Affine Kac Moody algebras and hyberbolic Kac Moody algebras were classified using Dynkin diagrams, like finite dimensional simple Lie algebras over $\mathbb{C}$ . Does this classification using Dynkin diagram extend further to a bigger class of Lie algebras? Or is there any other class of Lie algebras or Lie super algebras which were characterized in terms of some Dynkin diagrams?
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2$\begingroup$ Maybe i'm looking at this in the wrong way, but the reason the classifications 'work' is that you can start with a Dynkin diagram and build a Lie algebra out of it. So why not start with a Dynkin-like diagram, apply the same construction and see what you get? $\endgroup$– VincentCommented Oct 10, 2017 at 7:19
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1$\begingroup$ Kac-Moody algebras in general are constructed from an arbitrary generalized Cartan matrix, which you could think of as a generalization of an ordinary Dynkin diagram. Is this the sort of idea you have in mind? $\endgroup$– Tom De MedtsCommented Oct 10, 2017 at 12:45
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1$\begingroup$ I don't think it's accurate to say that affine Kac-Moody algebras were classified using Dynkin diagrams. The classification of the affine KM algebras is equivalent to the classification of all $n\times n$ generalized Cartan matrices which are positive semidefinite of rank $n-1$. We then use Dynkin diagrams to conveniently represent these generalized Cartan matrices (though new Dynkin diagrams had to be invented e.g. for twisted affine A2). $\endgroup$– Paul LevyCommented Oct 12, 2017 at 17:14
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