# Lie groups and Lie algebras: Questions 3

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. sl(2,C)

(Depends on the video where we worked out a similar example).

Let $\mathbf{C}^{2}$ be the standard representation of $\mathfrak{sl}(2,\mathbf{C})$ and let $X=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ and $Y=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ be the usual matrices. Calculate the action of $\mathrm{Sym}^{3}(X)$ and $\mathrm{Sym}^{3}(Y)$ on $\mathrm{Sym}^{3}(\mathbf{C}^{2})$ explicitly.

### 2. Exterior powers

(Depends on the video about symmetric powers.)

Given a representation $R\colon G\to GL(V)$ and an integer $n$ , define the alternating map $\mathrm{Alt}\colon V^{\otimes n}\to V^{\otimes n}$ by $\mathrm{Alt}(v_{1}\otimes\cdots\otimes v_{n})=\frac{1}{n!}\sum_{\sigma\in S_{n% }}\mathrm{sign}(\sigma)v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)}$ where $\mathrm{sign}(\sigma)$ is the sign of the permutation ($1$ for even permutations, $-1$ for odd permutations).

• Show that $\mathrm{Alt}$ is a morphism of representations. Its image is called the $n$ th exterior power of $V$ , written $\Lambda^{n}V$ . This is a subrepresentation of $V^{\otimes n}$ .

• Show that if $v_{i}=v_{j}$ for some $i\neq j$ then $\mathrm{Alt}(v_{1}\otimes\cdots\otimes v_{n})=0$ .

### 3. More on exterior powers

(Depends on previous question).

• Define $v\wedge w=v\otimes w-w\otimes v=2\mathrm{Alt}(v\otimes w)$ and more generally $v_{1}\wedge\cdots\wedge v_{n}=n!\mathrm{Alt}(v_{1}\otimes\cdots\otimes v_{n})$ . With this notation:

1. Let $V=\mathbf{C}^{2}$ . Write down a basis for $\Lambda^{2}V$ and show that $V^{\otimes 2}=\mathrm{Sym}^{2}(V)\oplus\Lambda^{2}(V)$ .

2. Let $W=\mathbf{C}^{3}$ . Write down a basis for $\Lambda^{2}W$ . Is it true that $W^{\otimes 2}=\mathrm{Sym}^{2}(W)\oplus\Lambda^{2}W$ ? (Hint: what are the dimensions of these vector spaces?)

3. Let $U=\mathbf{C}^{n}$ . Write down a basis of $\Lambda^{n}U$ .

• Given integers $k$ and $n$ , what is the dimension of $\Lambda^{k}\mathbf{C}^{n}$ ?

### 4. Even more on exterior powers

(Depends on previous question).

Let $e_{1},\ldots,e_{n}$ be a basis for $\mathbf{C}^{n}$ . If $v_{1}=\sum_{i=1}^{n}c_{1,i}e_{i}$ , $v_{2}=\sum_{i=1}^{n}c_{2,i}e_{i}$ , $\ldots$ , $v_{n}=\sum_{i=1}^{n}c_{n,i}e_{i}$ , show that $v_{1}\wedge\cdots\wedge v_{n}=\det(C)e_{1}\wedge\cdots\wedge e_{n}$ , where $C$ is the matrix with entries $c_{i,j}$ . (Hint: Look at the formula for the determinant in index notation.)

### 5. You're probably sick of them by now

(Depends on previous question).

Let $\mathbf{C}^{2}$ be the standard representation of $SU(2)$ . Find the weight space decomposition of $\Lambda^{2}\mathrm{Sym}^{2}(\mathbf{C}^{2})$ and hence show that $\Lambda^{2}\mathrm{Sym}^{2}(\mathbf{C}^{2})\cong\mathrm{Sym}^{2}(\mathbf{C}^{2})$ .

### 6. Pure tensors

(Depends on the video where we introduced tensor powers).

Let $V$ and $W$ be vector spaces. By tensor, I mean an element of $V\otimes W$ . A tensor of the form $v\otimes w$ is called a pure tensor.

Let $V=W=\mathbf{C}^{2}$ and let $e_{1},e_{2}$ be a basis.

• Show that $e_{1}\otimes e_{1}+e_{2}\otimes e_{1}-e_{1}\otimes e_{2}-e_{2}\otimes e_{2}$ is a pure tensor.

• If $v=ae_{1}+be_{2}$ and $w=ce_{1}+de_{2}$ , write out $v\otimes w$ .

• Using the previous part of the question, show that if $Pe_{1}\otimes e_{1}+Qe_{1}\otimes e_{2}+Re_{2}\otimes e_{1}+Se_{2}\otimes e_{2}$ is a pure tensor then $PS=QR$ .

• Write down a tensor which is not pure and justify your answer.

## 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-.

### 7. Morphisms, part 1

Suppose that $R\colon G\to GL(V)$ and $S\colon G\to GL(W)$ are representations of a group $G$ and that $L\colon V\to W$ is a morphisms of representations from $R$ to $S$ .

• Show that the kernel of $L$ is a subrepresentation of $V$ .

• Assuming there exists an invariant Hermitian inner product on $V$ , and assuming that $L$ is surjective, show that $V\cong\ker(L)\oplus W$ .

• The adjoint representation of a matrix group on its Lie algebra is the representation $\mathrm{Ad}\colon G\to GL(\mathfrak{g})$ defined by $\mathrm{Ad}(g)(X)=gXg^{-1}$ . Show that the trace map $\mathrm{Tr}\colon\mathfrak{g}\to\mathbf{R}$ is a morphism from the adjoint representation to the trivial 1-dimensional representation. Give an example of a group $G$ for which this morphism is zero and an example of a group $G$ for which this morphism is surjective.

### 8. Morphisms, part 2

(Depends on previous question and the video where we defined symmetric powers.)

• Recall that $v_{1}\cdots v_{p}$ is shorthand for $\mathrm{Av}(v_{1}\cdots v_{p})$ . Let $R\colon G\to GL(V)$ be a representation. Show that the map $F\colon\mathrm{Sym}^{p}(V)\otimes\mathrm{Sym}^{q}(V)\to\mathrm{Sym}^{p+q}(V),$ defined on pure tensors by $F((v_{1}v_{2}\cdots v_{p})\otimes(w_{1}w_{2}\cdots w_{q})=v_{1}v_{2}\cdots v_{% p}w_{1}w_{2}\cdots w_{q}$ , is a morphism of representations.

• Suppose that $V=\mathbf{C}^{2}$ is the standard representation of $SU(2)$ , that $p=1$ and $q=2$ . Write out explicitly the values of $F\colon\mathbf{C}^{2}\otimes\mathrm{Sym}^{2}(\mathbf{C}^{2})\to\mathrm{Sym}^{3% }(\mathbf{C}^{2})$ applied to a basis (e.g. elements like $e_{1}\otimes e_{1}e_{2}$ ).

• Hence find the kernel of $F$ .

• Deduce that $\mathbf{C}^{2}\otimes\mathrm{Sym}^{2}(\mathbf{C}^{2})\cong\mathbf{C}^{2}\oplus% \mathrm{Sym}^{3}(\mathbf{C}^{2})$ . (Hint: (a) Find the weight space decomposition of $\ker(F)$ . (b) Use the previous question. You may assume the existence of an invariant Hermitian inner product.)

### 9. Even more about the Heisenberg group

(Builds on Questions 4 and 11 on Sheet 2, but we recap the necessary ideas below).

Recall that $H_{3}$ is the group of 3-by-3 matrices $\begin{pmatrix}1&a&c\\ 0&1&b\\ 0&0&1\end{pmatrix}$ and that $N_{3}$ is the subgroup of matrices $\begin{pmatrix}1&0&c\\ 0&1&0\\ 0&0&1\end{pmatrix}$ with $c\in\mathbf{R}$ . We saw last time that:

1. elements of $N_{3}$ commute with everything in $H_{3}$ ,

2. if $R\colon H_{3}\to GL(V)$ is a representation then $\det(R(g))=1$ for all $g\in N_{3}$ .

Let $G_{3}\subset N_{3}$ be the subgroup of matrices for which $c\in\mathbf{Z}$ .

• Prove that $G_{3}$ is normal in $H_{3}$ (so that the quotient $H_{3}/G_{3}$ is a well-defined group).

• Why is $N_{3}/G_{3}$ isomorphic to $U(1)$ ?

• If $R\colon H_{3}/G_{3}\to GL(V)$ is a representation, let $V=\bigoplus W_{m}$ be the decomposition of $V$ into weight spaces for the action of $N_{3}/G_{3}$ . Prove that for any $g\in H_{3}/G_{3}$ and any $v\in W_{m}$ , $gv\in W_{m}$ (so that $W_{m}$ is a subrepresentation of $V$ ). (Hint: Use the fact that elements of $N_{3}$ commute with everything in $H_{3}$ . If you get stuck, take a look at the proof of the lemma in this future video: it uses the same idea you will need).

• Show that the only nonzero weight space is $W_{0}$ . (Hint: If $g\in N_{3}$ , what is the determinant of $R(g)|_{W_{m}}$ ?)

• Deduce that $H_{3}/G_{3}$ is not isomorphic to a subgroup of $GL(n,\mathbf{C})$ for any $n$ .

Even though it's not a matrix group, $H_{3}/G_{3}$ is a perfectly nice Lie group: it has local coordinates given by the three matrix entries.

### 10. Fourier theory again

As before, let $V$ be the (infinite-dimensional) vector space of complex-valued $2\pi$ -periodic functions $f(\theta)\mbox{ such that }f(\theta+2\pi)=f(\theta)$ and define the homomorphism $R\colon U(1)\to GL(V)$ by setting $R(e^{i\phi})$ to be the linear map which takes a function $f(\theta)$ to $f(\theta+\phi)$ .

Suppose that there is a function $\lambda(\phi)$ such that $f$ is a $\lambda(\phi)$ -eigenvector of $R(e^{i\phi})$ for all $\phi$ . How do we know that $\lambda(\phi)=e^{in\phi}$ for some $n\in\mathbf{Z}$ ?

### 11. Casimir, part 1

(Depends only on the video about $SL(2,\mathbf{C})$ .)

Let $X,Y,H$ be the usual elements of $\mathfrak{sl}(2,\mathbf{C})$ satisfying the commutation relations $[H,X]=2X,\quad[H,Y]=-2Y,\quad[X,Y]=H.$ Show that if $f\colon\mathfrak{sl}(2,\mathbf{C})\to\mathfrak{gl}(V)$ is a (complex linear) representation then the matrix $C:=f(X)f(Y)+f(Y)f(X)+\frac{1}{2}f(H)^{2}$ commutes with $f(M)$ for every $M\in\mathfrak{sl}(2,\mathbf{C})$ . (Hint: Check it for $M=X,Y,H$ . Is that enough?)

WARNING! Since $f$ is a representation of Lie algebras, we know that e.g. $f(H)=f([X,Y])=f(X)f(Y)-f(Y)f(X)$ . We do not know that $f(XY)=f(X)f(Y)$ (and usually this latter equation is not true, so don't use it!).

### 12. Casimir, part 2

(Depends on the previous question and the formula from the proof of the classification of irreducible $SU(2)$ -representations)

In the notation from the previous question, suppose that $V_{\lambda}\subset V$ is an eigenspace of $C$ with eigenvalue $\lambda$ . Show that $V_{\lambda}$ is a subrepresentation. If $V$ is irreducible and has highest weight $m$ , show that $C$ is the diagonal matrix $\left(m+\frac{m^{2}}{2}\right)I$ . (Hint: evaluate $C$ on the highest weight vector).