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In the Chapter 8 of Kac's book about Kac-Moody algebras there is a theorem which guarantee that we can work only with diagram automorphisms to construct twisted affine algebras. Specifically, he proves that if we start with a finite order automorphism, then the twisted affine algebra is isomorphic to another one constructed from a diagram automorphism. To prove this result, he uses that the base field is $\mathbb C$.

My question is: is there a version of this result for an algebraically closed field of characteristic zero?


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Silly remark: isn't it possible to write the whole thing in the first order language of the theory of fields? If so, then it will be automatically true in any algebraically closed field of characteristic 0. – Pasha Zusmanovich Apr 6 '12 at 8:40

It's an unfortunate tendency in textbooks on Lie algebras to assume that the base field is $\mathbb{C}$ even though virtually everything done in the classical structure theory and finite dimensional representation theory remains valid over an arbitrary algebraically closed field of characteristic 0. For the theory of affine Lie algebras the book by Kac is an essential foundational source, but a possibly more readable textbook account of his ideas was given by Roger Carter in his 2005 Cambridge University Press book Lie Algebras of Finite and Affine Type. Though Carter also tends to work by default just over the complex field, it's reasonably clear throughout that the arguments used are purely algebraic.

Concerning the introduction of twisted affine algebras (as in Carter's Chapter 18), there may be slightly different viewpoints. But for example Carter initially classifies generalized Cartan matrices and related data, then shows how to realize suitable Lie algebras having that data. In the process only the graph automorphisms play an essential role.

Maybe there are good alternatives to such books(?), but it's worthwhile anyway to consult Carter's book in order to clarify other issues you may encounter when reading denser parts of the Kac book. In principle they are both talking about the same objects.

ADDED: To respond in part to Angelo's comments, I'm not convinced that the limited context here requires any use of complex exponentials. First, it's generally agreed that the finite dimensional simple Lie algebras over $\mathbb{C}$ can be studied adequately over any algebraically closed field of characteristic 0.

But Kac needs to deal with the automorphism group of this Lie algebra as well. That can be studied from two viewpoints: complex Lie groups or algebraic groups over an algebraically closed field of characteristic 0. The automorphism group itself is essentially the same (up to base change) in either case, but it may be more convenient to construct "diagonal" automorphisms by direct use of exponentiation applied to elements of a Cartan subalgebra of the Lie algebra. From the algebraic viewpoint this is where it gets tricky, whereas automorphisms coming from root vectors can be studied using polynomial versions of exponential series. There was work by Chevalley and then by Steinberg sorting out such questions. When Kac works with automorphisms of finite order, he probably finds it more natural to work over $\mathbb{C}$ rather than get there less directly. But I still think it's a matter of taste and not related essentially to the outcome for affine Lie algebras. (That said, I've been away from this literature quite a while.)

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@Jim I agree with you and I tried many books, classics or not. When the approach is purely algebraically we can consider almost automatically for a general algebraically closed fields. However, the motivation for my question is because in the proof he uses the exponencial function of elements involving the complex number $i$. Of course we have $\sqrt{-1}$ in any algebraically closed field, but I need more faith that his demonstration works in general. – Angelo Feb 18 '12 at 20:21
@Angelo: I'd be surprised in this algebraic context if any serious use has to be made of complex exponentials, but that's not a proof. – Jim Humphreys Feb 18 '12 at 20:56
@Jim: Well, it is done in the page 126 of 3rd edition of Kac's book. – Angelo Feb 18 '12 at 21:19
@Jim: You motivated me to look at Carter's book once more. However, I am very surprised. Carter does not talk anything about the result that I am talking about! – Angelo Feb 21 '12 at 2:34
@Angelo: That's true, which suggested to me that the discussion by Kac is perhaps a side issue, using analytic language to discuss Lie algebra automorphisms. Carter does construct and classify the affine and twisted affine algebras, using algebraic methods. What I'm not sure about is how explicitly the automorphisms of finite order have been treated algebraically in the literature (Kac and Helgason being the usual sources). The automorphism groups are themselves algebraic groups. – Jim Humphreys Feb 21 '12 at 14:38

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