# ${\boldsymbol\pi}$-systems of symmetrizable Kac-Moody algebras

## TL;DR

This paper develops a comprehensive theory of $$-systems for symmetrizable Kac-Moody algebras, analyzing their properties, Weyl group actions, and classifying specific cases relevant to physics.

## Contribution

It generalizes the concept of $$-systems to all symmetrizable Kac-Moody algebras and determines the number of Weyl group orbits in key examples.

## Key findings

- Unique $$-system of type $HA_1^{(1)}$ in $E_{10}$ up to Weyl group action.
- Complete classification of $$-systems in many cases of interest in physics.
- Fundamental properties of $$-systems established for symmetrizable Kac-Moody algebras.

## Abstract

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $\pi$-system. This is a subset of the roots such that pairwise differences of its elements are not roots. These arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac-Moody algebras. In this work, we systematically develop the theory of $\pi$-systems of symmetrizable Kac-Moody algebras and establish their fundamental properties. We study the orbits of the Weyl group on $\pi$-systems, and completely determine the number of orbits in many cases of interest in physics. In particular, we show that there is a unique $\pi$-system of type $HA_1^{(1)}$ (the Feingold-Frenkel algebra) in $E_{10}$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.