TITLE: Indecomposable finite-dimensional representations of the nonabelian two-dimensional Lie algebra
QUESTION [5 upvotes]: Let $\mathfrak{g}$ be the nonabelian $2$-dimensional complex Lie algebra. It can be generated by two independent vectors $e_1,e_2$ such that $[e_1,e_2]=e_1$. Thus, $\mathfrak{g}$ is solvable and it follows that its irreducible representations are $1$-dimensional (I guess this is Lie's theorem). Also we may look at the adjoint representation: it is $2$-dimensional but it can be easily shown that it is indecomposable, however. So, is it true that for each $n\in \mathbb{N}$ there is a indecomposable representation of $\mathfrak{g}$ of dimension $n$? I will be grateful if you share the examples.

REPLY [2 votes]: Here is an answer derived from the answer of Stephen.
First, write the equivalent relation $[ - e_2, e_1]= e_1$ and denote $e = - e_2$, $f=  e_1$ to get $[e,f] = f$. 
For $\lambda$ a parameter ( take it $\text{mod}\  \mathbb{Z}$) consider the infinite dimensional vector space $V_\lambda$ spanned by $v_i$, $i \in \mathbb{Z}$. It is easy to check that the formulas below give an action  of our Lie algebra on $V_{\lambda}$:
\begin{eqnarray}
 e (v_i) &=& (\lambda + i ) \cdot v_i \\
f( v_i) &=& v_{i+1}
\end{eqnarray}
For $n$ in $\mathbb{Z}$ the subspace $V_{\lambda,N}$ spanned by $v_i$, $i \ge N$ is subrepresentation of $V_{\lambda}$. Let's show that there are no other subrepresentations. Indeed,  let $W\ne 0$ an invariant subspace of  $V_{\lambda}$. Let $v \in W$, $v = \sum_{i \ge i_0} a_i v_i$, $a_{i_0} \ne 0$.  Since all the $v_i$ are eigenvectors of $e$ for distinct eigenvalues and $W$ is invariant under $e$ it follows that $v_{i_0} \in W$. Now raise the indexes to conclude $W \supset V_{\lambda, i_0}$. 
The quotient representations $V_{\lambda, N} / V_{\lambda, N + n}$  ( a subquotient of $V_{\lambda}$ ) has dimension $n$ and all its subrepresentations are $V_{\lambda, N+m} / V_{\lambda, N + n}$ for $0\le m \le n$. Hence the subrepresentations of  $V_{\lambda, N} / V_{\lambda, N + n}$ form a complete flag and hence no two can decompose the space $V_{\lambda, N} / V_{\lambda, N + n}$ in a nontrivial way.
These $V_{\lambda, N} / V_{\lambda, N + n}$, let's denote them by $W_{\lambda+n-1,n}$ should give you all the finite dimensional representations of your algebra. The dimension is of $W_{\mu, n}$ is $n$ and the highest weight of $e$ is $\mu$.