Affine extensions of non-crystallographic Coxeter groups induced by projection

TL;DR

This paper demonstrates how affine extensions of non-crystallographic Coxeter groups can be derived from projections of affine extended root systems, linking physical applications with broader mathematical structures.

Contribution

It introduces a method to obtain affine extensions of non-crystallographic Coxeter groups via diagram foldings and projections from affine extended root systems, connecting physical models with advanced mathematical frameworks.

Findings

Affine extensions of H_4, H_3, H_2 derived from E_8, D_6, A_4

Connections to Kac-Moody extensions and physical applications

Potential for applications in hyperbolic geometry and conformal field theory

Abstract

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of the Kac-Moody-type extensions considered in Dechant et al. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals) and chemistry (fullerenes). By connecting these here to extensions of E_8, D_6 and A_4, we place them into the broader context of crystallographic lattices such as E_8, suggesting their potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.