Is it possible for a complex Lie group to be disconnected? What about a compact complex Lie group?
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Are there disconnected complex lie groups that are NOT finite groups? (p.s. any finite group as a complex Lie group (zero-dimensional one) is not so illuminating to me as a great example.) – annie marie cœur Jul 31 '21 at 15:06
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@anniemariecœur Yes. Take the direct product of any Lie group with any finite group. – Avi Steiner Aug 12 '21 at 17:24
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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.
Moishe Kohan
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5More generally, the direct product of an $n$-dimensional complex Lie group and a nontrivial finite group is a disconnected $n$-dimensional complex Lie group. – Jim Belk Jan 18 '14 at 00:05
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Are there disconnected complex lie groups that are NOT finite groups? – annie marie cœur Jul 31 '21 at 15:05
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p.s. any finite group as a complex Lie group (zero-dimensional one) is not so illuminating to me as a great example. – annie marie cœur Jul 31 '21 at 15:06
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@anniemariecœur Of course, just take the direct product of a connected one and a finite one. – Moishe Kohan Jul 31 '21 at 15:07
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Is it possible to have something more like semidirect product? nontrivial action between the twos? – annie marie cœur Aug 01 '21 at 16:19
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Dear @anniemariecœur, Yes, semidirect product you described exists, see Ch. III, sec 1, no.5, Proposition 7 of Groupes et Algèbres de Lie by Bourbaki. – Doug Jan 25 '23 at 16:44