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Affine extensions of non-crystallographic Coxeter groups induced by projection

Dechant, Pierre-Philippe; Boehm, Celine; Twarock, Reidun

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Authors

Pierre-Philippe Dechant

Celine Boehm

Reidun Twarock



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 those considered in [P.-P. Dechant, C. Bœhm, 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.

Citation

Dechant, P., Boehm, C., & Twarock, R. (2013). Affine extensions of non-crystallographic Coxeter groups induced by projection. Journal of Mathematical Physics, 54(9), Article 093508. https://doi.org/10.1063/1.4820441

Journal Article Type Article
Publication Date Sep 1, 2013
Deposit Date Jan 20, 2014
Publicly Available Date Jan 24, 2014
Journal Journal of Mathematical Physics
Print ISSN 0022-2488
Electronic ISSN 1089-7658
Publisher American Institute of Physics
Peer Reviewed Peer Reviewed
Volume 54
Issue 9
Article Number 093508
DOI https://doi.org/10.1063/1.4820441

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Copyright Statement
© 2013 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Dechant, Pierre-Philippe and Boehm, Celine and Twarock, Reidun (2013) 'Affine extensions of non-crystallographic Coxeter groups induced by projection.', Journal of mathematical physics., 54 (9). 093508 and may be found at http://dx.doi.org/10.1063/1.4820441





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