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In his paper on Kac-Moody groups Jacques Tits uses $\Phi$ to represent the set of real roots of a Kac-Moody algebra and in Section 3.7(a) he writes "Let $\Phi_{1}$ be a finite semisimple closed subsystem of $\Phi$." What does he mean by "semisimple"?

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2 Answers 2

Taken in isolation the question is reasonable, but it's important to add the explicit journal reference: J. Algebra 105 (1987), 542-573. This paper is rather technical and necessarily heavy on notation, starting with the usual definition of generalized Cartan matrix and corresponding Kac-Moody Lie algebra over $\mathbb{C}$ given by generators and relations. Tits usually refers just to "roots" or "real roots", the images of the fixed "simple" roots under the "Weyl group", since imaginary roots play little role in the paper. While he gives the usual definition of "closed" set of roots in 3.2 (a set containing the sums of any two of its roots which are again roots), he does not define explicitly what it means to be a finite "semisimple" closed subsystem. From the context, this would be a finite closed set of roots which is the usual root system of a finite dimensional semisimple subalgebra of the given Kac-Moody algebra (as Koen S suggests).

I should add that the study of Kac-Moody groups has become fairly broad and involves further work by Tits and many others such as Olivier Mathieu and Shrawan Kumar (who wrote a 2002 book on the subject). It's not clear to me how one would best approach the subject nowadays, but that's another question.

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I don't have the paper with me, but I would guess it means: "a closed subsystem corresponding to a semisimple subalgebra".

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