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 E8, D6, and A4. We show that the induced affine extensions of the non-crystallographic groups H4, H3, and H2 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 E8, D6, and A4, we place them into the broader context of crystallographic lattices such as E8, 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.
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September 2013
Research Article|
September 13 2013
Affine extensions of non-crystallographic Coxeter groups induced by projection
Pierre-Philippe Dechant;
Pierre-Philippe Dechant
a)
1Ogden Centre for Fundamental Physics, Department of Physics,
University of Durham
, South Rd., Durham DH1 3LE, United Kingdom
2Department of Mathematics, York Centre for Complex Systems Analysis,
University of York
, Heslington, York, United Kingdom
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Céline Bœhm;
Céline Bœhm
b)
1Ogden Centre for Fundamental Physics, Department of Physics,
University of Durham
, South Rd., Durham DH1 3LE, United Kingdom
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Reidun Twarock
Reidun Twarock
c)
2Department of Mathematics, York Centre for Complex Systems Analysis,
University of York
, Heslington, York, United Kingdom
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Pierre-Philippe Dechant
1,2,a)
Céline Bœhm
1,b)
Reidun Twarock
2,c)
1Ogden Centre for Fundamental Physics, Department of Physics,
University of Durham
, South Rd., Durham DH1 3LE, United Kingdom
2Department of Mathematics, York Centre for Complex Systems Analysis,
University of York
, Heslington, York, United Kingdom
a)
Electronic mail: [email protected]
b)
Electronic mail: [email protected]
c)
Electronic mail: [email protected]
J. Math. Phys. 54, 093508 (2013)
Article history
Received:
October 25 2011
Accepted:
August 22 2013
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Citation
Pierre-Philippe Dechant, Céline Bœhm, Reidun Twarock; Affine extensions of non-crystallographic Coxeter groups induced by projection. J. Math. Phys. 1 September 2013; 54 (9): 093508. https://doi.org/10.1063/1.4820441
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