It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan subalgebra of $L^{\phi}$ is a Cartan subalgebra in $L$. Moreover, we can always find a $\phi$-stable Cartan and a $\phi$-stable Borel subalgebra of $L$. For a reference see Chapter 8 of Kac's Book Infinite-dimensional Lie algebras.

It is too much to expect that the above results are true for a symmetrizable Kac-Moody Lie algebra. At least do we have similar theorems for affine Kac-Moody Lie algebras ? Is there a good reference where the finite order automorphisms of affine Kac-Moody Lie algebras are studied ?


Heintze, Ernst; Groß, Christian. Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category. (English summary) Mem. Amer. Math. Soc. 219 (2012), no. 1030, viii+66 pp. ISBN: 978-0-8218-6918-5

Edit: I recommend the above reference for the following reasons.

Through the work of F. Levstein, G. Rousseau, their collaborators and many others the classification of finite order automorphisms and real forms of affine Kac-Moody algebras has been completed (in the algebraic case), after 15 years of work. It is said to fill hundreds of pages.

The work above is supposed to present a much simpler approach which gives more complete results; it also works in the smooth category. In fact the interest of the authors in those questions came from differential geometry, namely in their study of infinite dimensional symmetric spaces originating from involutions of "smooth" affine Kac-Moody algebras.

Instead of using the structure of Kac-Moody algebras, the authors reduce the problems as fast as they can to the finite-dimensional case. They consider automorphisms of two kinds, according to whether they preserve or reverse the orientation of loops, and show that they have a normal form, which allows to describe them in terms of curves of automorphisms in the underlying finite-dimensional simple Lie algebra.

In the case of involutive automorphisms, they carry the analysis in detail and deduce a complete classification; this turns out to be an extension of É. Cartan's classification of symmetric spaces. They also apply their results in the complex case to conjugate linear involutions and obtain the classification of real forms.

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    $\begingroup$ It is especially for the non-experts very usful if you say something about the reference. What is in it? Why is this a good reference? Thank you. $\endgroup$ Sep 16 '20 at 15:15

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