Representations of two-dimensional Lie algebra

Question

It is widely known that there is only one $2$-dimensional non-abelian Lie algebra: it can be generated by two vectors $e_1$ and $e_2$ such that $[e_1,e_2]=e_1$. Let us lenote it by $L$. The question is how to describe all irreducible representations of $L$ (over complex numbers).

It means that we are to find all pairs of linear operators $A$ and $B$ in some vector space such that $AB-BA=A$, isn't it? I can give only one example: the adjoint representation $x\mapsto \mathrm{ad}\space x$.

Also I would like to note that I am to use as little techniques as possible.

Answer 1

By Lie's theorem all irreducible representations of a complex solvable Lie algebra are $1$-dimensional. This is a well-known statement concerning Lie's theorem: Let $\pi: \mathfrak{g}\rightarrow \mathfrak{gl}(V)$ be a representation of a complex solvable Lie algebra on a vector space $V$ . By Lie’s theorem there exists $v\in V$ such that for every $x\in \mathfrak{g}$ we have $\pi(x)(v)\in \mathbb{C}\cdot v$, so in particular $\mathbb{C}\cdot v$ is an invariant subspace of $V$.