Do disconnected complex lie groups exist?

Question

Is it possible for a complex Lie group to be disconnected? What about a compact complex Lie group?

Answer 1

Yes, every finite group is a complex Lie group (zero-dimensional one).

Edit. Incidentally, this wikipedia article is unaware of existence of nontrivial finite groups as it erroneously claims that every compact complex Lie group is a complex torus. The correct statement is that every connected compact complex Lie group is a complex torus. A proof can be found for instance here. The general statement is that for every compact complex Lie group $G$ there exists a short exact sequence $$ 1\to A \to G\to F\to 1 $$ where $A$ is abelian (a complex torus, the connected component of the identity in $G$) and $F$ is a finite group. Such sequence may or may not split.

Conversely, given a sequence as above, where $A$ is a complex torus and $F$ is a finite group whose action on $A$ is holomorphic, $G$ has natural structure of a compact complex Lie group. Such group is disconnected if (but not only if) $F$ is nontrivial and the sequence splits.