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According to https://en.wikipedia.org/wiki/Complex_Lie_group, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way $G × G → G$, $(x,y)↦xy^{-1}$ is holomorphic. Basic examples are
$\operatorname {GL} _{n}(\mathbb {C} )$, the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group $\mathbb {C} ^{*}$). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group.

Questions:

  1. Can we prove that "Complex Lie group that is connected but not a complex torus, must be non-compact?"

  2. A complex Lie group is a Lie group over the complex numbers. If I start with "a real Lie group, say SU(2), over the complex numbers"; do I get a complex Lie group? Is this how the $\operatorname {SL} _{2}(\mathbb {C} )$ be constructed? Is that I start with "a real Lie algebra, say $\mathfrak{su}(2)$, over the complex numbers"; do I get a complex Lie algebra $\mathfrak{sl}(2,\mathbb {C})$?

  3. What is that meant to say "not to be confused with the complex Lie group $\mathbb {C} ^{*}$" in the Wikipedia above? (A connected compact complex Lie group is ... "not to be confused with the complex Lie group $\mathbb {C} ^{*}$")

annie marie cœur
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  • (1) is immediate from real Lie groups, (2) please clarify, (3) "complex torus" has different meanings in different context --- abelian varieties here, but it can also mean products of $\mathbb{G}_m$ as in Hilbert-Mumford theorem. – user10354138 Jul 30 '21 at 14:52
  • thanks, my context here is the Lie group and geometry - same as the Wikipedia, I hope to get comments on the Wikipedia page – annie marie cœur Jul 30 '21 at 15:06
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    For item 1, see the reference in my answer here. The key is that if you have a compact complex Lie group, then the image of its adjoint representation is a compact complex submanifold in a complex vector space, hence, must be finite. For item 3 and the terminological confusion that Wikipedia warns you about, see my answer here. Question 2 is unclear (the word "get" is unclear). – Moishe Kohan Jul 30 '21 at 17:46
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    For question 2, read the answer here and the long discussion in comments. It is possible that you have the terminological problem as OP in that question. – Moishe Kohan Jul 30 '21 at 17:49
  • Yes this helps https://math.stackexchange.com/q/642140/141334 – annie marie cœur Jul 31 '21 at 15:03
  • 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 15:07
  • @anniemariecœur: Take any (connected) complex Lie group $G$ of dimension $n$ you like, and any non-trivial finite group $F$ you like, and consider the Lie group $G \times F$. It has dimension $n$ and it is not connected (topologically, it looks like $ord(F)$ many copies of $G$). – Torsten Schoeneberg Aug 02 '21 at 14:33

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