Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality

JOURNAL OF GEOMETRY AND PHYSICS

卷:114

Article

Khudaverdian, H. M.¹  Mkrtchyan, R. L.² 

[1] Univ Manchester, Sch Math, Oxford Rd, Manchester M13 9PL, Lancs, England

[2] Yerevan Phys Inst, 2 Alikhanian Br Str, Yerevan 0036, Armenia

关键词: Simple Lie algebras;    McKay correspondence;    Vogel's universality;    Diophantine equations;    Regular maps;   

DOI  :  10.1016/j.geomphys.2016.11.021

来源: Elsevier

【 摘 要 】

We notice that one of the Diophantine equations, knm = 2kn + 2km + 2nm, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes regular polyhedra with k edges in each vertex, n edges of each face, with total number of edges vertical bar m vertical bar, and Euler characteristics chi = +/- 2. In the case of negative m this equation corresponds to chi = 2 and describes true regular polyhedra, Platonic solids. The case with positive m corresponds to Euler characteristic chi = -2 and describes the so called equivelar maps (charts) on the surface of genus 2. In the former case there are two routes from Platonic solids to simple Lie algebras-abovementioned Diophantine classification and McKay correspondence. We compare them for all solutions of this type, and find coincidence in the case of icosahedron (dodecahedron), corresponding to E-8 algebra. In the case of positive k, n and m we obtain in this way the interpretation of (some of) the mysterious solutions (Y-objects), appearing in the Diophantine classification and having some similarities with simple Lie algebras. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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