\({\varvec{\pi }}\)-systems of symmetrizable Kac–Moody algebras


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.

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The authors are grateful to Thibault Damour for suggesting that we try to prove conjugacy of embeddings of \({A}^{++}_1\) into \(E_{10}\), and for many illuminating discussions. We also wish to thank Ling Bao, Paul Cook and Axel Kleinschmidt for helpful discussions.

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Correspondence to Sankaran Viswanath.

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The first author’s research is partially supported by the Simons Foundation, Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Award Number 422182. KNR and SV acknowledge support from DAE under a XII plan project. KR acknowledges SICI for an SRSF grant.

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Carbone, L., Raghavan, K.N., Ransingh, B. et al. \({\varvec{\pi }}\)-systems of symmetrizable Kac–Moody algebras. Lett Math Phys 111, 5 (2021). https://doi.org/10.1007/s11005-020-01345-2

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  • \(\pi \)-system 
  • Regular subalgebra
  • Weyl group orbits
  • Symmetrizable Kac–Moody algebras

Mathematics Subject Classification

  • 17B22 (17B67)