given by

(here, think of $v=\left(\begin{array}{c}x\\ y\end{array}\right)\in {ℂ}^2$ as a column vector).

Compute the derived action for the “standard” basis of $𝔰{𝔩}_2(ℂ)$ given by , , and . You should get something involving the partial derivatives $\frac{\partial }{\partial x}$ and $\frac{\partial }{\partial y}$.

Let $V={ℂ}^2$ be the standard representation of ${\mathrm{GL}}_2(ℂ)$.

1. 1.

   Show that ${\mathrm{\Lambda }}^2(V)\cong det$ as Lie group representations.

2. 2.

   Show that ${\mathrm{\Lambda }}^2(V)\cong \mathrm{tr}$ as representations of $𝔤{𝔩}_2(ℂ)$. (You could just ‘take the derivative’ of part (a), but please do it directly instead.)

3. 3.

   Find an explicit homomorphism $\rho :{\mathrm{GL}}_2(ℂ)\to {\mathrm{GL}}_3(ℂ)$ corresponding to ${\mathrm{Sym}}^2(V)$.

Let $V$, $W$ be representations of $𝔰{𝔩}_{2,ℂ}$. Let $v$ and $w$ be two weight vectors of $V$ and $W$ respectively with respective weights $\alpha$ and $\beta$. Show that

is a weight vector with weight $\alpha +\beta$, and that if $v$ and $w$ are highest weight vectors then so is $v\otimes w$.

Let $V$ be a finite-dimensional representation of $𝔰{𝔩}_{2,ℂ}$.

1. 1.

   What are the weights of the dual representation $V^{\ast }$?

2. 2.

   Deduce that $V\cong V^{\ast }$.

Let $(\pi ,V)$ be a finite-dimensional representation of $𝔰{𝔩}_{2,ℂ}$. Consider the Casimir element⁸⁸ 8 Conventions differ; it might be more usual to call $1+2𝒞$ the Casimir.

1. 1.

   Show $𝒞$ commutes with the action of $𝔰{𝔩}_{2,ℂ}$. Conclude that if $V$ is irreducible then $𝒞$ acts as a scalar.

2. 2.

   What is the scalar for $V={\mathrm{Sym}}^n({ℂ}^2)$, the irreducible representation of highest weight $n$?

3. 3.

   Compute the action of $𝒞$ on the space $V$ of polynomial functions $\varphi$ on ${ℂ}^2$, with action the derivative of $(g\varphi )(v)=\varphi (g^{-1}v)$ (see problem 68).

If $\lambda \in ℂ$, show that there is a (possibly infinite dimensional!) representation of $𝔰{𝔩}_{2,ℂ}$ with highest weight $\lambda$.

Consider $V={\mathrm{Sym}}^n({ℂ}^2)$, the irreducible representation of highest weight $n$ of $𝔰{𝔩}_{2,ℂ}$. Decompose the following representations into irreducibles, and find highest weight vectors for the irreducible constituents:

1. 1.

   ${\mathrm{Sym}}^2({\mathrm{Sym}}^2({ℂ}^2))$;

2. 2.

   ${\mathrm{\Lambda }}^2\left({\mathrm{Sym}}^2({ℂ}^2)\right)$;

3. 3.

   ${\mathrm{Sym}}^3({ℂ}^2)\otimes {\mathrm{Sym}}^2({ℂ}^2)$;

4. 4.

   ${\mathrm{Sym}}^3\left({\mathrm{Sym}}^2({ℂ}^2)\right)$.

For the third example, find bases for the irreducible subrepresentations.

1. 1.

   For $a\ge b$ integers, decompose the representation ${\mathrm{Sym}}^a{ℂ}^2\otimes {\mathrm{Sym}}^b{ℂ}^2$ of $𝔰{𝔩}_{2,ℂ}$ into irreducibles. (This is known as the Clebsch–Gordan formula).

2. 2.

   (+) Can you find a general expression for the highest weight vectors for the irreducible subrepresentations? What about for the weight bases?

Show that the real Lie algebras $𝔰{𝔩}_{2,ℝ}$ and $𝔰{𝔲}_2$ are not isomorphic. Hint: consider the adjoint action of an arbitrary element of $𝔰{𝔲}_2$.

We have that ${\mathrm{Sym}}^n({ℂ}^2)$ is the irreducible representation of $\mathrm{SU}(2)$ of dimension $n+1$. Let ${\chi }_n$ be its character. Every conjugacy class of $\mathrm{SU}(2)$ contains an element of the form

Show that

Let $(\rho ,V)$ be an irreducible representation of $𝔤{𝔩}_{2,ℂ}$, and let .

1. 1.

   Show that $\rho (Z)$ is a scalar.

2. 2.

   Show that the restriction of $V$ to $𝔰{𝔩}_{2,ℂ}$ is irreducible.

3. 3.

   Show that for every $\lambda \in ℂ$ and integer $n\ge 0$, there is a unique irreducible representation of $𝔤{𝔩}_{2,ℂ}$ of dimension $n+1$ with $\rho (Z)=\lambda I$.

4. 4.

   Which of these are derivatives of representations of ${\mathrm{GL}}_2(ℂ)$? Hence classify the finite dimensional holomorphic irreducible representations of ${\mathrm{GL}}_2(ℂ)$.

1. 1.

   Verify the formula

   $$r^2\mathrm{\Delta }=J_x^2+J_y^2+J_z^2+{\mathrm{\ell }}^2+\mathrm{\ell }$$

   as operators on ${𝒫}^{\mathrm{\ell }}$.

2. 2.

   Find the image of the Casimir element from problem 72 under our isomorphism $𝔰{𝔩}_{2,ℂ}\to 𝔰{𝔬}_{3,ℂ}$, and compare to part 1.

Let $\mathrm{\ell }\ge 1$.

1. 1.

   Verify that ${(x-iy)}^{\mathrm{\ell }}$ is a highest weight vector in ${\mathscr{H}}^{\mathrm{\ell }}$.

2. 2.

   By applying the lowering operator, find weight vectors of weights $i(\mathrm{\ell }-1)$ and $i(\mathrm{\ell }-2)$.

3. 3.

   Find a basis of weight vectors in ${\mathscr{H}}^{\mathrm{\ell }}$ when $\mathrm{\ell }=1$ and $\mathrm{\ell }=2$ (see example 7.40).

1. 1.

   Prove that, for $f\in {𝒫}^{\mathrm{\ell }}$,

   $$\mathrm{\Delta }(r^{2}f)=r^2\mathrm{\Delta }(f)+2(2\mathrm{\ell }+3)f.$$

2. 2.

   Find a similar formula for

   $$\mathrm{\Delta }(r^{2k}f)-r^{2k}\mathrm{\Delta }(f).$$

3. 3.

   Use this to give another proof that

   $${\mathscr{H}}^{\mathrm{\ell }}\cap r^2{𝒫}^{\mathrm{\ell }-2}=\{0\}.$$

   (Hint: if $f$ is in the intersection, let $f=r^{2k}g$, $g$ not divisible by $r^2$).

1. 1.

   Let $V$ be the standard — three-dimensional — representation of $𝔰{𝔬}_3$. Find a basis of weight vectors for ${\mathrm{Sym}}^2(V)$, and decompose it into irreducible subrepresentations.

2. 2.

   Let ${\mathscr{H}}^2$ be the five-dimensional representation of $\mathrm{SO}(3)$. Decompose ${\mathscr{H}}^2\otimes {\mathscr{H}}^2$ into irreducible representations.

Verify that

and

Consider, instead of $𝔰{𝔩}_{3,ℂ}$, $𝔰{𝔩}_{2,ℂ}$. What are the roots and root spaces? What is the relation between the weights (as linear functionals on $𝔥$) and between the weights defined in section 7?

Let $V={({ℂ}^3)}^{\ast }$ be the dual of the standard representation, with basis $e_1^{\ast },e_2^{\ast },e_3^{\ast }$ dual to the standard basis.

1. 1.

   Show that the $e_i^{\ast }$ are weight vectors with weights $-L_i$.

2. 2.

   Find the action of each $E_{ij}$ on $e_3^{\ast }$ and deduce that $e_3^{\ast }$ is a highest weight vector with weight $-L_3$.

Show that $a_1{L}_1+a_2{L}_2+a_3{L}_3\in {\mathrm{\Lambda }}_W$ if and only if $a_1-a_2,a_2-a_3\in \mathbb{Z}$. Must the $a_i$ be integers?

The root lattice ${\mathrm{\Lambda }}_R\subset {\mathrm{\Lambda }}_W$ is the subgroup of the weight lattice generated by the roots.

1. 1.

   Draw a picture showing the root lattice inside the weight lattice.

2. 2.

   Show that ${\mathrm{\Lambda }}_R$ has index three in ${\mathrm{\Lambda }}_W$ (i.e. the quotient ${\mathrm{\Lambda }}_W/{\mathrm{\Lambda }}_R$ has order three).

3. 3.

   What would the root lattice and weight lattice be for $𝔰{𝔩}_{2,ℂ}$? What is the index in this case?

4. 4.

   Let $V$ be a finite-dimensional irreducible representation of $𝔰{𝔩}_{3,ℂ}$. Show that any two weights of $V$ differ by an element of the root lattice.

Find the weights of ${\mathrm{Sym}}^3({ℂ}^3)$ and draw the weight diagram.

Using weights, or otherwise, show that

where $ℂ$ is the trivial representation and $𝔰{𝔩}_{3,ℂ}$ is the adjoint representation.

(non-examinable) Let $(\rho ,V)$ be a representation of $𝔰{𝔩}_{3,ℂ}$. As ${\mathrm{SL}}_3(ℂ)$ is simply-connected $\rho$ exponentiates to a representation, $\tilde{\rho }$, of ${\mathrm{SL}}_3(ℂ)$. Let

Show that, for every weight $\alpha$, $\tilde{\rho }({\sigma }_3)$ is an isomorphism

Here $s_3(a_1{L}_1+a_2{L}_2+a_3{L}_3)=a_1{L}_2+a_2{L}_1+a_3{L}_3$.

Give another proof of Theorem 8.26.

Let $a,b\ge 0$ be integers. Check that

is a highest weight vector with weight $a{L}_1-b{L}_3$.

Show that, if $V$ is a finite-dimensional representation of $𝔰{𝔩}_{3,ℂ}$ with a unique highest weight vector (up to scalar multiplication), then $V$ is necessarily irreducible.

Deduce that the standard representation, its dual, and the adjoint representation are irreducible.

1. 1.

   Find the weights of ${\mathrm{Sym}}^2({ℂ}^3)\otimes {({ℂ}^3)}^{\ast }$ and draw the weight diagram.

2. 2.

   Show that

   $$e_1^2\otimes e_1^{\ast }+e_1{e}_2\otimes e_2^{\ast }+e_1{e}_3\otimes e_3^{\ast }\in {\mathrm{Sym}}^2({ℂ}^3)\otimes {({ℂ}^3)}^{\ast }$$

   is a highest weight vector with weight $L_1$.

3. 3.

   Let $v=e_1^2\otimes e_3^{\ast }$. Calculate $E_{32}{E}_{21}v$ and $E_{21}{E}_{32}v$ and show that they are linearly independent.

4. 4.

   Show that

   $${\mathrm{Sym}}^2({ℂ}^3)\otimes {({ℂ}^3)}^{\ast }\cong V^{(2,1)}\oplus {ℂ}^3$$

   and find the weight diagram for $V^{(2,1)}$.

(harder!) The aim of this problem is to show that, for $n\ge 0$,

It suffices to show that ${\mathrm{Sym}}^n({ℂ}^3)$ is irreducible with highest weight $n{L}_1$.

1. 1.

   Show that ${\mathrm{Sym}}^n({ℂ}^3)$ has a basis of weight vectors

   $$\{e_1^a{e}_2^b{e}_3^c:a,b,c\ge 0,a+b+c=n\}$$

   and that these have distinct weights (so, every weight has multiplicity one).

2. 2.

   Show that $e_1^n$ is the unique highest weight vector in ${\mathrm{Sym}}^n({ℂ}^3)$, up to scalar multiplication.

3. 3.

   Deduce that ${\mathrm{Sym}}^n({ℂ}^3)$ is an irreducible representation with highest weight $n{L}_1$. See problem 92.

(monster!) Let $V={ℂ}^3$, let $W=V^{\ast }$, and let $a,b>0$. For $v\in V,w\in W$, define $(v,w)=w(v)$.

Let

be defined by

where ${\widehat{v}}_i$ means $v_i$ is omitted (and similarly for ${\widehat{w}}_j$).

1. 1.

   Show that $\varphi$ is an $𝔰{𝔩}_{3,ℂ}$-homomorphism.

2. 2.

   Show that ${\mathrm{Sym}}^a(V)\otimes {\mathrm{Sym}}^b(W)$ has a unique highest weight vector of weight $(a-i)L_1-(b-i)L_3$ for each $0\le i\le \min (a,b)$, and no other highest weight vectors.

3. 3.

   Show that the highest weight vector from the previous part is in $\ker (\varphi )$ if and only if $i=0$.

4. 4.

   Deduce that $\ker (\varphi )\cong V^{(a,b)}$ is the irreducible representation of highest weight $a{L}_1-b{L}_3$.

5. 5.

   Show that $\varphi$ is surjective, and hence decompose ${\mathrm{Sym}}^a(V)\otimes {\mathrm{Sym}}^b(V^{\ast })$ into irreducibles.

6. 6.

   Find the dimension of $V^{(a,b)}$. Find its weights.

This problem is hard! For a solution, see Fulton and Harris, section 13.2, but watch out for the unjustified ’clearly’ just before Claim 13.4.