Affine extensions of non-crystallographic Coxeter groups induced by projection
Affine extensions of non-crystallographic Coxeter groups induced by projection
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DOI：
10.1063/1.4820441
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发表时间：
2013-09-01
影响因子：
1.3
通讯作者：
Twarock, Reidun
中科院分区：
文献类型：
--
作者：
Dechant, Pierre-Philippe;Boehm, Celine;Twarock, Reidun
在本文中，我们证明了非晶体学Coxeter群的仿射扩张可以通过Coxeter-Dynkin图折叠和根系E-8，D-6和A（4）的仿射扩张版本的投影导出。我们证明了非晶体学群H-4，H-3和H-2的诱导仿射扩张对应于[P. -]中考虑的那些的一个特殊子集。P. Dechant，C. Boehm，和R. Twarock，J. Phys. A：Math. Theor. 45，285202（2012）]。这类扩展的动机是在生物学（病毒），物理学（准晶）和化学（富勒烯）中二十面体系统的物理应用。通过将它们与E-8、D-6和A（4）的扩展联系起来，我们将它们置于更广泛的晶体学晶格（如E-8）的背景下，表明它们在高能物理、可积系统和模形式理论中的应用潜力。通过反转投影，我们可以在Cartan矩阵中引入不同的数域，这可能会在双曲几何和有理共形场论中开辟诱人的可能性。(C)2013 AIP Publishing LLC.
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. (C) 2013 AIP Publishing LLC.