$${\varvec{\pi }}$$-systems of 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 set of roots such that pairwise differences of its elements are not roots. Such systems 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. For several Kac–Moody algebras with physical significance, we study the orbits of the Weyl group on $$\pi $$ -systems and completely determine the number of orbits. In particular, we show that there is a unique $$\pi $$ -system of type $${A}^{++}_1$$ (the Feingold–Frenkel rank 3 hyperbolic algebra) in $$E_{10}$$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.