Question

I'm aware that there is a classification of certain kinds of complex Lie groups like semisimple or compact. Is there a classification of the Lie groups in the 1-dimensional case? It seems to me that the only Lie groups are the complex plane ${\mathbb{C}}$, the multiplicative group of nonzero complex numbers ${\mathbb{C}}^*$ and the torus ${\mathbb{T}}$.

Answer 1

Think about the easier question of classifying one-dimensional complex Lie algebras: there's only one! -- namely, $\mathbb{C}$ with the trivial bracket. The simply connected Lie group with $\mathbb{C}$ as its Lie algebra is $\mathbb{C}$ itself, and thus all other connected complex Lie groups with Lie algebra $\mathbb{C}$ are quotients of $\mathbb{C}$ by a discrete subgroup $\Gamma \subset \mathbb{C}$. This subgroup $\Gamma$ is a lattice in $\mathbb{C}$ and is therefore classified by rank: if $\mathrm{rank} \, \Gamma = 1$, then $\mathbb{C}/\Gamma = \mathbb{C}^\ast$; and if $\mathrm{rank} \, \Gamma = 2$, then $\mathbb{C}/\Gamma$ is a Riemann surface of genus one (and by varying $\Gamma$ all such arise).

Answer 2

the answer is "no": if you construct $\mathbb C/(\mathbb Z + \mathbb Z\tau)$ (1 and $\tau$ are $\mathbb R$-linearly independent) you will get an "elliptic curve" that is also a Lie group. For different $\tau$ corresponding elliptic curves may not be equivalent as complex Lie groups