2022/04/12 by Pierre-Philippe Dechant, Dechant, Pierre-Philippe
Biochemistry, Genetics and Molecular Biology · Materials Science · Mathematics · Physics and Astronomy · #14E16 #15A66 #17B22 #20F55 #52B10 #52B11 #52B15 #97M50 #97M60 #97M80 #Biomolecules (q-bio.BM) #DNA and Nucleic Acid Chemistry #Enzyme Structure and Function #FOS: Biological sciences #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph) #Microtubule and mitosis dynamics #Quantum Algebra (math.QA) #math-ph #math.GR #math.MP #math.QA #msc:14E16 #msc:15A66 #msc:17B22 #msc:20F55 #msc:52B10 #msc:52B11 #msc:52B15 #msc:97M50 #msc:97M60 #msc:97M80 #q-bio.BM
paper · pdf · doi:10.48550/arxiv.2204.05718
22 pages; Proceedings of the Nankai Symposium on Geometry, Physics and Number Theory
arxiv created 2022/04/12 · openalex publication_date 2022/04/12 · arxiv updated 2022/04/13 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 H3. 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 E8 from H3 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.