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Amazing Diagrams Everywhere

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Amazing Diagrams Everywhere ADE Platonic solids Root systems Kostant's game Amazing Diagrams Everywhere Sira Gratz Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant's game Overview 1 ADE diagrams, and some first examples 2 Quiver representations 3 Cluster algebras Sira Gratz ADE We will call these diagrams the ADE diagrams. ADE Platonic solids Overview Root systems The diagrams Kostant's game The list Name Diagram An, n ≥ 1 • • • ::: • • • • • ::: • • Dn, n ≥ 4 • • • • • • E6 • • • • • • • E7 • • • • • • • • E8 • Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant's game The list Name Diagram An, n ≥ 1 • • • ::: • • • • • ::: • • Dn, n ≥ 4 • • • • • • E6 • • • • • • • E7 • • • • • • • • E8 • We will call these diagrams the ADE diagrams. Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant's game A sample of classifications ADE diagrams show up in the classification of Platonic solids finite subgroups of SO(3) complex simple Lie algebras connected graphs with least eigenvalue −2 quivers of finite representation type cluster algebras of finite type etc. Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Platonic solids Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Kepler's Solar System Sira Gratz ADE A platonic solid is a regular convex polyhedron. All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same. ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Definition Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices. Sira Gratz ADE All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same. ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Definition Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices. A platonic solid is a regular convex polyhedron. Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Definition Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices. A platonic solid is a regular convex polyhedron. All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same. Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game List https://commons.wikimedia.org/wiki/Category: Set_of_Platonic_solids;_green;_animated Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Hosohedron and dihedron Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Duality This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license. https://creativecommons.org/licenses/by-sa/4.0/deed.en Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Duality tetrahedron ! tetrahedron cube ! octahedron dodecahedron ! icosahedron regular dihedron (n vertices) ! regular hosohedron (n faces) Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Duality tetrahedron ! tetrahedron cube ! octahedron dodecahedron ! icosahedron regular n-dihedron ! regular n-hosohedron Sira Gratz ADE These are precisely the finite subgroups of SO(3). ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Classification ADE diagram Rotational symmetries of the . An \directed" regular n-hosohedron and n-dihedron Dn regular n-hosohedron and n-dihedron E6 tetrahedron E7 cube and octahedron E8 dodecahedron and icosahedron Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant's game Classification ADE diagram Subgroup of SO(3) An cyclic group Cn Dn dihedral group Dn E6 tetrahedral group E7 octahedral group E8 icosahedral group These are precisely the finite subgroups of SO(3). Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Reflections Definition n Let α 2 R . The reflection sα is the reflection along the n hyperplane perpendicular to α: For β 2 R we have (β; α) s (β) = β − 2 α α (α; α) Sira Gratz ADE (β,α) −2 (α,α) α sα(β) (β,α) jαj ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game The reflection sα β θ α Recall: (β; α) = jβjjαj cos(θ), and, in particular, jαj2 = (α; α). Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game The reflection sα (β,α) −2 (α,α) α β sα(β) θ α (β,α) jαj Recall: (β; α) = jβjjαj cos(θ), and, in particular, jαj2 = (α; α). Sira Gratz ADE We call an element α 2 Φ a root of Φ. ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Definition Definition n A finite set Φ of non-trivial vectors in R is a (crystallographic) root system if it satisfies the following n Φ spans R The only scalar multiples of α 2 Φ that belong to Φ are ±α For all α 2 Φ we have sα(Φ) = Φ If α; β 2 Φ, then sα(β) = β − mα with m 2 Z. Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Definition Definition n A finite set Φ of non-trivial vectors in R is a (crystallographic) root system if it satisfies the following n Φ spans R The only scalar multiples of α 2 Φ that belong to Φ are ±α For all α 2 Φ we have sα(Φ) = Φ If α; β 2 Φ, then sα(β) = β − mα with m 2 Z. We call an element α 2 Φ a root of Φ. Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Example π π 3 3 Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game This file is licensed under the Creative Commons Attribution 3.0 Unported license. https://creativecommons.org/licenses/by/3.0/deed.en Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Possible angles Question Let Φ be a (crystallographic) root system, and let α; β 2 Φ. What are the possible values for the angle θ between α and β? Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Possible angles Question Let Φ be a (crystallographic) root system, and let α; β 2 Φ. What are the possible values for the angle θ between α and β? (α; β) (β; α) (α; β)2 2 2 = 4 = 4 cos2(θ) 2 (α; α) (β; β) jαj2jβj2 Z | {z } | {z } 2Z 2Z Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Possible angles Question Let Φ be a (crystallographic) root system, and let α; β 2 Φ. What are the possible values for the angle θ between α and β? 2 4 cos (θ) 2 Z; cos(θ) 2 [−1; 1] p p 1 2 3 ) cos(θ) 2 f0; ± ; ± ; ± ; ±1g 2 2 2 Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Possible angles Question Let Φ be a (crystallographic) root system, and let α; β 2 Φ. What are the possible values for the angle θ between α and β? p p 1 2 3 cos(θ) 2 f0; ± 2 ; ± 2 ; ± 2 ; ±1g π π 2π π 3π π 5π , θ 2 f 2 ; 3 ; 3 ; 4 ; 4 ; 6 ; 6 ; 0; πg Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Simple roots Definition n Let Φ ⊆ R be a root system. A subset ∆ ⊆ Φ is called a set of simple roots of Φ, if the following axioms are satisfied. n ∆ = fα1; : : : ; αng is a basis of R Every β 2 Φ can be written as n X β = mi αi i=1 such that for all i = 1;:::; n we have either mi 2 Z≥0 or mi 2 Z≤0. Sira Gratz ADE We set Φ+ = fpositive roots of Φg; Φ− = fnegative roots of Φg: ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Positive and negative roots Definition Let Φ be a root system, and ∆ = fα1; : : : ; αng ⊆ Φ a choice of simple roots of Φ. If in the expression n X β = mi αi i=1 we have mi 2 Z≥0 for all i = 1;:::; n, we say that β is positive, and else, if mi 2 Z≤0 for all i = 1;:::; n we say that it is negative. Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant's game Positive and negative roots Definition Let Φ be a root system, and ∆ = fα1; : : : ; αng ⊆ Φ a choice of simple roots of Φ. If in the expression n X β = mi αi i=1 we have mi 2 Z≥0 for all i = 1;:::; n, we say that β is positive, and else, if mi 2 Z≤0 for all i = 1;:::; n we say that it is negative.
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