π -systems of symmetrizable Kac–Moody algebras

Lisa Carbone, K. N. Raghavan, Biswajit Ransingh, Krishanu Roy, Sankaran Viswanath

Research output: Contribution to journalArticlepeer-review

Abstract

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a π-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 π-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 π-systems and completely determine the number of orbits. In particular, we show that there is a unique π-system of type A1++ (the Feingold–Frenkel rank 3 hyperbolic algebra) in E10 (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

Original languageEnglish (US)
Article number5
JournalLetters in Mathematical Physics
Volume111
Issue number1
DOIs
StatePublished - Feb 2021

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Regular subalgebra
  • Symmetrizable Kac–Moody algebras
  • Weyl group orbits
  • π-system

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