Root systems & Clifford algebras: from symmetries of viruses to $E_8$ & ADE correspondences

TL;DR

This paper explores the use of Clifford algebras to analyze root systems and reflection groups, revealing connections between viral symmetries, exceptional Lie algebras, and geometric structures like $E_8$ and ADE correspondences.

Contribution

It introduces a Clifford algebra framework for root systems and reflection groups, enabling new proofs and explicit connections between polytopes, Lie algebras, and symmetry groups.

Findings

Constructed $E_8$ from $H_3$ root systems.

Established $ADE$ correspondences via induction proofs.

Linked viral symmetry structures to mathematical root systems.

Abstract

In this paper we discuss reflection groups and root systems, in particular non-crystallographic ones, and a Clifford algebra framework for both these concepts. A review of historical as well as more recent work on viral capsid symmetries motivates the focus on the icosahedral root system $H_3$. We discuss a notion of affine extension for non-crystallographic groups with applications to fullerenes and viruses. The icosahedrally ordered component of the nucleic acid within the virus capsid and the interaction between the two have shed light on the viral assembly process with interesting applications to antiviral therapies, drug delivery and nanotechnology. The Clifford algebra framework is very natural, as it uses precisely the structure that is already implicit in this root system and reflection group context, i.e. a vector space with an inner product. In addition, it affords a uniquely simple reflection formula, a double cover of group transformations, and more insight into the geometry, e.g. the geometry of the Coxeter plane. This approach made possible a range of root system induction proofs, such as the constructions of $E_8$ from $H_3$ and the exceptional 4D root systems from 3D root systems. This makes explicit various connections between what Arnold called Trinities (sets of three exceptional cases). In fact this generalises further since the induction construction contains additional cases, namely two infinite families of cases. It therefore actually yields $ADE$ correspondences between three sets of different mathematical concepts that are usually thought of separately as polytopes, subgroups of $SU(2)$ and $ADE$ Lie algebras. Here we connect them explicitly and in a unified way by thinking of them as three different sets of root systems.