Reducible but not decomposable loop algebra representations Let $\mathfrak{g}$ be a simple, finite-dimensional Lie algebra. We define its loop algebra $L\mathfrak{g} = \mathfrak{g} \otimes \mathbb{C}\lbrack t,t^{-1} \rbrack$ with Lie bracket $\lbrack x \otimes t^n, y \otimes t^m \rbrack = \lbrack x,y \rbrack \otimes t^{n+m}$, where the bracket to the right is the one on $\mathfrak{g}$. I would like to find a representation of $L\mathfrak{g}$ that is reducible but not decomposable. Surprisingly, I could not find any answers to this in the literature, except for a small mention here [1], though this proved far too advanced for me.
[1] https://pdfs.semanticscholar.org/8c44/102fa07290bb35516cefcdc1b7d5fada7b7d.pdf
 A: From my old homework solutions:
Let $\mathfrak{g}$ be any simple finite-dimensional Lie algebra. Let $W$ be a
nontrivial finite-dimensional irreducible representation of $\mathfrak{g}$.
[Such a representation $W$ exists, because otherwise all
finite-dimensional irreducible representations of $\mathfrak{g}$ would be
trivial, so that the adjoint representation would have a filtration with
all subquotients being trivial, and thus $\mathfrak{g}$ would be nilpotent, which
contradicts to $\mathfrak{g}$ being simple.]
Consider the polynomial ring
$\mathbb{C}\left[  \varepsilon\right]$
(where $\varepsilon$ is an indeterminate)
and its quotient ring
$\mathbb{C}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)$
(a $2$-dimensional $\mathbb{C}$-vector space, with algebra
structure similar to
the dual numbers
but defined over $\mathbb{C}$).
Given any $u \in \mathbb{C}\left[\varepsilon\right]$, we let $\overline{u}$ be the canonical projection of $u$ onto $\mathbb{C}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)$.
We notice that we can consider the Lie algebra $\mathfrak{g}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  :=\mathfrak{g}
\otimes\mathbb{C}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)
$ as a quotient Lie algebra of
$L\mathfrak{g}$ by means of the Lie algebra homomorphism $L\mathfrak{g}
=\mathfrak{g}\otimes\mathbb{C}\left[  t,t^{-1}\right]  \rightarrow
\mathfrak{g}\otimes\mathbb{C}\left[  \varepsilon\right]  /\left(
\varepsilon^{2}\right)  $ obtained by tensoring $\operatorname*{id}
:\mathfrak{g}\rightarrow\mathfrak{g}$ with the $\mathbb{C}$-algebra
homomorphism
\begin{align*}
\mathbb{C}\left[  t,t^{-1}\right]   &  \rightarrow\mathbb{C}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  ,\\
t  &  \mapsto\overline{1+\varepsilon}.
\end{align*}
Thus, in order to construct a reducible but not decomposable
finite-dimensional representation of $L\mathfrak{g}$, it is enough to
construct a reducible but not decomposable finite-dimensional representation
of $\mathfrak{g}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)  $.
We let $\mathfrak{g}\left[  \varepsilon\right]  /\left(  \varepsilon
^{2}\right)  $ act on $W\oplus W$ as follows:
\begin{align}
\left(  a+b\overline{\varepsilon}\right)  \rightharpoonup\left(  v,w\right)
=\left(  a\rightharpoonup v,\ b\rightharpoonup v+a\rightharpoonup w\right)  .
\end{align}
(Here, $g \rightharpoonup w$ means the result of a Lie algebra element
$g$ acting on a module element $w$.)
It is clear that this representation of $\mathfrak{g}\left[  \varepsilon
\right]  /\left(  \varepsilon^{2}\right)  $ is reducible (it has $0\oplus W$
as a subrepresentation). Now, we are going to show that it is indecomposable.
In fact, it is easy to see that every nonzero $\mathfrak{g}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  $-subrepresentation of
$W\oplus W$ contains a nonzero element of $0\oplus W$.
[Proof. Let $V$ be a nonzero $\mathfrak{g}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  $-subrepresentation of
$W\oplus W$. Then, there exists a nonzero $\left(  v,w\right)  \in V$. We want
to prove that $V$ contains a nonzero element of $0\oplus W$.
If $v=0$, then we are done (because then, $\left(  v,w\right)  $ is a nonzero
element of $0\oplus W$). So assume $v\neq0$.
There exists a $b\in\mathfrak{g}$ such that $b\rightharpoonup v\neq0$ (because
otherwise, $\mathfrak{g}v$ would be $0$, so that $W$ would contain a trivial
subrepresentation, and thus be trivial itself (since $W$ is irreducible),
contradicting the fact that $W$ is nontrivial). For this $b$, the vector
$\left(  b\overline{\varepsilon}\right)  \rightharpoonup\left(  v,w\right)
=\left(  0,\ b\rightharpoonup v\right)  $ is a nonzero element of $0\oplus W$
contained in $V$ (since $\left(  v,w\right)  \in V$). Thus, we have proven
that $V$ contains a nonzero element of $0\oplus W$, qed.]
Thus, every nonzero $\mathfrak{g}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  $-subrepresentation of
$W\oplus W$ contains $0\oplus W$.
[Proof. Let $V$ be a nonzero
$\mathfrak{g}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)
$-subrepresentation of $W\oplus W$. We have already shown that $V$ contains a
nonzero element of $0\oplus W$. In other words, there exists a nonzero $w\in
W$ such that $\left(  0,w\right)  \in V$. But since $W$ is irreducible and
$w\neq0$, we have $U\left(  \mathfrak{g}\right)  \cdot w=W$
(where $U\left(\mathfrak{h}\right)$ denotes the universal enveloping algebra
of a Lie algebra $\mathfrak{h}$). Now, since
$\left(  0,w\right)  \in V$ and since $V$ is a $\mathfrak{g}\left[
\varepsilon\right]  /\left(  \varepsilon^{2}\right)  $-representation, we have
\begin{align}
V &\supseteq\underbrace{U\left(  \mathfrak{g}\left[  \varepsilon\right]
/\left(  \varepsilon^{2}\right)  \right)  }_{\substack{\supseteq U\left(
\mathfrak{g}\right)  \\\text{(where we canonically include }\mathfrak{g}\text{
in }\mathfrak{g}\left[\varepsilon\right]/\left(\varepsilon^2\right)
\\\text{by sending }x\text{ to }
x+0\overline{\varepsilon}\text{)}}}\cdot\left(  0,w\right)  \supseteq U\left(
\mathfrak{g}\right)  \cdot\left(  0,w\right)  =\left(  0,\underbrace{U\left(
\mathfrak{g}\right)  \cdot w}_{=W}\right)  \\
&=\left(  0,W\right)  =0\oplus W.
\end{align}
Thus, $V$ contains $0\oplus W$, qed.]
Hence, if we were able to decompose
the $\mathfrak{g}\left[  \varepsilon\right]  /\left(  \varepsilon^{2}\right)
$-representation $W\oplus W$ into two nonzero addends, then
both of these addends
would contain $0\oplus W$, which is absurd. We are thus done proving the
indecomposability of $W\oplus W$.
