Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Root systems & Clifford algebras: from symmetries of viruses to $E_8$ &
  ADE correspondences   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.

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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin