From the Trinity $(A_3,B_3,H_3)$ to an ADE correspondence

Abstract

In this paper we present novel $ADE$ correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids $(A_3,B_3,H_3)$ and the Trinity of exceptional 4D root systems $(D_4,F_4,H_4)$ to an explicit Clifford algebraic construction linking the two ADE sets of root systems $(I_2(n),A_1\times I_2(n),A_3,B_3,H_3)$ and $(I_2(n),I_2(n)\times I_2(n),D_4,F_4,H_4)$. The latter are connected through the McKay correspondence with the ADE Lie algebras $(A_n,D_n,E_6,E_7,E_8)$. We show that there are also novel indirect as well as direct connections between these ADE root systems and the new ADE set of root systems $(I_2(n),A_1\times I_2(n),A_3,B_3,H_3)$, resulting in a web of three-way ADE correspondences between three ADE sets of root systems.

25 pages, 1 figure, 3 tables

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• Notes on the K3 Surface and the Mathieu group M_24
• Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
• The $E_8$ geometry from a Clifford perspective
• The Birth of $E_8$ out of the Spinors of the Icosahedron
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Cited by in corpus (3)

• Cluster Algebras: Network Science and Machine Learning
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• Machine Learning Clifford invariants of ADE Coxeter elements