Lie groups and Lie algebras: Questions 3

1. sl(2,C)

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

Let ${𝐂}^2$ be the standard representation of $𝔰𝔩(2,𝐂)$ and let and be the usual matrices. Calculate the action of ${\mathrm{Sym}}^3(X)$ and ${\mathrm{Sym}}^3(Y)$ on ${\mathrm{Sym}}^3({𝐂}^2)$ explicitly.

2. Exterior powers

(Depends on the video about symmetric powers.)

Given a representation $R:G\to GL(V)$ and an integer $n$ , define the alternating map $$\mathrm{Alt}:V^{\otimes n}\to V^{\otimes n}$$ by $$\mathrm{Alt}(v_1\otimes \mathrm{\cdots }\otimes v_n)=\frac{1}{n!}\sum _{\sigma \in S_n}\mathrm{sign}(\sigma )v_{\sigma (1)}\otimes \mathrm{\cdots }\otimes v_{\sigma (n)}$$ 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 ${\mathrm{\Lambda }}^{n}V$ . This is a subrepresentation of $V^{\otimes n}$ .

• Show that if $v_i=v_j$ for some $i\ne j$ then $\mathrm{Alt}(v_1\otimes \mathrm{\cdots }\otimes v_n)=0$ .

3. More on exterior powers

(Depends on previous question).

• Define $v\wedge w=v\otimes w-w\otimes v=2\mathrm{A}\mathrm{l}\mathrm{t}(v\otimes w)$ and more generally $v_1\wedge \mathrm{\cdots }\wedge v_n=n!\mathrm{Alt}(v_1\otimes \mathrm{\cdots }\otimes v_n)$ . With this notation:

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

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

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

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

4. Even more on exterior powers

(Depends on previous question).

Let $e_1,\mathrm{\dots },e_n$ be a basis for ${𝐂}^n$ . If $v_1={\sum }_{i=1}^n{c}_{1,i}{e}_i$ , $v_2={\sum }_{i=1}^n{c}_{2,i}{e}_i$ , $\mathrm{\dots }$ , $v_n={\sum }_{i=1}^n{c}_{n,i}{e}_i$ , show that $v_1\wedge \mathrm{\cdots }\wedge v_n=det(C)e_1\wedge \mathrm{\cdots }\wedge e_n$ , 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 ${𝐂}^2$ be the standard representation of $SU(2)$ . Find the weight space decomposition of ${\mathrm{\Lambda }}^2{\mathrm{Sym}}^2({𝐂}^2)$ and hence show that ${\mathrm{\Lambda }}^2{\mathrm{Sym}}^2({𝐂}^2)\cong {\mathrm{Sym}}^2({𝐂}^2)$ .

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 $V=W={𝐂}^2$ 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 $v=a{e}_1+b{e}_2$ and $w=c{e}_1+d{e}_2$ , write out $v\otimes w$ .

• Using the previous part of the question, show that if $P{e}_1\otimes e_1+Q{e}_1\otimes e_2+R{e}_2\otimes e_1+S{e}_2\otimes e_2$ is a pure tensor then $PS=QR$ .

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

7. Morphisms, part 1

(Depends on the video in which we defined morphisms of representations.)

Suppose that $R:G\to GL(V)$ and $S:G\to GL(W)$ are representations of a group $G$ and that $L:V\to W$ 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 $\mathrm{Ad}:G\to GL(𝔤)$ defined by $\mathrm{Ad}(g)(X)=gX{g}^{-1}$ . Show that the trace map $\mathrm{Tr}:𝔤\to 𝐑$ 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 $v_1\mathrm{\cdots }{v}_p$ is shorthand for $\mathrm{Av}(v_1\mathrm{\cdots }{v}_p)$ . Let $R:G\to GL(V)$ be a representation. Show that the map $$F:{\mathrm{Sym}}^p(V)\otimes {\mathrm{Sym}}^q(V)\to {\mathrm{Sym}}^{p+q}(V),$$ defined on pure tensors by $F((v_1{v}_2\mathrm{\cdots }{v}_p)\otimes (w_1{w}_2\mathrm{\cdots }{w}_q)=v_1{v}_2\mathrm{\cdots }{v}_p{w}_1{w}_2\mathrm{\cdots }{w}_q$ , is a morphism of representations.

• Suppose that $V={𝐂}^2$ is the standard representation of $SU(2)$ , that $p=1$ and $q=2$ . Write out explicitly the values of $$F:{𝐂}^2\otimes {\mathrm{Sym}}^2({𝐂}^2)\to {\mathrm{Sym}}^3({𝐂}^2)$$ applied to a basis (e.g. elements like $e_1\otimes e_1{e}_2$ ).

• Hence find the kernel of $F$ .

• Deduce that ${𝐂}^2\otimes {\mathrm{Sym}}^2({𝐂}^2)\cong {𝐂}^2\oplus {\mathrm{Sym}}^3({𝐂}^2)$ . (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 and that $N_3$ is the subgroup of matrices with $c\in 𝐑$ . We saw last time that:

1. elements of $N_3$ commute with everything in $H_3$ ,

2. if $R:H_3\to GL(V)$ 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 $c\in 𝐙$ .

• 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 $R:H_3/G_3\to GL(V)$ is a representation, let $V=\oplus W_m$ 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 $GL(n,𝐂)$ 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

(Depends on Question 12 on Sheet 2 and on the video about representations of $U(1)$ .)

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

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

11. Casimir, part 1

(Depends only on the video about $SL(2,𝐂)$ .)

Let $X,Y,H$ be the usual elements of $𝔰𝔩(2,𝐂)$ satisfying the commutation relations $$[H,X]=2X,[H,Y]=-2Y,[X,Y]=H.$$ Show that if $f:𝔰𝔩(2,𝐂)\to 𝔤𝔩(V)$ 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 $M\in 𝔰𝔩(2,𝐂)$ . (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!).

There is an in-depth project which investigates this phenomenon in more detail.

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).