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I have been reading about the classification of real simple Lie algebras, and one of the key theorems states:

Let $S$ be a semisimple complex Lie algebra. The map $$\Psi:\Bigg\{\begin{split}&\text{Isomorphism classes}\\&\text{of real forms of $S$}\end{split}\Bigg\}\longrightarrow\Bigg\{\begin{split}&\text{Conjugacy classes in}\\&\text{ $\operatorname{Aut}_\mathbb{C}S$ of involutions}\end{split}\Bigg\}$$ by sending $[\sigma]$ (an equivalence class of conjugations associated to $S$) to $[\sigma\tau]$, where $\tau$ is a compact conjugation that commutes with $\sigma$. Then $\Psi$ is well-defined and bijective.

I believe the inverse map to this is by taking the fixed subalgebra $S^{\theta}:=\{x\in S:\theta(x)=x\}$ given an involution, which is a real form of $S$. This reminds me of the fundamental theorem of Galois theory, but with "real forms of $S$" viewed as a "subfields" of $S$.

Since this is coming from an amateur in Galois theory, are there actually some connections within this realisation? More generally, is there an "extension of Lie algebras" analogous to field extensions?

blastzit
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  • An example for an extension of Lie algebras is given, say, here. But I think you mean something different. In fact, Galois cohomology plays a role to describe the forms for Lie algebras. See for example this MO-post. – Dietrich Burde Jun 13 '22 at 14:41
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    I think the "extension" you are looking for as inverse of real forms is complexification, or more general scalar extension, cf. Fact 2 in https://math.stackexchange.com/a/4184237/96384. However, this needs field extensions as prerequisite. Quite generally, like Dietrich says, for a Galois extension $E\vert K$ the classification of $K$-forms of (semisimple) LAs over $E$ involves Galois cohomology. Very vaguely speaking, one has to check in what ways $Gal(E\vert K)$ can act on the automorphism group of our LA. I.e. we need to know two groups, and how they relate. – Torsten Schoeneberg Jun 13 '22 at 16:51
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    Two more remarks: 1) For the above, I advertise section 4.2 of my thesis, but again I stress this only covers semisimple LAs. I have no idea what happens beyond that (in particular, solvable ones). 2) I think the right framework to view all that is Galois descent, which is a vast theory mostly beyond my understanding as well. Cf. https://mathoverflow.net/q/22032/27465, but also K. Conrad's very user-friendly notes of basic "algebraic" cases, https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisdescent.pdf . – Torsten Schoeneberg Jun 13 '22 at 16:59

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