Affine extensions of non-crystallographic Coxeter groups induced by projection

TY - JOUR

T1 - Affine extensions of non-crystallographic Coxeter groups induced by projection

AU - Dechant, Pierre-Philippe

AU - Bœhm, Céline

AU - Twarock, Reidun

N1 - (c) 2013 AIP Publishing LLC. This is an author produced version of a paper published in Journal of Mathematical Physics. Uploaded in accordance with the publisher's self-archiving policy.

PY - 2013/9/11

Y1 - 2013/9/11

N2 - 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 those considered in [P.-P. Dechant, C. Boehm, and R. Twarock, J. Phys. A: Math. Theor. 45, 285202 (2012)]. 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.

AB - 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 those considered in [P.-P. Dechant, C. Boehm, and R. Twarock, J. Phys. A: Math. Theor. 45, 285202 (2012)]. 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.

UR - https://blogs.ams.org/visualinsight/2015/01/01/icosidodecahedron-from-projected-d6-root-polytope/

U2 - 10.1063/1.4820441

DO - 10.1063/1.4820441

M3 - Article

SN - 0022-2488

VL - 54

SP - 1

EP - 22

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 9

M1 - 93508

ER -