TITLE: What are the simply-connected two-dimensional Lie groups?
QUESTION [5 upvotes]: I would like to know what the simply-connected Lie groups of dimension $2$ are.
It is well-known that for every Lie algebra, there is exactly one simply-connected Lie group having it as its Lie algebra. I know that there are two two-dimensional Lie algebras, namely the abelian Lie algebra and another Lie algebra with basis ${x, y}$ such that $[x, y]=x$. The simply-connected Lie group corresponding to the abelian Lie algebra is $\mathbb{R}^2$. Can you tell me, what the Lie group corresponding to the other Lie algebra is?

REPLY [2 votes]: Here is perhaps a more explicit answer. Since $G$ is simply connected, it suffices to look at the Lie algebra. Let $G$ be a Lie group with a two-dimensional Lie algebra $\mathfrak{g}$ spanned by vectors
$X,A.$ If all Lie brackets of $\mathfrak{g}$ are trivial then $\mathfrak{g}$
is just a commutative Lie algebra and there is really nothing else to say.
Suppose next that $\mathfrak{g}$ is not commutative. Next, let us consider the
adjoint representation of $\mathfrak{g}$ with respect to the ordered
basis $\mathcal{B}=\left(  X,A\right)  .$ Then the matrix representation of
$ad\left(  X\right)  $ with respect to this fixed basis is given by
$$
\left[  ad\left(  X\right)  \right]  _{\mathcal{B}}=\left[
\begin{array}
[c]{cc}%
0 & a\\
0 & b
\end{array}
\right]
$$
Similarly, the matrix representation of $ad\left(  A\right)  $ with respect to
our fixed basis is given by
$$
\left[  ad\left(  A\right)  \right]  _{\mathcal{B}}=\left[
\begin{array}
[c]{cc}%
c & 0\\
d & 0
\end{array}
\right]  .
$$
Next, $\left[  ad\left(  A\right)  \right]_{\mathcal{B}}$ has for eigenvalues $c,0.$ Note
that $c$ cannot be zero. Otherwise, we could then find a basis for the Lie
algebra such that the endomorphism $adA$ is zero; contradicting the fact that $\mathfrak{g}$ is not commutative. Assuming next
that $c$ is not zero, then $c^{-1}ad\left(  A\right)  $ has for eigenvalue
$1,0.$ As such, $c^{-1}ad\left(  A\right)  $ is diagonalizable and its Jordan
canonical form is
$$
\left[
\begin{array}
[c]{cc}%
1 & 0\\
0 & 0
\end{array}
\right]  .
$$
Consequently, there exists a basis $\left(  Y,B\right)  $ for the Lie algebra
$\mathfrak{g}$ such that the only non-trivial Lie brackets are given by $\left[  B,Y\right]  =Y.$ This completes the classification of all two-dimensional Lie algebras.