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\begin{center}
Introduction to Lie Theory\\
Homework \#6\\
\end{center}


\begin{enumerate}
\item
Let 
$\mathfrak{g} = \mathfrak{g}_1\oplus \mathfrak{g}_2$ be a direct sum
of two Lie algebras.
Let $V$ be a finite-dimensional $\mathfrak{g}$-module.
Prove that $V$ is completely reducible as a $\mathfrak{g}$-module if and only it is completely
reducible
both as a $\mathfrak{g}_1$-module and as a $\mathfrak{g}_2$-module.

({\em Hint.} Consider the finite-dimensional algebra $A$ that is the
image of $U(\mathfrak{g})$ under the induced algebra homomorphism 
$\rho:U(\mathfrak{g}) \rightarrow \End_{\C}(V)$, use Wedderburn theorem.)
\item
Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra with
Killing form $\kappa$.
We showed in L6-2 that $\mathfrak{g}$ is the
direct sum $\mathfrak{g}_1\oplus\cdots\oplus
\mathfrak{g}_n$ of its simple ideals, with $\mathfrak{g}_i \perp_\kappa
\mathfrak{g}_j$ for $i \neq j$.
Let $\beta$
be some  other invariant bilinear form on $\mathfrak{g}$.
Prove:
\begin{enumerate}
\item $\beta|_{\mathfrak{g}_i}$ is a multiple of the Killing form
$\kappa|_{\mathfrak{g}_i}$ for each $i=1,\dots,n$.
\item $\mathfrak{g}_i \perp_\beta \mathfrak{g}_i$
for $i \neq j$.
\item $\beta$ is a symmetric bilinear form.
\end{enumerate}
\end{enumerate}
The remaining questions require the notion of a {\em toral subalgebra} $\mathfrak{t}$
of a finite-dimensional semisimple Lie algebra $\mathfrak{g}$. Recall that this is a subalgebra
consisting entirely of semisimple elements of $\mathfrak{g}$. As 
established in L7-1, such a subalgebra is necessarily
Abelian. Hence, there is a {\em root space decomposition}
\begin{equation*}
\mathfrak{g} = \mathfrak{g}_0 \oplus \bigoplus_{\alpha \in R}
\mathfrak{g}_\alpha
\end{equation*}
where $R = \{0 \neq \alpha \in \mathfrak{t}^*\:|\:
\mathfrak{g}_\alpha \neq 0\}$ is the set of {\em roots} of
$\mathfrak{g}$ with respect to $\mathfrak{t}$. Here, 
$$\mathfrak{g}_\alpha = \{x \in \mathfrak{t}\:|\:[t,x] =
\alpha(t) x\text{ for all }t \in \mathfrak{t}\}
$$
for any $\alpha \in \mathfrak{t}^*$.
\begin{itemize}
\item[3.]
Suppose that $\mathfrak{t}$ is a toral subalgebra of a
finite-dimensional semisimple Lie algebra $\mathfrak{g}$
such that $\mathfrak{t}$ is equal to its centralizer
$\mathfrak{c}_{\mathfrak{g}}(\mathfrak{t})$.
Why does this imply that $\mathfrak{t}$ is actually a {\em maximal}
toral subalgebra of $\mathfrak{g}$? 
Use this observation to show that the subalgebra of $\mathfrak{sl}_n(\C)$
consisting of all diagonal matrices of trace zero is a maximal toral
subalgebra.
\end{itemize}
Now let $\mathfrak{g} = \mathfrak{sp}_{2n}(\C)$ using the
 coordinates chosen in L3-2. Let $\mathfrak{t}$ be the subalgebra
  of all diagonal matrices in $\mathfrak{g}$, and note that
any element $t \in \mathfrak{t}$ can be written as
$t = \operatorname{diag}(t_1,\dots,t_n,-t_n,\dots,-t_1)$ for $t_1,\dots,t_n \in \C$.
Let $\eps_i \in \mathfrak{t}^*$ be the linear map sending $t \mapsto
t_i$ for $i=1,\dots,n$, so that $\eps_1,\dots,\eps_n$ give a basis for $\mathfrak{t}^*$.
\begin{itemize}
\item[4.] Describe the root space decomposition 
of  $\mathfrak{g} = \mathfrak{sp}_{2n}(\C)$ with respect to
$\mathfrak{t}$ 
using the notation just
  introduced.
You should show in particular that
$$
R = \{\pm \eps_i \pm \eps_j, \pm 2 \eps_k\:|\:1 \leq i < j \leq n, 1
\leq k \leq n\},
$$
that $\mathfrak{g}_\alpha$ is one-dimensional for each $\alpha \in R$,
and that $\mathfrak{g}_0 = \mathfrak{t}$. Deduce that $\mathfrak{t}$
is a maximal toral subalgebra.
\item[5.]
Let $V = \C^{2n}$ be the natural representation of $\mathfrak{g} =
\mathfrak{sp}_{2n}(\C)$
denoting its standard ordered basis $v_1,\dots,v_n, v_{-n}, \dots, v_{-1}$.
\begin{itemize}
\item[(a)] We noted in L5-3 that $V$ is an irreducible
  $\mathfrak{g}$-module. Convince yourself of this again! It follows
  that $\mathfrak{g}$ is indeed a semisimple Lie algebra.
\item[(b)]
Show that $v_i$ is of weight $\eps_i$ and $v_{-i}$ is of
  weight $-\eps_i$, where $\eps_1,\dots,\eps_n \in \mathfrak{t}^*$ are
  as defined above.
\item[(c)] Describe how $S^2(V)$ decomposes into weight spaces with
  respect to $\mathfrak{t}$. Then show that $S^2(V) \cong \mathfrak{g}$ (the adjoint
  representation) as $\mathfrak{g}$-modules.
\item[(d)] Show that $S^2(V)$ is an irreducible
  $\mathfrak{g}$-module.
Deduce that $\mathfrak{g}$ is a simple Lie algebra.
\end{itemize}
({\em Hint.} For (a) and/or (d) you might find it easier to prove
irreducibility as representations of the algebraic group
$G = Sp_{2n}(\C)$.)
\end{itemize}
\end{document}
