Abstract :

The derived algebra of a symmetrizable Kac--Moody Lie algebra is generated (as a Lie algebra) by its real root spaces. In this talk, we address the natural converse question: given any symmetric subset of real root vectors, is the Lie subalgebra generated by the subset again a derived algebra of a some Kac--Moody Lie algebra? We call such Lie algebras root generated and give an affirmative answer to the above question. Moreover, we show that the exact analog of Dynkin's famous result (proved for finite-dimensional semisimple Lie algebras) holds for symmetrizable Kac--Moody algebras namely, there exist one-to-one correspondences between the root generated subalgebras, real closed subroot systems, and the real $\pi$-systems containing the positive roots. We shall discuss how the above results are related to the graded embedding problem for Kac--Moody Lie algebras. Finally, we apply our results to the untwisted affine case to classify the symmetric regular subalgebras (introduced by Dynkin in the finite-dimensional setting). We also show that in the untwisted affine case, any root-generated subalgebra associated with a maximal closed subroot system can be embedded into a unique maximal symmetric regular subalgebras. This is a joint work with Deniz Kus and R. Venkatesh.