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Root Systems and Clifford Algebras Dechant, Pierre-Philippe ORCID: https://orcid.org/0000-0002-4694-4010 (2016) A new construction of E8 and the other exceptional root systems. In: Algebra Seminar, 18th January 2016, Queen Mary University of London. (Unpublished) Downloaded from: http://ray.yorksj.ac.uk/id/eprint/4025/ Research at York St John (RaY) is an institutional repository. It supports the principles of open access by making the research outputs of the University available in digital form. Copyright of the items stored in RaY reside with the authors and/or other copyright owners. Users may access full text items free of charge, and may download a copy for private study or non-commercial research. For further reuse terms, see licence terms governing individual outputs. Institutional Repository Policy Statement RaY Research at the University of York St John For more information please contact RaY at [email protected] Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay correspondence E8 from the icosahedron H4 as a rotation group II: The Coxeter plane AnewconstructionofE8 and the other exceptional root systems Pierre-Philippe Dechant Department of Mathematics (and Biology), University of York QMUL algebra seminar – January 18, 2016 Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay correspondence E8 from the icosahedron H4 as a rotation group II: The Coxeter plane Main results Each 3D root system induces a 4D root system H3 (icosahedral symmetry) induces the E8 root system Cli↵ord algebra is a very natural framework for root systems and reflection groups Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Overview 1 Root systems and Cli↵ord algebras Root systems Cli↵ord Basics 2 H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay correspondence 3D to 4D spinor induction Trinities and McKay correspondence 3 E8 from the icosahedron 4 H4 as a rotation group II: The Coxeter plane Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Root systems Root system Φ: set of a2 a1 + a2 vectors a in a vector space with an inner product such that 1. Φ = , Φ a a1 Ra a a a − 1 \ {− }8 2 2. s Φ=Φ a Φ a 8 2 Simple roots:express (a1 + a2) a2 every element of Φvia a − − Z-linear combination. (v a) reflection/Coxeter groups s : v s (v)=v 2 | a a ! a − (a a) | Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Cartan Matrices (ai ,aj ) aj Cartan matrix of ai sis Aij =2 =2| | cosqij (a ,a ) a i i | i | 2 1 A2: A = − 12 Coxeter-Dynkin diagrams: node✓− = simple◆ root, no link = roots p p orthogonal, simple link = roots at 3 ,linkwithlabelm = angle m . 4 5 n A3 B3 H3 I2(n) Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Lie groups to Lie algebras to Coxeter groups to root systems Lie group: manifold of continuous symmetries (gauge theories, spacetime) Lie algebra: infinitesimal version near the identity Non-trivial part is given by a root lattice Weyl group is a crystallographic Coxeter group: An,Bn/Cn,Dn,G2,F4,E6,E7,E8 generated by a root system. So via this route root systems are always crystallographic. Neglect non-crystallographic root systems I2(n),H3,H4 . Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Non-crystallographic Coxeter groups H2 H3 H4 t(a1 + a2) ⇢ ⇢ a1 + ta2 ta + a a2 1 2 a1 a − 1 (ta + a ) a2 − 1 2 − (a + ta ) − 1 2 t(a + a ) − 1 2 H H H : 10, 120, 14,400 elements, the only Coxeter groups 2 ⇢ 3 ⇢ 4 that generate rotational symmetries of order 5 linear combinations now in the extended integer ring 1 p Z[t]= a + tb a,b Z golden ratio t = (1 + p5) = 2cos { | 2 } 2 5 2 1 2p x = x +1 t0 = s = (1 p5) = 2cos t + s =1,ts = 1 2 − 5 − Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane The Icosahedron Rotational icosahedral group is I = A5 of order 60 Full icosahedral group is H3 of order 120 (including reflections/inversion); generated by the root system icosidodecahedron Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Cli↵ord Algebra and orthogonal transformations Form an algebra using the Geometric Product for two vectors ab a b + a b ⌘ · ^ Inner product is symmetric part a b = 1 (ab + ba) · 2 Reflecting a in b is given by a0 = a 2(a b)b = bab (b and − · − b doubly cover the same reflection) − Via Cartan-Dieudonn´e theorem any orthogonal (/conformal/modular) transformation can be written as successive reflections x0 = n n ...n xn ...n n = AxA˜ ± 1 2 k k 2 1 ± Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKayRoot correspondence systems E8 from the icosahedron Cli↵ord Basics H4 as a rotation group II: The Coxeter plane Cli↵ord Algebra of 3D E.g. Pauli algebra in 3D (likewise for Dirac algebra in 4D) is 1 e ,e ,e e e ,e e ,e e I e e e { } { 1 2 3} { 1 2 2 3 3 1} { ⌘ 1 2 3} 1scalar 3vectors 3bivectors 1trivector We can|{z}multiply| together{z } | root vectors{z in this} | algebra{z ai a} j ... A general element has 8 components, even products (rotations/spinors) have four components: R = a + a e e + a e e + a e e RR˜ = a2 + a2 + a2 + a2 0 1 2 3 2 3 1 3 1 2 ) 0 1 2 3 So behaves as a 4D Euclidean object – inner product 1 (R ,R )= (R R˜ + R R˜ ) 1 2 2 2 1 1 2 Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Overview 1 Root systems and Cli↵ord algebras Root systems Cli↵ord Basics 2 H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay correspondence 3D to 4D spinor induction Trinities and McKay correspondence 3 E8 from the icosahedron 4 H4 as a rotation group II: The Coxeter plane Check axioms: 1. Φ Ra = a,a a Φ \ {− }8 2 2. s Φ=Φ a Φ a 8 2 Proof: 1. R and R are in a spinor group by construction − (double cover of orthogonal transformations), 2. closure under reflections is guaranteed by the closure property of the spinor group (with a twist: R R˜ R ) − 1 2 1 Induction Theorem: Every rank-3 root system induces a rank-4 root system (and thereby Coxeter groups) Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay3D to correspondence 4D spinor induction E8 from the icosahedron Trinities and McKay correspondence H4 as a rotation group II: The Coxeter plane Induction Theorem – root systems Theorem: 3D spinor groups give 4D root systems. Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Proof: 1. R and R are in a spinor group by construction − (double cover of orthogonal transformations), 2. closure under reflections is guaranteed by the closure property of the spinor group (with a twist: R R˜ R ) − 1 2 1 Induction Theorem: Every rank-3 root system induces a rank-4 root system (and thereby Coxeter groups) Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay3D to correspondence 4D spinor induction E8 from the icosahedron Trinities and McKay correspondence H4 as a rotation group II: The Coxeter plane Induction Theorem – root systems Theorem: 3D spinor groups give 4D root systems. Check axioms: 1. Φ Ra = a,a a Φ \ {− }8 2 2. s Φ=Φ a Φ a 8 2 Pierre-Philippe Dechant A new construction of E8 and the other exceptional root systems Root systems and Cli↵ord algebras H4 as a rotation group I: 3D to 4D spinor induction, Trinities and McKay3D to correspondence 4D spinor induction E8 from the icosahedron Trinities and McKay correspondence H4 as a rotation group II: The Coxeter plane Induction Theorem – root systems Theorem: 3D spinor groups give 4D root systems.
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