Talk:Complex Lie group

Is this right?

The section Linear algebraic group associated to a complex semisimple Lie group begins as follows:

"Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows:let $A$ be the ring of holomorphic functions f on G such that $G\cdot f$ spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: $g\cdot f(h)=f(g^{-1}h)$)."

But then (g₁ g₂)⋅ f(h) = f((g₁ g₂)^-1h) = f((g₂^-1g₁^-1h)).

But by the definition of a left action, (g₁ g₂)⋅ f(h) = (g₁)⋅ (g₂⋅f(h)) for all g₁, g₂, h in G. But then we would have f((g₂^-1g₁^-1 h)) = f((g₁^-1g₂^-1 h)) for all g₁, g₂, h in G, which is not generally true. 2601:200:C000:1A0:C820:3D3D:D3F2:D688 (talk) 01:42, 24 May 2022 (UTC)

$(g_{1}g_{2}\cdot f)(h)=(g_{1}\cdot (g_{2}\cdot f))(h)=(g_{2}\cdot f)(g_{1}^{-1}h)=f(g_{2}^{-1}g_{1}^{-1}h)$. So it looks ok to me. Remember you unwind the outmost action first. -- Taku (talk) 08:34, 24 May 2022 (UTC)

How about more basic examples?

This article inludes a link to the article Table of Lie groups, which contains a fairly short table of the most essential complex Lie groups.

It would immeasurably improve this article if that table or a similar one were reproduced here. 2601:200:C000:1A0:5DA5:7EC9:7DD6:DDD1 (talk) 19:35, 3 August 2022 (UTC)

Unclear statement in Examples

The section Examples includes this one:

"Let X be a compact complex manifold. Then, as in the real case, $\operatorname {Aut} (X)$ is a complex Lie group whose Lie algebra is $\Gamma (X,TX)$."

But it is not at all clear what "$\operatorname {Aut} (X)$" would mean "in the real case", since it is unclear what *structure* is assumed to exist on the real maniifold.

Hence nobody knows what "the real case" means.

I agree the statement can use a clarification on the meaning of automorphism group. (I know I am the one who added the statement but I have no idea what I meant.) By the real case, I think it refers to the diffeomorphism group, but that’s generally an infinite-dimensional Lie group. —- Taku (talk) 07:04, 17 March 2023 (UTC)