CRYSTALLOGRAPHIC ROOT SYSTEMS MATH61202 MSc Single Project Spring Semester - 2014 Aram Dermenjian Student ID: 9071063 Supervisor: Sasha Premet School of Mathematics Contents Abstract 4 1 Introduction - Reflection Groups 5 1.1 Euclidean Space[6],[7] ............................ 5 1.2 Reflection Groups[8],[9] ........................... 7 1.2.1 I2(m)................................ 9 1.2.2 An .................................. 10 1.2.3 Bn .................................. 12 1.2.4 Dn .................................. 12 2 Lie Algebras 14 2.1 History[2],[11] ................................. 14 2.2 Definitions[5],[8] ............................... 15 2.3 Solvability and Nilpotency[8],[13] ...................... 16 2.4 Representations[5],[8] ............................. 19 2.5 Killing Form[5],[8] .............................. 21 2.6 Cartan Decompositions[8] .......................... 23 2.7 Weight Spaces[8] ............................... 27 2.8 A new Euclidean space[8] .......................... 28 3 Root Systems 31 3.1 Definitions[8],[9] ............................... 31 3.2 Positive and Simple Roots[9],[15] ...................... 32 3.3 Coxeter Groups[3],[9] ............................. 35 3.4 Fundamental Domains[9] .......................... 41 3.4.1 Parabolic Subgroups . 41 2 3.4.2 Fundamental Domains . 42 3.5 Irreducible Root Systems[8] ......................... 44 4 Classification of Root Systems 46 4.1 Coxeter Graphs[9] .............................. 46 4.2 Crystallographic Root Systems[5],[8] .................... 47 4.3 Dynkin Diagrams[8] ............................. 49 4.4 Classification . 50 4.4.1 Classification of Dynkin Diagrams[4] ................ 51 4.4.2 Classification of Coxeter Graphs[9] ................. 57 5 Simple Lie Algebras 64 5.1 Connections with Crystallographic Root Systems[3],[9] .......... 64 5.2 Simple Lie Algebras Constructed[3],[8],[9] .................. 67 5.2.1 An .................................. 67 5.2.2 Bn .................................. 68 5.2.3 Cn .................................. 68 5.2.4 Dn .................................. 69 5.2.5 G2 .................................. 69 5.2.6 F4 .................................. 70 5.2.7 E8 .................................. 71 5.2.8 E7 .................................. 71 5.2.9 E6 .................................. 72 6 Conclusion 73 Word count 20,276 3 Abstract This project shows how the attempt to classify all semisimple Lie algebras in an alge- braically closed field of characteristic 0 led to the development of a rich area of study called root systems. Root systems have many applications, not only in Lie algebra, but also in geometry and combinatorics. We’ll examine what is needed in order to classify all crystallographic root systems and will briefly touch on noncrystallographic ones as well. Additionally we will explain how this classification relates to semisimple Lie algebras and also take a slight detour midway to show that root systems are a powerful theory in their own right through the use of Coxeter groups. A list of references used in this project can be found in the bibliography. Each section will be cited with a list of the one or two most dominantly used references for that section. No definitions are my own (unless otherwise stated) and come from the references as outlined in the section header. Each proof replicated from a source are reworded to fit better with the structure provided, and referenced accordingly to show the proof idea was not my original work. 4 Chapter 1 Introduction - Reflection Groups 1.1 Euclidean Space[6],[7] Let V be a nonempty set. We say that V is a vector space over an arbitrary field F if it is an abelian group under addition and for all a, b F and ⌫,µ V we have 2 2 (i) a⌫ V 2 (ii) a(⌫ + µ)=a⌫ + aµ (iii) (a + b)⌫ = a⌫ + b⌫ (iv) a(b⌫)=(ab)⌫ (v) 1⌫ = ⌫ We can see that Rn := R R ... R is a vector space over a field F by letting the ⇥ ⇥ ⇥ n n elements of R be ordered n-tuples (⌫1, ⌫2,...,⌫n)suchthateach⌫i F and by taking | {z } 2 addition and scalar multiplication to be defined as: (⌫1, ⌫2,...,⌫n)+(µ1,µ2,...,µn)=(⌫1 + µ1, ⌫2 + µ2,...,⌫n + µn) a(⌫ , ⌫ ,...,⌫ )=(a⌫ ,a⌫ ,...,a⌫ ) a F 1 2 n 1 2 n 2 Definition 1.1.1. We say that a vector space V over a field F is an Euclidean space (or an inner product space)ifwecandefineapositivedefinitesymmetricbilinear form ( , )(calledtheinner product)onV . In other words for any ⌫,µ,! V and − − 2 a, b F : 2 5 (i) (⌫, ⌫) 0and(⌫, ⌫) = 0 if and only if ⌫ =0 (positivedefinite) ≥ (ii) (⌫,µ)=(µ, ⌫)(symmetric) (iii) (a⌫ + b!,µ)=a(⌫,µ)+b(!,µ)(bilinear) As (⌫, ⌫)isalwayspositivewecandefinethelength of a vector as ⌫ = (⌫, ⌫). k k p We also define the angle ✓(⌫,µ)betweenanytwovectors⌫ and µ as (⌫,µ) cos ✓(⌫,µ)= . ⌫ µ k kk k To see that Rn is an Euclidean space we need an inner product on it. We thus define the dot product of any two vectors ⌫ and µ in Rn as follows: (⌫,µ) ⌫ µ = ⌫ µ + ⌫ µ + ...+ ⌫ µ 7! · 1 1 2 2 n n It is easy to see the dot product is an inner product on Rn thus making Rn an Euclidean space. We call Rn the real Euclidean space. Similarly we can define an inner product on the complex numbers which allow us to call Cn the complex Euclidean space. We start looking at me features this inner product allows us to have. Definition. Any two vectors are orthogonal if (⌫,µ)=0. It’s easy to notice that orthogonality implies that the angle ✓ between two vectors ⇡ must be 2 . If we have a set of vectors one would expect multiple vectors to potentially be orthogonal to any given one motivating us to define the following. Definition. If W is a subspace of an Euclidean space V then the orthogonal com- plement of W is the set W ? := ⌫ V (⌫, !)=0forall! W { 2 | 2 } We easily see that W ? is a subspace of V . We next state an important theorem. A proof can be found in [6]. 6 Theorem 1.1.2 (Kernel-image theorem). Let V and W be vector spaces over some field F and φ be a linear map from V to W . Then dim(im(φ)) + dim(ker(φ)) = dim(V ) By the Kernel-image theorem we see that an inner product allows us to create a map from V to W such that dim(W )+dim(W ?)=dim(V ). Definition 1.1.3. If we are given any vector space V over a field F and we take the set of all linear transformations from V to F then that set is known as the dual space of V . The dual space is denoted as V ? and is a vector space over F with the following operations (φ + )(⌫)=φ(⌫)+ (⌫) (aφ)(⌫)=a(φ(⌫)) for all φ, V ?, ⌫ V and a F . 2 2 2 It can be shown that for finite dimensional vector spaces there is a basis of elements that generate the whole dual space. This basis is called the dual basis. It can also be shown that dim V ? = dim V . We state without proof the following theorem which has a pleasant corollary. A proof can be found in [6]. Theorem 1.1.4. If V and W are vector spaces over a field F such that dimV = dimW then V and W are isomorphic, written V ⇠= W . ? Corollary 1.1.5. V ⇠= V 1.2 Reflection Groups[8],[9] Definition 1.2.1 (Reflection). A reflection in real Euclidean space V can be thought of as a linear transformation s on V which sends a nonzero vector ↵ to its negative and fixes point-wise the hyperplane H↵ (a subspace with codimension 1). The standard formula for describing a reflection is 2(⌫, ↵) s (⌫)=⌫ ↵. ↵ − (↵, ↵) 7 To see that our formula coincides with our definition we recall that two vectors are orthogonal if (⌫,µ)=0.Let’ssuppose↵ is a nonzero vector we are sending to its negative, and ⌘ is a nonzero vector in the hyperplane of ↵ (⌘ H ). 2 ↵ 2(↵, ↵) 2(⌘, ↵) s (↵)=↵ ↵ s (⌘)=⌘ ↵ ↵ − (↵, ↵) ↵ − (↵, ↵) 0 = ↵ 2↵ = ⌘ ↵ − − (↵, ↵) = ↵ = ⌘ − We see that all ⌘ H are fixed and all ↵ are reflected to their negative. Thus our 2 ↵ definition of reflection matches our formula for reflections. Notice also that for any nonzero c R 2 2(⌫,c↵) s (⌫)=⌫ c↵ c↵ − (c↵,c↵) 2cc(⌫, ↵) = ⌫ ↵ − cc(↵, ↵) 2(⌫, ↵) = ⌫ ↵ − (↵, ↵) = s↵(⌫). Therefore scaling ↵ has no e↵ect on the reflection. We also see that reflections preserve orthogonality as the inner product is symmetric and bilinear 2(µ, ↵) 2(⌫, ↵) (s (µ),s (⌫)) = (µ ↵, ⌫ ↵) ↵ ↵ − (↵, ↵) − (↵, ↵) 2(⌫, ↵) 2(µ, ↵) 2(µ, ↵) 2(⌫, ↵) =(µ, ⌫) (µ, ↵) ( ↵, ⌫)+( ↵, ↵) − (↵, ↵) − (↵, ↵) (↵, ↵) (↵, ↵) 2(⌫, ↵) 2(µ, ↵) 4(µ, ↵)(⌫, ↵) =(µ, ⌫) (µ, ↵) (↵, ⌫)+ (↵, ↵) − (↵, ↵) − (↵, ↵) (↵, ↵)(↵, ↵) 4(µ, ↵)(⌫, ↵) 4(µ, ↵)(⌫, ↵) =(µ, ⌫) + − (↵, ↵) (↵, ↵) =(µ, ⌫). 2 Finally note that s↵ =1andthereforeeveryreflectionhasorder2. Definition 1.2.2. A reflection group is a finite group generated by reflections. We next describe some interesting reflection groups. 8 Hγ ↵2 ↵1 Hβ Figure 1.1: Reflections of D4 1.2.1 I2(m) The first reflection group we will look at is the dihedral group. The dihedral group of order 2m consists of reflections and rotations of a regular polygon with m labelled vertices. As an example if we take a square and label its vertices we can permute the labels by either reflecting or rotating the square. In order to make the dihedral group into a reflection group we must convert all rotations into reflections so that only reflections remain in our group. In order to do this we first notice that a rotation can be made into a product of two reflections. Looking at figure 1.1, we see that in order to rotate a vector ↵1 to ↵2 we can compose two reflections by first taking a reflection sγ (over hyperplane Hγ)andthentakingsβ (over hyperplane Hβ). From here we can easily see the following: Proposition 1.2.3. Let Dm be the dihedral group of order 2m. Over Dm reflections form two conjugacy classes when m is even and one conjugacy class when m is odd. Proof. We first claim that if φ is the angle between two vectors ↵ and β then the reflection s s rotates a vector by 2φ.