A Characterization of Minimal Extended Affine Root Systems (Relations to Elliptic Lie Algebras)

Abstract

Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called “minimal”, turns out to be of special interest, mostly because of the geometric properties of their Weyl groups; they possess the so-called presentation by conjugation. In this work, we characterize minimal extended affine root systems in terms of “minimal reflectable bases”, which resembles the concept of the “base” for finite and affine root systems. As an application, we construct elliptic Lie algebras by means of Serre-type generators and relations. © 2025 Research Institute for Mathematical Sciences, Kyoto University.