Exercises Weeks 3 & 4

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

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 $\underbrace{D^r \oplus \dots \oplus D^r}_{n}$ .
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 $F$ .
Is the ring $\mathbb{Z}$ semisimple? (we did this one in class)
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:

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?
Show that the character of $\rho_1 \otimes \rho_2$ is $\chi(g_1,g_2)=\chi_1(g_1) \chi_2(g_2)$ .
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.
Verify and experiment with the examples in my notes (to be posted soon-ish).

From Reuben’s talk on Lie algebras:

Show there is a bijection between representations of $L$ , and left $L$ -modules.
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 $X+X^T=0$ .
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 $\circ_1 = \circ_2$ ?
Find an example of an indecomposable $L$ -module which is not irreducible (i.e. contains nontrivial proper submodules).
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.)
Show that Lie algebras consisting of upper triangular matrices are solvable, and strictly upper triangular matrices are nilpotent.
Show that every irreducible representation of a solvable Lie algebra is one-dimensional.

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

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 $\mathfrak{g}/\textrm{Rad}(\mathfrak{g}) \simeq \mathfrak{sl}_n$ .
Prove the following direction of the Cartan-Killing criterion: If $\mathfrak{g}$ is a semisimple Lie algebra, then its Killing form is nondegenerate.
Prove that $\mathfrak{sl}_n$ , $\mathfrak{so}_n$ , $\mathfrak{sp}_{2n}$ are simple Lie algebras.