# Exercises From Cartan Subalgebras and Root Systems

1. Why are diagonal matrices their own normaliser?
2. How you do define determinant for a general linear operator? (i.e without defining a basis)
3. Why is the exponential map well defined for matrices? $exp: gl_{n} \rightarrow GL(g)$ Where the exponential map takes a matrix to its exponential Taylor series.
4. Consider $gl_{2} \& gl_{3}$, what is $\mathrm{ad}(x)$, and what is $e^{\mathrm{ad}(x)}$ ?
5. Show the rank of  $gl_{n}$ is n, where rank is the dimension of a Cartan Subalgebra.
6. Show that the regular elements in $gl_{n}$ are diagonal matrices with no repeated entries.
7. If we define the map $-1 : V \rightarrow V$ which takes a root $\alpha \rightarrow -\alpha$, For which root systems is this in the Weyl Group?
8. (Very optional) Why is the Dynkin diagram for $E_{8}$ not allowed to have a connection on the central root?

Don’t forget that next week is the problem session!

# Exercises Week 5

Complex Lie Algebras in Dimensions 1, 2, 3:

1. Let $L_2$ be the unique non-abelian Lie algebra of dimension 2. What is the realisation of $L_2$ as a Lie subalgebra of a matrix algebra?
2. Suppose we have a Lie algebra of dimension 3 and rank 1. Show that we can choose a basis $\{X, Y, Z\}$ such that $[X, Y] = [X, Z] = 0$. We know that the resulting Lie algebra has $[Y, Z] = X$ (WLOG). Show this is isomorphic to the Lie algebra of strictly upper-triangular $3 \times 3$ matrices. Write an explicit isomorphism.
3. Let $\mathfrak{g}_\alpha$ be the dimension 3 Lie algebra with $[X, Y] = Y$, $[X, Z] = \alpha Z$, $[Y, Z] = 0$. Show that $\mathfrak{g}_\alpha \simeq \mathfrak{g}_{\alpha'}$ iff $\alpha = \alpha'$ or $\alpha = \frac{1}{\alpha'}$. What is the realisation of this as a Lie subalgebra of a matrix algebra?
4. What is the realisation of the dimension 3 Lie algebra with $[X, Y] = Y$, $[X, Z] = Y + Z$, $[Y, Z] = 0$? Show that this is not isomorphic to the above.

Rep Theory of $\mathfrak{sl}_2(\mathbb{C})$:

1. Construct one infinite-dimensional irreducible representation of $\mathfrak{sl}_2(\mathbb{C})$.
2. Show that the standard representation, with $H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, $X= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, is isomorphic to $V(1)$.
3.  Show that $\textrm{Sym}^n S \simeq V(n)$.

# Exercises Weeks 3 & 4

From Mitchell’s talk on $\mathbb{C}[G]$-modules:

1. If $D$ is a division algebra, then check that every $D$-module is of the form $D^r$, and as a corollary, all $M_n(D)$-modules are of the form $\underbrace{D^r \oplus \dots \oplus D^r}_{n}$.
2. Prove the following lemma: If $F$ is a field, then $V$ is a (finitely generated) $F[G]$-module, then $V$ is a (finite dimensional) vector space over $F$.
3. Is the ring $\mathbb{Z}$ semisimple? (we did this one in class)
4. Prove the following theorem of Wedderburn: If $F$ is algebraically closed, then any semisimple algebra over $F$ deomposes as $\bigoplus_{i=1}^r M_{n_i}(F)$. As a corollary, this holds for $F[G]$. What does $r$ represent? What does this mean in the context of representations?

From my talk on representations of group products, and induced representations:

1. Let $\rho_1: G_1 \to GL(V_1), \rho_2: G_2 \to GL(V_2)$ be representations. Consider the representation $\rho_1 \oplus \rho_2 : G_1 \times G_2 \to GL(V_1 \oplus V_2)$ defined appropriately. Is it interesting?
2. Show that the character of $\rho_1 \otimes \rho_2$ is $\chi(g_1,g_2)=\chi_1(g_1) \chi_2(g_2)$.
3. We have shown that if $\rho_1, \rho_2$ are irreducible, $\rho_1 \otimes \rho_2$ is irreducible. Show then that in fact every irreducible representation of $G_1 \times G_2$ is of this form.
4. Verify and experiment with the examples in my notes (to be posted soon-ish).

From Reuben’s talk on Lie algebras:

1. Show there is a bijection between representations of $L$, and left $L$-modules.
2. Show that $\mathbb{R}^3$ under the cross product can be realised (by an isomorphism) as $\mathfrak{so}_2(\mathbb{R})$, the Lie subalgebra of $\mathfrak{gl}_2(\mathbb{R})$ containing matrices satisfying $X+X^T=0$.
3. Suppose there exist $G, \circ_1, \circ_2$ such that $(G, \circ_1)$ and $(G, \circ_2)$ are groups, and $(g \circ_1 h) \circ_2 k = g \circ_1 (h \circ_2 k)$. What can we say about $G$? Does it necessarily hold that $\circ_1 = \circ_2$?
4. Find an example of an indecomposable $L$-module which is not irreducible (i.e. contains nontrivial proper submodules).
5. Prove the following lemmas: For a Lie algebra $L$, and $m, n \geq 1$, $[L^m, L^n] \subseteq L^{m+n}$; and for $n \geq 0$, $L^{(n)} \subseteq L^{2^n}$. (It follows directly that nilpotent Lie algebras are solvable.)
6. Show that Lie algebras consisting of upper triangular matrices are solvable, and strictly upper triangular matrices are nilpotent.
7. Show that every irreducible representation of a solvable Lie algebra is one-dimensional.

And finally, from Chris’ talk on semisimple Lie algebras:

1. For $\mathfrak{g} = \mathfrak{gl}_n$, show that the radical is given by the centre, the subalgebra of nonzero scalar matrices, of the form $z I_n, z \in F^*$. Show also that $\mathfrak{g}/\textrm{Rad}(\mathfrak{g}) \simeq \mathfrak{sl}_n$.
2. Prove the following direction of the Cartan-Killing criterion: If $\mathfrak{g}$ is a semisimple Lie algebra, then its Killing form is nondegenerate.
3. Prove that $\mathfrak{sl}_n$, $\mathfrak{so}_n$$\mathfrak{sp}_{2n}$ are simple Lie algebras.

# Week 2 Exercises

Some questions we left open last time:

1. Let $\rho: \mathbb{Z}/n \to \mathbb{C}^*$ be given by $1 \mapsto \zeta$ for some $n$-th root of unity $\zeta$. Show that $\rho^*$ is $1 \mapsto \bar{\zeta}$. Conclude that $\rho, \rho^*$ are not isomorphic.
2. Let $A$ be an $n \times n$ matrix of finite order ($A^m = I$ for some $m \in \mathbb{Z}$). Show that $tr(A^{-1}) = \overline{tr(A)}$, without using eigenvalues.
3. In the proof of Prop. 1 of Serre (see Section 2.1), he states that “… $\rho_s$ can be defined to be a unitary matrix”. Show that every representation of a finite group is unitary. (As part of this exercise, give an appropriate definition of a unitary representation.)
4. Prove that the character of $V_1 \otimes V_2$ is given by $\chi_1 \cdot \chi_2$. If $\rho$ is a representation, prove that the character of a dual representation is $\chi_{\rho^*} = \overline{\chi_\rho}$. Find the characters of $\textrm{Sym}^2(\rho)$ and $\rho \wedge \rho = \textrm{Alt}^2(\rho)$.
5. If $f: (\rho_1, V_1) \to (\rho_2, V_2)$ is a map of representations, show that the kernel and image of $f$ are subrepresentations of $(\rho_1, V_1), (\rho_2, V_2)$ respectively.
6. Midterm question: Give your own proof of Serre Thm. 3 (section 2.3), that characters of irreducible representations form an orthonormal system.
7. Prove Prop. 6 of Serre (section 2.5): If $(\rho, V)$ is an irreducible representation on $G$ of degree $n$ and character $\chi$, and $f$ is a class function on $G$, then $\rho_f := \sum_{t \in G} f(t)\rho(t)$ is a homothety of ratio $\frac{1}{n} \sum_{t \in G} f(t) \chi(t)$.
8. Prove that the number of degree 1 representations on a finite group $G$ is $|G^{ab}|$.

# Exercises from Week 1

Here’s a list of some questions which had been left unanswered from the week 1 meeting:

1. Consider representations on vector spaces where the base field is not $\mathbb{C}$ (e.g. vector spaces over $\mathbb{F}_2$). What can go wrong?
2. If you choose two different characters (degree 1 representations) of a group, why aren’t they isomorphic?
3. Let $G$ be a finite group. How many different degree 1 representations does $G$ have?
4. Consider a group $G$ which acts on a set $X$, and show that this induces a corresponding action on $Fun(X)$ (the set of all functions mapping from $X$ into some other set, say $\mathbb{C}$). Show that the regular representation of $G$ is a particular case of this.
5. Theorem 1 of Serre states: Let $\rho: G\to GL(V)$ be a representation, and let $W$ be a subspace of $V$ which is stable under $\rho$; then there is a complement space $W^c$ which is also stable.
Write a proof of this theorem in your own words. Highlight where we use the following facts: (1) $G$ is finite; (2) $V$ is finite-dimensional; (3) $V$ is over the field $\mathbb{C}$.
For each of the three conditions, find a counter-example to show that the theorem doesn’t hold if the condition is dropped.
(This will also be a midterm exam question.)
6. If $G$ is a group, why is $F(G)$ (the free vector space on $G$) the same as the regular representation of $G$?
7. For vector spaces $U,V,W,X$, does it hold that $\textrm{Hom}(U,W)\otimes\textrm{Hom}(V,X) \simeq \textrm{Hom}(U\otimes V,W\otimes X)$?
8. If $U$ and $V$ are finite dimensional vector spaces, prove that $(U \otimes V)^* = U^* \otimes V^*$. Show that this does not hold in general when $U, V$ are infinite-dimensional.
9. For vector spaces $U, V, W$, prove that $\textrm{Hom}(U\otimes V,W)=\textrm{Hom}(U,\textrm{Hom}(V,W))$.
10. Let $V$ be a finite dimensional vector space, and $V^*$ denote its dual and $V^{**}=(V^*)^*$ denote its double dual. One knows that $V$ is isomorphic to $V^*$ and to $V^{**}$. However, the isomorphism $V\simeq V^{**}$ is much better than the isomorphism $V\simeq V^*$!! Here is one reason:
Suppose $G$ acts on $V$; i.e., we have a representation $\rho:G\to GL(V)$. Show that $G$ acts on $V^*$ and $V^{**}$. Let us denote these representations by $\rho^*$ and $\rho^{**}$. Show that the representations $(\rho, V)$ and $(\rho^{**}, V^{**})$ are always isomorphic, where as, it could happen that $(\rho,V)$ and $(\rho, V^*)$ are not isomorphic!!
One says that the isomorphism $V \simeq V^{**}$ is canonical or functorial, where as the isomorphism $V\simeq V^*$ is not (it depends on a choice of basis).