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arXiv · 2311.12583

Root generated subalgebras of symmetrizable Kac-Moody algebras

Irfan Habib·Deniz Kus·R. Venkatesh
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Abstract

The derived algebra of a symmetrizible Kac-Moody algebra $\lie g$ is generated (as a Lie algebra) by its root spaces corresponding to real roots. In this paper, we address the natural reverse question: given any subset of real root vectors, is the Lie subalgebra of $\lie g$ generated by these again the derived algebra of a Kac-Moody algebra? We call such Lie subalgebras root generated, give an affirmative answer to the above question and show that there is a one-to-one correspondence between them, real closed subroot systems and $\pi$-systems contained in the positive system of $\lie g$. Finally, we apply these identifications to all untwised affine types in order to classify symmetric regular subalgebras first introduced by Dynkin in the finite-dimensional setting. We show that any root generated subalgebra associated to a maximal real closed subroot system can be embedded into a unique maximal symmetric regular subalgebra.

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Irfan Habib, Deniz Kus, R. Venkatesh. 2023-11-21. Root generated subalgebras of symmetrizable Kac-Moody algebras. https://arxiv.org/abs/2311.12583

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