On the Root System of E8

Bachelor Thesis

On the root system of E8

Author: Guido Baardink

Supervisor: Dr. Andr´e Henriques

Abstract

We ﬁnd several subsystems of the root system of the exceptional Lie algebra E8 that are isomorphic to (direct products of) root systems of the classical Lie algebras An and/or Dn. By exploiting these subsystems we obtain six distinct ways to explicitly express the root system of E8.

January 2014 Contents

1 Introduction to Lie Groups ...... 2 2 Examples of Simple Lie Groups ...... 4 2.1 The root system of the special linear group ...... 4 2.2 The root system of the special orthogonal group ...... 7 3 Root Systems ...... 10 3.1 The Weyl Group ...... 10 3.2 Positive and Simple Roots ...... 10 3.3 Recovering a root system from its simple roots ...... 11 3.4 Dynkin diagrams ...... 12 3.5 The simple roots of sln+1 and so2n ...... 13 3.6 The Weyl group of sln+1 and so2n ...... 15 3.7 Classiﬁcation of Simple Lie Algebras ...... 17 3.8 Simply laced root systems ...... 17 4 Describing E8 ...... 18 4.1 The lowest root of E8 ...... 19 4.2 Describing E8 in the coordinates of A8 ...... 21 4.3 Describing E8 in the coordinates of D8 ...... 24 4.4 Describing E8 in the coordinates of A4 × A4 ...... 25 4.5 Describing E8 in the coordinates of A7 × A1 ...... 27 4.6 Describing E8 in the coordinates of A5 × A2 × A1 ...... 28 4.7 Describing E8 in the coordinates of D5 × A3 ...... 30 4.8 Overview of the root system identities ...... 32 5 Discussion ...... 33

Preface

The aim of this thesis is to give several explicit expressions for the root system of the exceptional Lie algebra E8. The first and third chapter will be used to introduce some important tools used in the study of Lie algebras and root systems in general. It should however be realized that this field is too broad to adequately cover in such a short space. Hence, although the author has endeavoured to present all the necessary theory as completely and succinctly as possible, the interested reader will be strongly advised to read some additional material on Lie algebras in order to fully appreciate the material presented in this thesis. The second chapter and part of the third chapter will be spent applying the introduced machinery to the Lie algebras of SLnC and SOnC. Apart from serving as perfect examples to get a grasp of the abstract theory, it turns out that the root systems of these algebras play a vital role in studying the root system of E8. In the last chapter we finally turn to the object of study. Starting off by giving an introduction on the method of analysis, we spent most of the chapter applying this method for various subsystems of the root system of E8.

Guido Baardink

1 1 Introduction to Lie Groups

In the first chapter we provide a compact introduction to the theory of Lie groups. We will introduce the most important tools that will be used throughout this thesis. But first let us give a definition of Lie groups: Definition 1.1. A Lie group is a group (G, ·) that is also a smooth manifold in such a way that the operations p : G × G → G :(g, h) 7→ g · h and ι : G → G : g 7→ g−1 are smooth maps.

The smoothness of G allows us to talk about its tangent space at the identity. It turns out that this tangent space is of such great importance that we reserve a special symbol (or rather a speciﬁc font) to denote it: TeG = g. This vector space is equipped with an additional structure deﬁned by considering the following functions:

Ψ: G → Aut(G): g 7→ Ψg −1 Ψg : G → G : h 7→ ghg Ad : G → Aut(g): g 7→ Ad g (1) Adg := d(Ψg)e : g → g : X 7→ AdgX

ad := d(Ad)e : g → End(g): X 7→ adX

adX : g → g : Y 7→ adX Y

As it turns out the last function, called the adjoint map of G, is linear in both its arguments, is alternating and 2 satisfies the Jacoby identity. To ease notation we write: [·, ·]: g → g :(X,Y ) 7→ adX (Y ). Definition 1.2. A Lie algebra is a vector space g together with a bilinear, alternating operator [·, ·]: g × g → g that satisfies the Jacobi identity. Hence we see that the tangent space at the identity of any Lie group equipped with the adjoint map forms a Lie algebra. It should be clear why this algebra is such an important tool in the study of Lie groups: by shifting arguments to the tangent space one can translate problems on manifolds to their problems on vector spaces. In particular it can be shown1 that any connected and simply connected Lie group can be completely reconstructed from its associated Lie algebra, essentially creating a one-to-one relation between Lie groups and Lie algebras. The reader might complain that the above definition of the adjoint map is very abstract and thus not very practical. However, if the group can be represented by matrices, we can give a more straightforward definition 0 of the bracket. Suppose we have two arcs A, B : R → GLnC such that A(0) = B(0) = In. Denoting X = A (0) 0 and Y = B (0) we have by definition {X,Y } ⊂ TeGLnC. Since GLnC is open and dense in the vector space of all n by n matrices with complex coefficients MnC, we see that X,Y ∈ MnC. Hence we see:

∂ ∂ −1 [X,Y ] := adX Y = A(t)B(τ)A (t) ∂t t=0 ∂τ τ=0

∂ 0 −1 = A(t)B (0)A (t) (2) ∂t t=0 = A0(0)B0(0)A−1(0) + A(0)B0(0).[−A−1(0)A0(0)A−1(0)] = X.Y.I − I.X.I.Y.I = X.Y − Y.X

Thus the Lie product on matrix groups is simply the commutator. Let us now introduce the notion of simple Lie groups. Deﬁnition 1.3. A Lie algebra g is called simple if dim g > 1 and it contains no non-trivial ideals.

Here, a subalgebra g0 ⊂ g is called an ideal if it satisﬁes:

∀X ∈ g0,Y ∈ g :[X,Y ] ∈ g0. (3)

1See for example §8.3 of [1].

2 One can deﬁne the related notion of a simple Lie group, as a Lie group of dimension greater than one that contains no proper, positive dimensional normal subgroups. It can be shown that the Lie algebra of any simple Lie group is simple, and vice versa.2 All groups treated in this thesis will be simple, as can be checked by considering the classiﬁcation of simple Lie algebras mentioned in section 3.

Now we introduce the notion of representations: Definition 1.4. A representation of a Lie algebra g is a vector space V together with a Lie algebra homomorphism ρ : g → End(V ): X 7→ ρX . Here, a Lie algebra homomorphism is defined to be a map between Lie algebras satisfiying:

∀X,Y ∈ g : ρ[X,Y ] = [ρX , ρY ] = ρX ρY − ρY ρX (4)

Note in particular that the vector space g with the homomorphism ρ : g → End(g): X 7→ adX form a representation of g, called the adjoint representation. Next we introduce the Cartan subalgebra of a simple Lie group. Deﬁnition 1.5. A subalgebra h of a simple Lie group that is abelian, acts diagonally and that is maximal with respect to those properties, is called a Cartan subalgebra of g. Let us elaborate on the terms used in the deﬁnition: • Abelian: the Lie bracket of any two elements X,Y ∈ h is zero.

• Acting diagonally: there exists a basis for g in which adX is a diagonal matrix for all X ∈ h. • Maximal: there is no abelian subalgebra h0 that properly contains h.

Note in particularly that over the ground field C, all Cartan subalgebras of a certain matrix Lie algebra are related by a change of basis. Hence we can proceed to define the root systems of a Lie algebra uniquely, up to change of a change of basis. For this we will introduce a new notion of eigenvectors and eigenvalues. Let (V, ρ) be a representation of a Lie algebra g and let α ∈ h∗ be a linear functional on the Cartan subalgebra, such that the following holds: ∃v ∈ V \{0} s.t. ∀H ∈ h : ρH (v) = α(H)v (5) It is customary to refer to these eigenvalues α as weights and to refer to the corresponding eigenvectors as weight vectors. In the special case of the adjoint representation we introduce the following terminology: Definition 1.6. The root system R(g) of a simple Lie algebra g is the set of eigenvalues of the adjoint action of the Cartan subalgebra, excluding zero. Explicitly we can write: R(g) := {α ∈ h∗ | ∃X ∈ g \{0}, ∀H ∈ h :[H,X] = α(H).X}\{0} (6) Naturally the elements α ∈ R(g) will be called the roots of the Lie algebra g. Finally we would like to extend our notion of root systems to direct products of simple Lie algebras. Consider two simple Lie algebras g1, g2 with respective Cartan subalgebras denoted by h1, h2. Observe that the following set satisfies all requirements for the Cartan subalgebra of g1 × g2:

h12 = [h1 × {0}] ∪ [{0} × h2] (7)

∗ ∗ Furthermore we note that all linear functionals β ∈ h12 can be written using linear functionals α1 ∈ h1 and ∗ α2 ∈ h2: β(H1,H2) = (α1(H1), α2(H2)) (8) Hence we conclude that the root system can be written as:

R(g1 × g2) = R(g1) ∪ R(g2) (9) Now that we have introduced the general machinery for studying simple Lie groups, we will proceed with two examples of simple Lie groups that will play a vital role later on.

2See §9.3 of [1].

3 2 Examples of Simple Lie Groups

We will now use the tools intoduced in the previous section to work out two examples. These groups will become very important later. But let us not digress and focuss on the examples at hand.

2.1 The root system of the special linear group First of all we will investigate the special linear group:

SLnC := {A ∈ MnC | det A = 1} (10)

Proposition 2.1. The Lie Algebra of SLnC is the space of traceless n by n matrices.

Proof: We ﬁrst show that every element X = (xij) ∈ slnC has trace zero. We consider curves of the form: 0 γ : R → SLnC s.t. γ(0) = I and γ (0) = X (11) n Since γ(t) ∈ SLnC we have det(γ(t)) = 1 for all t. By taking {ek} to denote the standard basis of C we observe: n n n ^ ^ ^ [γ(t)](ek) = det(γ(t)) ek = ek (12) k=1 k=1 k=1 Diﬀerentiating both sides with respect to t we get:   n n n d ^ X l−1 0 ^ 0 = [γ(t)](ek) = (−1) [γ (0)](el) ∧  [γ(0)](ek) dt   t=0 k=1 l=1 k=1 k6=l     n n n  n  n X l−1  ^  X l−1 X  ^  = (−1) X(el) ∧  I(ek) = (−1)  xljej ∧  ek (13) l=1 k=1 l=1 j=1 k=1 k6=l k6=l   n n n n n ! n X X  l−1 ^  X ^ ^ = xlj · (−1) ej ∧ ek = xll · ek = T r(X) · ek l=1 j=1 k=1 l=1 k=1 k=1 k6=l

Thus we observe that slnC is a subset of the collection of matrices with trace zero. Observing that dim(SLnC) = 2 dim{X ∈ MnC | T r(X) = 0} = n − 1 we conclude that this inclusion is in fact an equality.

This way, the algebra can be expressed using the elementary matrices Ekl ∈ MnC deﬁned by (Ekl)ij = δkiδlj:  n n n )  X X X slnC X = xijEij xkk = 0 (14)

 i=1 j=1 k=1

A straightforward choice of Cartan subalgebra h ⊂ slnC are the diagonal matrices of trace zero. ( n n )

X X h = H := hkEkk hk = 0 (15) k=1 k=1 Obviously this algebra is abelia;, its maximality can be shown by considering the action of an arbitrary element H ∈ h on an arbitrary element X ∈ glnC:

[H,X]ij = (HX)ij − (XH)ij = hiXij − hjXij = (hi − hj)Xij (16)

If the {hi} are chosen in such a way that hi 6= hj for distinct i, j then:

[H,X] = 0 ⇐⇒ ∀i 6= j : Xij = 0, (17)

4 which implies that X ∈ h as deﬁned in Eq.15. Furthermore we see from Eq.16 that the following elements are eigenvectors for the adjoint action of the Cartan subalgebra:

X = xklEkl for two distinct integers 1 ≤ k, l ≤ n (18)

Since they form a basis of slnC \ h we conclude that they constitute the complete set of eigenvectors. By the fact that they form a basis we can write slnC as a direct sum of eigenspaces in the following sense:    M  slnC = h ⊕  gkl where gkl := Span(Ekl) (19) 1≤k,l≤n  k6=l

From the eigenvalue-equation (Eq.16) we deduce that all eigenvalues for the action of h on slnC form the following root system: R(slnC) = {Li − Lj | 1 ≤ i, j ≤ n, i 6= j} (20) ∗ where Li ∈ h is deﬁned by: n X Li : h → C : hkEkk 7→ hi (21) k=1 At this point it is useful to talk about the geometry of this root system. First of all notice that we can identify the space of all n × n diagonal matrices with Cn. By doing so we observe that our Cartan subalgebra h can n P n be thought of as a linear subspace of C . The standard bilinear form ha|bi = i aibi on C restricts to a non-degenerate bilinear form on h that we can use to identify h and h∗ obtaining:

( n n )

∗ X X h = λkLk λk = 0 (22) k=1 k=1 However we notice that our base vectors, in which we defined our root system, do not lie in h∗. While this is not problematic, it is useful to point out that we can define base vectors within h∗ such that the above definition ∗ ∼ n 1 of our root system still holds. Observe that h = C /Cnˆ wheren ˆ = n (1, 1, 1, ..., 1). In that sense we can make the following identification: n 1 X L˜ := L − hL |nînˆ = L − L (23) i i i i n k k=1 ∗ Where we note that L˜i ∈ h and mark in particular that: X X X λkL˜k = λkLk iff λk = 0 (24) k k k

∗ Hence all vectors in h are invariant under interchanging Li and L˜i. Hence we can switch between the orthogonal ∗ basis Li and their projection on h . To get a feeling for this new notation let us work out the following inner products where i 6= j:

˜ ˜ 1 1 2 n n − 1 hLi|Lii = hLi − ΣkLk|Li − ΣkLki = 1 − + 2 = n n n n n (25) 1 1 2 n 1 hL˜ |L˜ i = hL − Σ L |L − Σ L i = 0 − + = − i j i n k k j n k k n n2 n Thus we have set of n vectors of equal length and equal angle spanning a (n − 1)-dimensional space. Hence they form the vertices of a regular (n − 1)-dimensional simplex. The above process is illustrated in Fig.1 for the case n = 3.

5 L1

L 3 L L3 2

(a) The projection of the base vectors from the (b) The root system of sl3C expressed in the pro- 3 ambient space C onto the linear subspace that jected base vectors. can be identiﬁed with h∗.

Figure 1

˜ Finally we will show some properties of R(slnC). First of all note that we can redeﬁne it in terms of Li as: ˜ ˜ R(slnC) = {Li − Lj | 1 ≤ i, j ≤ n, i 6= j} (26) Furthermore note that it spans a vector space of dimension n − 1 and the cardinality equals:

|R(slnC)| = n(n − 1) (27)

Lastly note that the inner products between any two roots in R(slnC) are given by:

hL˜i − L˜j | L˜i − L˜ji = 2

hL˜i − L˜j | L˜i − L˜ki = 1 (28) hL˜i − L˜j | L˜k − L˜ji = −1

hL˜i − L˜j | L˜k − L˜li = 0

For the sake of clarity we summarize all inner products in the table below, where i 6= j and αi, αj ∈ R(slnC):

2 ˜ 2 n−1 2 ||Li|| = 1 ||Li|| = n ||αi|| = 2

˜ ˜ 1 hLi | Lji = 0 hLi | Lji = − n hαi | αji ∈ {−1, 0, 1}

Table 1: Overview of the geometric properties of the objects of interest for the study of R(slnC).

6 2.2 The root system of the special orthogonal group Let us start from the deﬁnition of the orthogonal group:

T −1 OnC := {A ∈ GLnC | A = A } (29) From this deﬁnition we observe that det(A) = ±1. We deﬁne the special orthogonal group to be the subgroup of OnC corresponding to the positive determinant:

SOnC := {A ∈ OnC | det(A) = 1} (30)

Note that SOnC and OnC \ SOnC are disjoint, nonempty, open sets. Hence we observe that the connected component of the identity of OnC agrees with the connected component of the identity of SOnC. Thus we conclude that their tangent spaces at the origin agree. This allows us to disregard the constraint det(A) = 1 in calculating the Lie algebra of SOnC.

Proposition 2.2. The Lie Algebra of SOnC is the space of n by n matrices which are equal to the negative of their transpose.

T Proof: We ﬁrst show that every element X ∈ SOnC obeys X = −X . We consider curves of the form:

0 γ : R → SOnC s.t. γ(0) = I and γ (0) = X (31)

T −1 Since γ(t) ∈ SOnC we have γ (t) = γ (t) for all t. Hence we observe:

T 0 T d T d −1 −1 0 −1 −1 −1 X = [γ (0)] = γ (t) = γ (t) = −γ (0)γ (0)γ (0) = −I .X.I = −X (32) dt t=0 dt t=0

T Thus we see that sonC is a subset of {X ∈ MnC | X = −X}. Observing that dim(SOnC) = dim{X ∈ T n2−n MnC | X = −X} = /2, we conclude that this inclusion is in fact an equality.

Notice, however that sonC does not contain any non-zero diagonal matrices. As a consequence it becomes quite troublesome to work out the Cartan subalgebra and its eigenvalues. Hence for our current purposes it is practical to deﬁne SOnC in a more general way:

0 T −1 SOnC := {B ∈ GLnC | B M = MB } (33)

0 where M ∈ MnC is a ﬁxed symmetric matrix of maximal rank. This new group SOnC is isomorphic to the above deﬁned SOnC by a change of basis:

0 −1 T ρ : SOnC → SOnC : A → H AH where H H = M (34) We will shortly see why this second representation is more useful for current purposes. First we calculate in similar manner the tangent space to the identity.

0 T Proposition 2.3. The Lie Algebra of SOnC is exactly {Y ∈ MnC | Y M = −MY }. 0 T Proof: Again we show that every element Y ∈ SOnC obeys Y M = −MY . We consider the curves of the form: 0 0 β : R → SOnC s.t. β(0) = I and β (0) = Y (35) −1 0 Furthermore denoting γ = ρ ◦ β, there exists an X ∈ SOnC such that γ (0) = X. In this fashion we note: 0 d −1 −1 d −1 Y := β (0) = H γ(t)H = H γ(t) H = H XH (36) dt t=0 dt t=0

T Since X ∈ sonC we can employ the relation X = −X to derive a relation on Y : Y T M = [H−1XH]T HT H = HT XT [(HT )−1HT ]H = −HT [HH−1]XH = −HT H[H−1XH] = −MY (37)

7 0 T By comparing dimensions we ﬁnd that sonC = {X ∈ MnC | Y M = −MY }.

For the rest of this chapter we will drop the accent and restrict our case to the even dimensions, writing:

T so2nC = {X ∈ M2nC | Y M = −MY } (38) Where we specify the matrix M by:3 0 I M := n (39) In 0

Writing each element X ∈ so2nC in terms of four n × n-matrices we see:

T T T 0 In AB A C 0 In 0 = MX + X M = + T T In 0 CD B D In 0 (40) C + CT D + AT = A + DT B + BT

This gives us the following relations:

Aij = −Dji Cij = −Cji Bij = −Bji (41)

Thus all entries of D are fully determined by the entries of A, all n diagonal elements of both B and C are zero, furthermore all n(n−1)/2 lower triangle elements are fully deﬁned by the n(n−1)/2 upper triangle elements. Hence we can write every element X ∈ so2nC explicitly as:   α11 α12 α13 . . . α1n 0 γ12 γ13 . . . γ1n  α21 α22 α23 . . . α2n −γ12 0 γ23 . . . γ2n     . .   α31 α32 α33 . −γ13 −γ23 0 .     ......   ......     αn1 αn2 ...... αnn −γ1n −γ2n ...... 0    X =   (42)    0 β12 β13 . . . β1n −α11 −α21 −α31 ... −αn1    −β12 0 β23 . . . β2n −α12 −α22 −α32 ... −αn2     . .  −β13 −β23 0 . −α13 −α23 −α33 .     ......   ......  −β1n −β2n ...... 0 −α1n −α2n ...... −αnn

Now we can quite easily construct a Cartan subalgebra by considering the diagonal matrices: Λ 0 h = Λ = diag(h1, h2, . . . , hn) (43) 0 −Λ

Calculating the action of the Cartan on the whole algebra gives ∀X ∈ so2nC and ∀H ∈ h: Λ 0 AB AB Λ 0 [Λ,A]ΛB + BΛ [H,X] = − = (44) 0 −Λ C −AT C −AT 0 −Λ −(ΛC + CΛ) [Λ,AT ]

Since the elements of A, B and C can be independently chosen, we must separate our argument about eigenvalues for each of these three cases. First let’s consider A. We proceed as in section 2.1 to obtain:

T T [Λ,A]ij = (hi − hj)Aij hence also: [Λ,A ]ji = (hj − hi)Aij = (hj − hi)(−A )ji (45)

√ I I 3For the interested, in this case we have H := 2 n n 2 iIn −iIn

8 Thus if we remember our Eij matrices from our argument for slnC, it is easy to see that for i 6= j: Λ 0 Eij 0 Eij 0 Eij 0 , = (hi − hj) = (Li − Lj)(H) (46) 0 −Λ 0 −Eji 0 −Eji 0 −Eji

∗ where Li ∈ h is deﬁned as: diag(h1, h2, . . . , hn) 0 Li : h → C : 7→ hi (47) 0 − diag(h1, h2, . . . , hn)

Similarly for B and C, where again i 6= j:

Λ 0 0 E 0 E 0 E , ij = (h + h ) ij = (L + L )(H) ij 0 −Λ 0 0 i j 0 0 i j 0 0 (48) Λ 0 0 0 0 0 0 0 , = −(hi + hj) = −(Li + Lj)(H) 0 −Λ Eij 0 Eij 0 Eij 0

Since these eigenvectors span the whole of so2n \ h, we can give a complete picture of the root system of so2nC:

R(so2nC) = {±Li ± Lj | 1 ≤ i, j ≤ n i 6= j} (49)

Since we can identify h with Cn we ﬁnd that:

hLi | Lji = δij (50)

Hence we can easily visualize the root system of so2nC for small n. For the case n = 2 this is done in Fig.2.

Figure 2: Root system of so4C

As in the case of the special linear group we end with a short summary of lengths and inner products.

2 2 ||Li|| = 1 ||αi|| = 2

hLi | Lji = 0 hαi | αji ∈ {−1, 0, 1}

Table 2: Overview of the geometric properties of the objects of interest for the study of R(so2nC).

9 3 Root Systems

In this Section we introduce some tools for analyzing root systems, culminating in a classification of all simple Lie algebras. Since the subject of root systems is quite extensive, we will state many theorems without proof for the sake of brevity.4 Let us first of all recall the definition of the root system of an arbitrary Lie Algebra g:

R(g) := {α ∈ h∗ | ∃X ∈ g \{0} s.t. ∀H ∈ h :[H,X] = α(H).X}\{0} (51)

One of the most important tools we will introduce is the Weyl group of a root system.

3.1 The Weyl Group In analyzing root systems it turns out to be fruitful to consider the following linear maps, for α ∈ R: hβ | αi W : h∗ → h∗ : β 7→ β − 2 α (52) α hα | αi

∗ Deﬁnition 3.1. The Weyl group is the group of linear maps of h generated by the reﬂections Wα:

∗ ∗ W := < Wα : h → h | α ∈ R > (53) We will proceed noting the following theorem without proof: Theorem 3.1. The Weyl group is a subgroup of the isometry group of the root system, i.e. W < Iso(R).

Let us prove a few basic properties of the elementary reﬂections Wα:

Lemma 3.1. Wα is an involution Proof: Take any two elements α, β ∈ R and writingα ˆ = α/||α|| observe:

Thus WαWα = Id : R → R.

Lemma 3.2. Wα is an isometry Proof: Take any two elements α, β ∈ R observe:

Thus ||Wα(β)|| = ||β||.

3.2 Positive and Simple Roots From Theorem 3.1 we deduce that, for any root α ∈ R, its negative is also in R since:

−α = Wα(α) ∈ R (56) This motivates us to decompose R in positive roots and their negative counterparts. We can do so by choosing a linear functional l : h∗ → C in such a way that l(α) ∈ R \ 0 for all α ∈ R. Using this linear functional we introduce the following important notion. Deﬁnition 3.2. A root α ∈ R is called positive if l(α) > 0. Negative roots are then deﬁned as the roots for which l(α) < 0. Denoting the positive and negative root systems by R+ resp. R− observe that by our choice of l:

R = R+ ∪ R− (57)

Finally observe that α ∈ R+ implies that −α ∈ R− and vice versa. Next we deﬁne the notion of simple roots:

4All omitted proofs can be found in chapters 14 and 21 of [1].

10 Definition 3.3. A positive root is simple if it cannot be expressed as a sum of two positive roots. Let S(g) denote the set of simple roots in R(g). Then, since there are finitely many roots, the above definition implies that any positive root β ∈ R can be expressed as a linear combination of simple roots αi ∈ S: X X β = αik = miαi mi ∈ N0 (58) k i Note in particular that this implies that negative roots can be written as a linear combination of simple roots with negative coefficients. Observing that the simple roots are linearly independent, which follows from Thm. 3.4 and the fact that the simple roots all lie in the same halfspace, we find that no root is a linear combination of simple roots with coefficients of mixed sign. Finally we define the level of any positive root as the sum over its coefficients: X level(β) = mi (59) i

3.3 Recovering a root system from its simple roots Considering a few properties of the Weyl group we will show how to reconstruct the whole root system from knowing solely the inner product between all of its simple roots. We deﬁne the subgroup of the Weyl group generated by the set A ⊂ R as WA :=< Wα | α ∈ A > (60)

In particular we are interested in the subgroup generated by the simple roots WS. By considering this group we will come to appreciate the practical utility of the simple roots through showing that W = WS and WS(S) = R.

Lemma 3.3. For any positive root β, there exists a simple root γ such that Wγ (β) is a root of lower level. P Proof: Let β = i miαi be a positive root, i.e. ∀i : mi ≥ 0, and let γ = αj be a simple root. We then observe that: hβ|αji X Wαj (β) = mj − 2 αj + miαi (61) hαj|αji i:i6=j Thus we see:

level(β) − level(Wαj (β) = 2hβ|αˆji (62)

Hence Wαj (β) is of lower level if and only if hβ|αji > 0. The existence of such a simple root αj can be shown by contradiction. Suppose that for all αi ∈ S : hβ|αii ≤ 0 than we see by positivity of the coeﬃcients mi of β that: X 2 mihαi|βi = hβ|βi = ||β|| ≤ 0 (63) i

Proving the existence of a simple root αj such that hβ|αji < 0, thus proving the lemma.

Using the above lemma we can show that R = WS(S):

Theorem 3.2. Any root β can be written as β = W (α) for some α ∈ S and W ∈ WS. Proof: First of all note that we can restrict our proof to the positive roots, since if β = W (α) ∈ R+ then:

−β = −W (α) = W (−α) = W ◦ Wα(α) (64) P + We prove the theorem for β = i miαi ∈ R by induction on the level of β. The initial case is provided by noting that Id ∈ WS. By Lemma 3.3 we observe that for any non-simple positive root β there exists a simple root γ such that Wγ (β) is a positive root of lower level. Invoking the induction hypothesis gives the existence of an α ∈ S and a W ∈ WS such that Wγ (β) = W (α). Hence β = Wγ ◦ W (α) as required, proving that indeed R = WS(S).

The above theorem enables us to reconstruct any root system from the conﬁguration of its simple roots. In fact, we only have to know the values of inner products between all two pairs of simple roots. After all, calculating Wα(β) only requires knowledge of hα|βi and hα|αi.

11 Theorem 3.3. The Weyl group is generated by the reﬂections in the simple roots.

Proof: We show that the set of generators of the Weyl group is itself generated by the reﬂections in the simple roots. Take an arbitrary generator Wβ, then Theorem 3.2 tells us that ∃U ∈ WS such that β = U(α). Using furthermore that all elements of the Weyl group are isometries we observe:

−1 Since U, U and Wα are all elements of WS we conclude that W = WS.

3.4 Dynkin diagrams We begin our discussion of Dynkin diagrams and their application by the following observation, again stated without proof:

Lemma 3.4. For all α, β ∈ R, we have: hβ|αi n := 2 ∈ (66) αβ hα|αi Z Notice that Eq.66 puts a strong restriction on the geometry of the roots, since:

||β|| ||α|| n n = 2 cos θ 2 cos θ = 4 cos2 θ ∈ [0, 4] (67) αβ βα ||α|| ||β||

But we observe that nαβnβα is a product of integers and thus an integer itself, hence:

2 4 cos θ ∈ [0, 4] ∩ Z = {0, 1, 2, 3, 4} (68) Thus we note that θ can only take very few values, enumerated in Table 3.

θ 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π √ √ √ √ cos θ 1 3/2 2/2 1/2 0 −1/2 − 2/2 − 3/2 −1 √ √ √ √ q = ||β||/||α|| 1 3 2 1 ∗ 1 2 3 1

Table 3: Possible conﬁgurations of any two roots α, β ∈ R

Theorem 3.4. The angle between two simple roots cannot be acute. Proof: Let α, β be two simple roots with acute angle θ. By reading Table 3 we notice that we have only to consider three cases. However note that in all three cases we have: cos θ/(||β||/||α||) = 1/2. Hence we have:

hα|βi ||β||2 cos θ W (α) = α − 2 β = α − 2 · β = α − β (69) β hβ|βi ||β||2 ||β||/||α||

Thus β = Wβ(α) + α. Since Wα(β) lies in the positive half-space, we see that β cannot be simple.

This allows for an elegant visual representation of the root system in terms of the angle between any two simple roots. In the study of root systems it is customary to use the Dynkin diagrams deﬁned by the rules in Fig.3. In this ﬁgure the dots represent simple roots and the arrows in the last two diagrams point from the bigger root to the smaller root.

12 if θ = π/2

if θ = 2π/3

if θ = 3π/4

if θ = 5π/6

Figure 3: Elementary constituents of Dynkin diagrams

We can use this visual tool to compactly convey all the information to fully deﬁne a Lie algebra and its corresponding group. Moreover we can use this system to classify all simple Lie algebras, as will be presented in section 3.7. But before that let us apply the new machinery to our examples.

3.5 The simple roots of sln+1 and so2n Let us take a moment to apply the machinery constructed above to our examples. First we consider the special linear group. Recall that (Eq.20):

R(sln+1) = {L˜i − L˜j | 1 ≤ i, j ≤ (n + 1), i 6= j} (70) The positive roots can be deﬁned using the following natural linear functional on h∗:

n+1 n+1 ∗ X ˜ X l : h (sln+1) → C : λkLk 7→ kλk (71) k=1 k=1 Since all roots have integer coeﬃcients, we observe that this linear functional restricts to a real-valued linear functional on the root system, as required. Hence we observe that for any L˜i − L˜j ∈ R(sln+1):

l(L˜i − L˜j) = i − j (72)

Since i 6= j we see that l(α) is nonzero for all α ∈ R(sln+1) as required. Furthermore we ﬁnd that l(α) > 0 if and only if i > j. Hence we ﬁnd:

+ − R (sln+1) = {L˜i − L˜j | i > j} and R (sln+1) = {L˜i − L˜j | i < j} (73) Now it is easy to see that the positive roots that can not be written as the sum of two positive roots are precisely:

S(sln+1) = {L˜i+1 − L˜i | 1 ≤ i ≤ n} (74) From Theorem 3.2 we know that we can fully characterize a root system by the inner products between all the simple roots. Hence we calculate:

Hence by the orthonormality of the Li we ﬁnd that:   2 if: i = j hL˜i+1 − L˜i | L˜j+1 − L˜ji = −1 if: i = j ± 1 (76)  0 else

Thus we see that for any two distinct roots in α, β ∈ S(sln+1) we ﬁnd: hα | βi −1/2 ⇒ ∠(α, β) = 2π/3 cos(∠(α, β)) = = (77) ||α|| · ||β|| 0 ⇒ ∠(α, β) = π/2

13 Thus we can draw the Dynkin diagram of sln+1 as shown in Fig.4.

˜ ˜ L˜2−L˜1 L˜4−L˜3 Ln−Ln−1 ··· ˜ ˜ L˜3−L˜2 Ln+1−Ln

Figure 4: The Dynkin Diagram of sln+1.

Now for the special orthogonal group, recall that (Eq.49):

R(so2n) = {±Li ± Lj | 1 ≤ i, j ≤ n, i 6= j} (78)

For this algebra we can use a similar linear functional:

n n ∗ X X l : h (so2n) → C : λkLk 7→ kλk (79) k=1 k=1

Thus l(±Li ± Lj) = ±i ± j and since i 6= j, this functional is real and nonzero over all of R. Hence we have:

+ R (so2n) = { Li + Lj | i > j} ∪ {Li − Lj | i > j} − (80) R (so2n) = {−Li − Lj | i < j} ∪ {Li − Lj | i < j}

+ + We recognize R (sln−1) as a subset of R (so2n). Again it is easy to see that:

S(so2n) = {L1 + L2} ∪ {Li+1 − Li | 1 ≤ i ≤ (n − 1)} (81)

It is useful to note that the last part in the above equation is isomorphic to S(sln+1). Hence we know all inner products on the system S(so2n) \{L1 + L2} and need only to calculate:   0 if: i = 1 hL1 + L2 | Li+1 − Lii = hL1 | Li+1i − hL1 | Lii + hL2 | Li+1i − hL2 | Lii = −1 if: i = 2 (82)  0 else

π Thus we ﬁnd that ∠(L1 + L2,Li+1 − Li) 6= /2 if and only if i = 2, in which case we ﬁnd:

2π ∠(L1 + L2,L3 − L2) = /3 (83)

Hence we can visualize so2n by its Dynkin diagram as shown in Fig.5.

L˜1+L˜2

˜ ˜ L˜2−L˜1 L˜4−L˜3 Ln−1−Ln−2 ··· ˜ ˜ L˜3−L˜2 Ln−Ln−1

Figure 5: The Dynkin Diagram of so2n.

14 3.6 The Weyl group of sln+1 and so2n Now that we have found the simple roots of our two simple Lie groups, we can use these to construct their Weyl groups. First we consider the involutions generated by the elements in R(sln+1), by writing down the following inner products:

hL˜i − L˜j | L˜ki = 0

hL˜i − L˜j | L˜i + L˜ji = 0 (84)

hL˜i − L˜j | L˜i − L˜ji = 2

Let Wij denote the Weyl group element associated with L˜i − L˜j ∈ R(sln+1), then the above translates to:

Wij(L˜k) = L˜k

Wij(L˜i + L˜j) = L˜i + L˜j = L˜j + L˜i (85)

Wij(L˜i − L˜j) = −(L˜i − L˜j) = L˜j − L˜i

Noting that Wij is a linear operator, we observe for a general root in the span of the simple roots S(sln+1):

n+1 ! n+1 n+1 X X X Wij λkL˜k = Wij(λkL˜k) = Wij(λiL˜i + λjL˜j) + Wij(λkL˜k) k=1 k=1 k=1 k6∈{i,j} n+1 λi + λj λi − λj X = W (L˜ + L˜ ) + (L˜ − L˜ ) + λ L˜ ij 2 i j 2 i j k k k=1 k6∈{i,j} (86) n+1 λi + λj λi − λj X = (L˜ + L˜ ) + (L˜ − L˜ ) + λ L˜ 2 i j 2 j i k k k=1 k6∈{i,j} n+1 X = λjL˜i + λiL˜j + λkL˜k k=1 k6∈{i,j}

th th Thus we find that reflecting in the element L˜i − L˜j ∈ R(sln+1) effectively switches the i and j coordinates. Consequently we find that the Weyl group WS(sln+1) = WR(sln+1) is the group that acts on its span by coordinate-wise permutation. To summarize:

Proposition 3.1. The Weyl group of sln+1 can be identiﬁed with the symmetric group on n + 1 letters in the following way: ( n+1 n+1 ) X X W = W : λ L˜ 7→ λ L˜ σ ∈ S (87) sln+1 k k σ(k) k n+1 k=1 k=1 At this point it turns out to be useful to introduce the following notation Deﬁnition 3.4. The set of all coordinate-wise permutations of an arbitrary vector in an n-dimensional vector space with respect to a base {Lk}1≤k≤n is denoted by:

n ! ( n )

X X P λkLk = λσ(k)Lk σ ∈ Sn (88) k=1 k=1 Correspondingly we denote, for any set A ⊂ R: [ P(A) = P(α) (89) α∈A

15 Applying this notation we ﬁnd by Thm.3.2:

R(sln+1) = Wsln+1 (S(sln+1)) = P[S(sln+1)] (90) However we notice that all simple roots are coordinate-wise permutations of each other, thus we can write:

R(sln+1) = P(α) ∀α ∈ S(sln+1) (91) which can be checked by comparing Eq.70 and Eq.74. Now for so2n observe that: ∼ S(so2n) = {L1 + L2} ∪ {Li+1 − Li | 1 ≤ i ≤ (n − 1)} = {L1 + L2} ∪ S(sln) (92)

Thus we ﬁnd that the Weyl group is generated by the following generators

Wso2n := < Wα | α ∈ S(so2n) > = < W | W ∈ Wsln ∨ W = WL1+L2 > (93)

Since we already know how S(sln) acts, we consider the remaining generator:

hL1 + L2 | Lki = 0 for k 6∈ {1, 2}

hL1 + L2 | L1 + L2i = 2 (94)

hL1 + L2 | L1 − L2i = 0

This translates to:

WL1+L2 (Lk) = Lk

WL1+L2 (L1 + L2) = −L1 − L2 (95)

WL1+L2 (L1 − L2) = L1 − L2 Again we can work this out on a general element from the ambient space of the root system:

n ! n n X X X WL1+L2 λkLk = WL1+L2 (λkLk) = WL1+L2 (λ1L1 + λ2L2) + WL1+L2 (λkLk) k=1 k=1 k≥3 n λ1 + λ2 λ1 − λ2 X = W (L + L ) + (L − L ) + λ L L1+L2 2 1 2 2 1 2 k k k≥3 n (96) λ1 + λ2 λ1 − λ2 X = − (L + L ) + (L − L ) + λ L 2 1 2 2 1 2 k k k≥3 n X = −λ2L1 − λ1L2 + λkLk k≥3

Hence we see that this element of the Weyl group switches the ﬁrst and second coordinate and inverses the sign of those two coordinates. Hence we ﬁnd a description for the complete Weyl group:

Proposition 3.2. The Weyl group of so2n can be written as:

( n n ) X X W = W : λ L 7→ ±λ L σ ∈ S (97) so2n k k σ(k) k n k=1 k=1 where the number of minus signs is even.

So we can express the root system R(so2n), using any Li − Lj ∈ S(so2n), by:

R(so2n) = Wso2n (S(so2n)) = P(Li + Lj) ∪ P(Li − Lj) ∪ P(−Li − Lj) (98) which can be checked by comparing Eq.78 and Eq.81.

16 3.7 Classiﬁcation of Simple Lie Algebras Omitting a simple but lengthy proof we state the following theorem, classifying all the simple Lie Algebras. Lemma 3.5. The Dynkin diagrams describing the root systems of simple Lie Algebras are precisely:

An ··· E6

E7 Bn ···

E8 Cn ···

Dn ··· G2

As it turns out the series of Dynkin diagrams on the left hand side correspond to well-understood “classical” simple Lie groups, two of which we already encountered in Fig.4 and Fig.5:

R(An) = R(sln+1) R(Bn) = R(so2n+1) (99) R(Cn) = R(sp2n) R(Dn) = R(so2n)

Since we are mainly interested in sln+1 and so2n we will not bother proving the other identities. Apart from these four classical series the classiﬁcation encompasses ﬁve “exceptional” simple Lie groups, the largest of which is E8.

3.8 Simply laced root systems Deﬁnition 3.5. A root system of a simple Lie algebra is simply laced if the angles between any two simple roots is either 90◦ or 120◦.

Hence observe that the simply laced root systems are exactly An, Dn, E6, E7 and E8. Let us consider the following theorem for these groups: Theorem 3.5. All roots in a simply laced root system have the same length. The angle between any two such roots is one of the following angles: 0◦, 60◦, 90◦, 120◦ or 180◦. Proof: Let R be a simply laced root system and let v, w ∈ R. By Lemma 3.2 we know that we can write these roots as: v = W (α) and w = W 0(β) (100) 0 ◦ ◦ where α, β ∈ S and W, W ∈ W. By deﬁnition of simply laced root systems we have that ∠(α, β) ∈ {90 , 120 }, hence we read from Table 3 that ||α|| = ||β||. Invoking Lemma 3.2 gives:

||v|| = ||W (α)|| = ||α|| = ||β|| = ||W 0(β)|| = ||w|| (101)

Again referring to Table 3, we conclude that equality of lengths of v and w indeed implies that:

◦ ◦ ◦ ◦ ◦ ∠(v, w) ∈ {0 , 60 , 90 , 120 , 180 } (102)

17 4 Describing E8

In the previous Section we have been introduced to the Lie Group E8 by its Dynkin diagram. In principle, this Dynkin diagram tells us the mutual relations between the simple roots from which we could deduce the configuration ∠(u, v) hu, vi Wv(u) of the whole root system and consequently give an abstract account of the algebra itself. 0◦ 2 −u However, rather than abstractly constructing the algebra, we will try to ex- 60◦ 1 u − v press E8 in terms of the already familiar An and Dn algebras. This can be done by applying the so-called lowest root method, which finds us a familiar 90◦ 0 u system of simple roots within R(E ). As it turns out this enables us to ex- 8 120◦ −1 u + v press the simple roots of E8 in the coordinates of the familiar systems and subsequently makes it possible to express the whole root system in terms of 180◦ −2 −u the root systems of sln+1 and so2n. Before elaborating on the method, we would like to point out that E8 is simply laced. Thus by Theorem 3.5 we Table 4: All possible rela- can enumerate all possible relations between two vectors in R(E8) as done in tions between two roots u, v ∈ Table 4. R(E8) where we have taken the Let us now describe the method. It revolves around finding a specific element lengths of the√ roots to satisfy: of the root system v0 ∈ R(E8) called the lowest root, defined in terms of the ||u|| = ||v|| = 2. maximal root of a root system. Definition 4.1. A root α ∈ R is a maximal root if for any simple root β ∈ S we have hα|βi ≥ 0

We will assume existence and uniqueness of this maximal root in any simply laced root system, since proof of this is beyond the scope of this paper. Building upon this existence and uniqueness, note the following: Lemma 4.1. In any simply laced root system, the maximal root is exactly the root that has the highest level. Proof: Let α ∈ R denote the root of highest level. Proceeding towards a contradiction we assume the existence of a simple root β ∈ S such that hα|βi < 0 in which case we have:

level (Wβ(α)) − level(α) = −hα|βi > 0 (103)

Which forces us to conclude that α is not of highest level. Employing the existence and uniqueness of the maximal root we are have to conclude that the highest root α ∈ R is exactly the maximal root.

In the case of R(E8) we will make a habit of denoting the maximal root by vmax. Consequently, we deﬁne the lowest root simply as: v0 := −vmax (104) In practice we can construct the maximal root of any simply laced root system by the following algorithm.

Theorem 4.1. The maximal root can be constructed by repeated reﬂection of a simple root w1 ∈ S in the fashion noted below, until we reach an integer m for which there is no v ∈ S such that hv|wmi = −1. Then we have vmax = vm.

wk+1 = Wvk (wk) for vk ∈ S s.t. hvk|wk−1i = −1 (105)

+ Proof: Consider any root wk ∈ R \{vmax}. Now the deﬁnition of the maximal root implies the existence of a simple root vk ∈ S such that: hwk|vki < 0 (106)

In particular for simply laced Lie algebras this implies that hwk|vki = −1, hence we have:

level (Wvk (wk)) = level(wk) + 1 (107)

18 So as long as wk 6= vmax the algorithm produces a root of higher level. Since we start from an arbitrary simple root w1 ∈ S we will have level(wk) = k. Now we denote the level of the maximal root by m := level(vmax) ∈ N. Since the maximal root has the highest level, the algorithm cannot continue after m steps. By considering the uniqueness of the maximal root we then conclude that wm = wmax.

We will shortly use this algorithm to calculate this maximal root. For now, we will get ahead of ourselves for a bit by depicting the relation between v0 and the simple roots of E8 in Fig.6.

v7 v6 v5 v4 v3 v2 v1 v0

Figure 6: The relation between the lowest root and the simple roots.

Note that the above Dynkin diagram does not correspond to any simple Lie algebra. In fact, note that v0 is not linearly independent of the simple roots. However, this diagram tells us a few interesting properties of E8. By removing one simple root from the diagram we obtain a linearly independent system of roots that we can identify with the simple roots of (the product of) other simple Lie algebras, as shown in table 5.

v8 v8

E7 × A1 E6 × A2 v7 v6 v5 v4 v3 v2 v0 v7 v6 v5 v4 v3 v1 v0

v8 v8

D5 × A3 A4 × A4 v7 v6 v5 v4 v2 v1 v0 v7 v6 v5 v3 v2 v1 v0

v8 v8

A5 × A2 × A1 A7 × A1 v7 v6 v4 v3 v2 v1 v0 v7 v5 v4 v3 v2 v1 v0

D8 A8 v7 v6 v5 v4 v3 v2 v1 v0 v7 v6 v5 v4 v3 v2 v1 v0

Table 5: The Dynkin diagram of several root systems that can be embedded in R(E8).

Apart from E6 and E7, we are familiar with all the mentioned groups. Hence we will utilize these subgroups to get a better understanding of the structure of E8. But ﬁrst let us calculate the lowest root to verify the statement made in ﬁgure 6.

4.1 The lowest root of E8

Denoting the simple roots of E8 with S(E8) = {vi | 1 ≤ i ≤ 8}, we will now execute the construction of the lowest root in R(E8) following Thm. 4.1. In our case we start out with the simple root v5, using the following notational convention:

µ8 8 X := µkvk µ7 µ6 µ5 µ4 µ3 µ2 µ1 k=0

19 With this notational convention and starting out from v5 we get to the maximal root by applying the algorithm as shown below:

1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 3 1 2 2 2 2 2 1 1 1 2 4 1 2 3 2 2 2 1 5 1 2 3 3 3 2 1 6 1 2 3 2 2 2 1 2 2 2 7 1 2 4 3 3 2 1 8 1 3 4 3 3 2 1 9 2 3 4 3 3 2 1 2 2 3 10 2 3 4 4 3 2 1 11 2 3 5 4 3 2 1 12 2 4 5 4 3 2 1 3 3 3 13 2 3 5 4 3 2 1 14 2 4 6 4 3 2 1 15 2 4 6 5 3 2 1 3 3 3 16 2 4 6 5 4 2 1 17 2 4 6 5 4 3 1 18 2 4 6 5 4 3 2

All reﬂections are given verbally in the enumeration below, along with a justiﬁcation of the mirroring by a short calculation of inner products.

(1) Mirror in v8, v6, v7, v4, v3, v2 then v1. (10) Mirror in v4. Since 2 · 0 − 1 · (1 + 0) = −1 Since 2 · 3 − 1 · (4 + 3) = −1 (2) Mirror in v5. (11) Mirror in v5. Since 2 · 1 − 1 · (1 + 1 + 1) = −1 Since 2 · 4 − 1 · (4 + 3 + 2) = −1 (3) Mirror in v6, v4, v3 then v2. (12) Mirror in v6. Since 2 · 1 − 1 · (2 + 1) = −1 Since 2 · 2 − 1 · 5 = −1 (4) Mirror in v5. (13) Mirror in v8. Since 2 · 2 − 1 · (2 + 2 + 1) = −1 Since 2 · 3 − 1 · (5 + 2) = −1 (5) Mirror in v4 then v3. (14) Mirror in v5. Since 2 · 1 − 1 · 3 = −1 Since 2 · 5 − 1 · (4 + 4 + 3) = −1 (6) Mirror in v8. (15) Mirror in v4. Since 2 · 2 − 1 · (3 + 2) = −1 Since 2 · 4 − 1 · (6 + 3) = −1 (7) Mirror in v5. (16) Mirror in v3. Since 2 · 3 − 1 · (3 + 2 + 2) = −1 Since 2 · 3 − 1 · (5 + 2) = −1 (8) Mirror in v6. (17) Mirror in v2. Since 2 · 2 − 1 · (4 + 1) = −1 Since 2 · 2 − 1 · (4 + 1) = −1 (9) Mirror in v7. (18) Mirror in v1. Since 2 · 1 − 1 · 3 = −1 Since 2 · 1 − 1 · 3 = −1

Thus we ﬁnd that we can express the lowest root v0 explicitly as:

v0 = −vmax = −(2v1 + 3v2 + 4v3 + 5v4 + 6v5 + 4v6 + 2v7 + 3v8) (108)

Let us brieﬂy check its expected relations to all the simple vectors:

hv1, v0i = −2hv1, v1i − 3hv1, v2i = −1 hv2, v0i = −2hv2, v1i − 3hv2, v2i − 4hv2, v3i = 0 hv3, v0i = −3hv3, v2i − 4hv3, v3i − 5hv3, v4i = 0 hv4, v0i = −4hv4, v3i − 5hv4, v4i − 6hv4, v5i = 0 hv5, v0i = −5hv5, v4i − 6hv5, v5i − 4hv5, v6i − 3hv5, v8i = 0 hv6, v0i = −6hv6, v5i − 4hv6, v6i − 2hv6, v7i = 0 hv7, v0i = −4hv7, v6i − 2hv7, v7i = 0 hv8, v0i = −6hv8, v5i − 3hv8, v8i = 0

20 4.2 Describing E8 in the coordinates of A8

v7 v6 v5 v4 v3 v2 v1 v0

α8 α7 α6 α5 α4 α3 α2 α1

Figure 7: The subset of S(E8) generating a copy of R(A8).

Having found an explicit expression for our lowest root, we can use it to investigate the whole root system of E8 using the coordinates of the root systems of An and Dn we found in Section 3. At ﬁrst we consider the A8-subset. We observe that: ∼ S(A8) = {v0, v1, v2, v3, v4, v5, v6, v7} ⊂ R(E8) (109)

Thus we can simplify a great deal of the notation by equating S(A8) to {vk}0≤k≤7 without losing consistency. In this sense we immediately observe that:

R(A8) = WS(A8)(S(A8)) ⊂ R(E8) (110)

Hence we can identify all but one of the simple roots of E8 with simple roots from A8 by:

vk = αk+1 = L˜k+2 − L˜k+1 for 1 ≤ k ≤ 7 (111)

Then using Eq.108 we can write the remaining simple root of R(E8) as follows:

1 v8 = − 3 (v0 + 2v1 + 3v2 + 4v3 + 5v4 + 6v5 + 4v6 + 2v7) 1 = − 3 (α1 + 2α2 + 3α3 + 4α4 + 5α5 + 6α6 + 4α7 + 2α8) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ (112) 1 (L2 − L1) + 2(L3 − L2) + 3(L4 − L3) + 4(L5 − L4) = − 3 +5(L˜6 − L˜5) + 6(L˜7 − L˜6) + 4(L˜8 − L˜7) + 2(L˜9 − L˜8)

1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ = + 3 (L1 + L2 + L3 + L4 + L5 + L6 − 2L7 − 2L8 − 2L9) Recall from Eq.24 that:

n+1 n+1 n+1 X X X ∀u = λkL˜k ∈ h(An): λk = 0 ⇒ u = λkLk (113) k=1 k=1 k=1 where {Lk} forms an orthogonal base. Since every root in R(E8) can be written as a linear combination of simple roots from E8, and since each of those simple roots can be expressed as a linear combination of simple roots from A8 we have that

R(E8) ⊂ SpanR(S(A8)) = h(A8) (114)

Hence we can write all w ∈ R(E8) in coordinates with respect to an orthogonal basis:

9 9 X X w = λkL˜k = λkLk =: (λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8) (115) k=1 k=1

21 In that notation we ﬁnd:

v0 = (−1, 1, 0, 0, 0, 0, 0, 0, 0 )

v1 = ( 0, 1, −1, 0, 0, 0, 0, 0, 0 )

v2 = ( 0, 0, 1, −1, 0, 0, 0, 0, 0 )

v3 = ( 0, 0, 0, 1, −1, 0, 0, 0, 0 )

v4 = ( 0, 0, 0, 0, 1, −1, 0, 0, 0 ) (116)

v5 = ( 0, 0, 0, 0, 0, 1, −1, 0, 0 )

v6 = ( 0, 0, 0, 0, 0, 0, 1, −1, 0 )

v7 = ( 0, 0, 0, 0, 0, 0, 0, 1, −1 ) 1 v8 = /3( 1, 1, 1, 1, 1, 1, −2, −2, −2 )

Let us try to recover the whole root system from these simple roots. As we have stated in Section 3, this entails

ﬁnding WS(S). As we know, each element in WS is a composition of involutions Wvi for vi ∈ S(E8). Thus we have:

WS = W{vk}1≤k≤8 = W{vk}0≤k≤8 = WS(A8)∪{v8} (117)

In this case we need to consider the involutions generated by Wα for either α ∈ S(A8) or α = v8. We will ﬁrst of all focus on the reﬂections in the simple roots of A8. Recall from Eq.?? that:

WS(An)(α) = P(α) (118)

where the deﬁnition of the right-hand side is given in Eq.88. By observing that −v8 = Wv8 (v8) ∈ R(E8) and denoting WA8 = WS(A8) we ﬁnd that the following sets are subsets of R(E8):

R(A8) = WA8 ( v0 ) = P( 1, −1, 0, 0, 0, 0, 0, 0, 0 ) 1 R(v8) = WA8 ( v8 ) = + 3 P( 1, 1, 1, 1, 1, 1, −2, −2, −2 ) (119) 1 R(−v8) = WA8 (−v8) = − 3 P( 1, 1, 1, 1, 1, 1, −2, −2, −2 )

We are about to show that the above is a partition of R(E8). Note that clearly the sets in Eq.119 are disjoint, so we are left to prove that their union equals R(E8). Equivalently we prove that this union contains all simple roots and is invariant under the full Weyl group of E8. Let us denote the union by:

R∪ = R(A8) ∪ R(v8) ∪ R(−v8) (120)

In order to prove R∪ := R(E8), we consider the following lemmas:

Lemma 4.2. The image of the set R(v8) under the involution Wv8 is contained in R∪.

Proof: Let β ∈ R(v8). We split the proof in four cases:

Case 1: β = v8. In this case we have Wv8 (v8) = −v8 ∈ R(−v8) ⊂ R∪.

Case 2: β diﬀers in exactly two coordinates from v8. Without loss of generality we can calculate:

1 /3( 1, 1, 1, 1, 1, 1, −2, −2, −2 ) 1 hv8, βi = = [5 − 4 + 8] = 1 (121) 1/3( 1, 1, 1, 1, 1, −2, 1, −2, −2 ) 9

1 Thus Wv8 (β) = β − 1 · v8 = 3 (0, 0, 0, 0, 0, 3, −3, 0, 0) ∈ R(A8) ⊂ R∪.

Case 3: β diﬀers in exactly four coordinates from v8. Again without loss of generality we calculate:

1 /3( 1, 1, 1, 1, 1, 1, −2, −2, −2 ) 1 hv8, βi = = [4 − 8 + 4] = 0 (122) 1/3( 1, 1, 1, 1, −2, −2, 1, 1, −2 ) 9

22 Thus Wv8 (β) = β − 0 · v8 = β ∈ R(A8) ⊂ R∪.

Case 4: β diﬀers in exactly six coordinates from v8.

1 /3( 1, 1, 1, 1, 1, 1, −2, −2, −2 ) 1 hv8, βi = = [3 − 12] = −1 (123) 1/3( 1, 1, 1, −2, −2, −2, 1, 1, 1 ) 9

1 Thus Wv8 (β) = β − (−1)v8 = 3 (2, 2, 2, −1, −1, −1, −1, −1, −1) ∈ R(−v8) ⊂ R∪

Conclusion: Wv8 ◦ WA8 (v8) ⊂ R∪

The above lemma can be used to prove a slightly stronger lemma:

Lemma 4.3. The image of the set R∪ under the involution Wv8 is contained in R∪ itself.

Proof: Showing the invariancy of R∪ comes down to showing that the images of R(v8), R(−v8) and R(A8) under Wv8 are all contained in R∪. The ﬁrst of which we have proven already in Lemma 4.2 above. Since R∪ is invariant under negation it follows that Wv8 (R(−v8)) ⊂ R∪. Lastly let α ∈ R(A8). Using Table 4 and the fact that α 6= ±v8, we can proceed by cases:

◦ ∠(α, v8) = 60 ⇒ Wv8 (α) = α − v8 = −Wα(v8) ∈ R(−v8) ⊂ R∪ ◦ (124) ∠(α, v8) = 120 ⇒ Wv8 (α) = α + v8 = Wα(v8) ∈ R(v8) ⊂ R∪

In the case that α and v8 are orthogonal the reﬂection keeps α invariant.

With the last lemma we are ﬁnally ready to prove the theorem:

Theorem 4.2. The set R∪ is exactly the root system of E8, i.e. R(E8) = R(A8) ∪ R(v8) ∪ R(−v8).

Proof: First of all we observe that R∪ is invariant under WA8 by deﬁnition. This, combined with Lemma 4.3, allows us to state that R∪ is invariant under the whole Weyl group of E8:

WE8 (R∪) ⊂ R∪ (125)

Also, since S(E8) ⊂ R∪ ⊂ R(E8), we see that:

WE8 (S(E8)) ⊂ WE8 (R∪) (126)

Noting that R(E8) = WE8 (S(E8)) and that R∪ ⊂ R(E8), the above two equations yield the following inclusion:

R(E8) ⊂ R∪ ⊂ R(E8) (127)

Hence we have proven that R∪ = R(E8).

More explicitly, we have found that:

∼ 1 R(E8) = R(A8) ∪ ± 3 P( 1, 1, 1, 1, 1, 1, −2, −2, −2 ) (128) Since the subsets are mutually disjoint we can calculate the cardinality of the full root system:

9 9 |R(E )| = |R(A )| + 2|R(v )| = + 2 = 240 (129) 8 8 8 2 3

23 4.3 Describing E8 in the coordinates of D8

v7 v6 v5 v4 v3 v2 v1 v0

δ8

δ1 δ2 δ3 δ4 δ5 δ6 δ7

Figure 8: The subset of S(E8) generating a copy of R(D8).

We denote the simple roots of Dn as follows: Lk+1 − Lk for 1 ≤ k ≤ (n − 1) δk = (130) L1 + L2 for k = n

We can identify the simple roots of D8 with roots in E8 in the following manner: δ7−k for 0 ≤ k ≤ 6 vk = (131) δ8 for k = 8

Then using Eq.108 we can write the remaining simple root of R(E8) as follows:

1 v7 = − 2 (v0 + 2v1 + 3v2 + 4v3 + 5v4 + 6v5 + 4v6 + 3v8) 1 = − 2 (δ7 + 2δ6 + 3δ5 + 4δ4 + 5δ3 + 6δ2 + 4δ1 + 3δ8) (132) 1 (L8 − L7) + 2(L7 − L6) + 3(L6 − L5) + 4(L5 − L4) = − 2 +5(L4 − L3) + 6(L3 − L2) + 4(L2 − L1) + 3(L1 + L2)

1 = − 2 (L8 + L7 + L6 + L5 + L4 + L3 + L2 − L1) Which we can write out coordinate-wise as:

v0 = ( 0, 0, 0, 0, 0, 0, 1, −1 )

v1 = ( 0, 0, 0, 0, 0, 1, −1, 0 )

v2 = ( 0, 0, 0, 0, 1, −1, 0, 0 )

v3 = ( 0, 0, 0, 1, −1, 0, 0, 0 )

v4 = ( 0, 0, 1, −1, 0, 0, 0, 0 ) (133)

v5 = ( 0, 1, −1, 0, 0, 0, 0, 0 )

v6 = ( 1, −1, 0, 0, 0, 0, 0, 0 ) 1 v7 = − /2(−1, 1, 1, 1, 1, 1, 1, 1 )

v8 = ( 1, 1, 0, 0, 0, 0, 0, 0 )

Observing that the ﬁrst seven roots {vi | 0 ≤ i ≤ 6} are isomorphic to S(A7) we recognize by invoking Eq.?? that the group generated by reﬂections in these roots is homomorphic to the symmetric group on eight elements, in the following way:

8 ! 8 X X ∀W ∈ W{vk}0≤k≤6 : ∃σ ∈ S8 s.t. W λkLk = λσ(k)Lk (134) k=1 k=1 Thus we ﬁnd that: ∀α ∈ R(E8): P(α) ⊂ R(E8) (135)

24 The action of v8 is calculated in Eq.??: simultaneously switching the sign of the ﬁrst and second coordinate. This allows us to reach the following element:

1 Wv8 ◦ Wv5 ◦ Wv6 (v7) = − /2(−1, −1, −1, 1, 1, 1, 1, 1)

Thus by considering the coordinate-wise permuting action of the roots {δk}1≤k≤7 we ﬁnd the following sets to be subsets of R(E8): WD v0 = P(−1, 1, 0, 0, 0, 0, 0, 0) ∪ ±P(1, 1, 0, 0, 0, 0, 0, 0) = R(D8) 8 (136) 1 1 WD8 v7 = ± 2 P(−1, 1, 1, 1, 1, 1, 1, 1) ∪ ± 2 P(−1, −1, −1, 1, 1, 1, 1, 1) However observe that these sets are disjoint, hence we can calculate the cardinality of their union: WD8 v0 ∪ WD8 v7) = |R(D8)| + WD8 v8 87 8 8 8 = + 2 · + 2 · + 2 · (137) 1 1 2 1 3

= 112 + 128 = 240

This agrees with Eq.129 and so we have found all the roots of E8: ∼ R(E8) = WD8 v0 ∪ WD8 v7 (138) Or more speciﬁcally: ∼ 1 1 R(E8) = R(D8) ∪ ± 2 P(−1, 1, 1, 1, 1, 1, 1, 1) ∪ ± 2 P(−1, −1, −1, 1, 1, 1, 1, 1) (139)

4.4 Describing E8 in the coordinates of A4 × A4

v7 v6 v5 v4 v3 v2 v1 v0 0 α1

0 0 0 α4 α3 α2 α4 α3 α2 α1

Figure 9: The subset of S(E8) generating a copy of R(A4 × A4).

Now we focus on the collection of roots (S(E8) ∪ {v0}) \{v4}. It can be shown that this subset generates a copy of R(A4 × A4) in R(E8) by the following identiﬁcation:  αk+1 for 0 ≤ k ≤ 3  0 vk = αk−3 for 5 ≤ k ≤ 7 (140) 0  α1 for k = 8

Then using Eq.108 we can write the remaining simple root of R(E8) as follows: 1 v4 = − 5 (v0 + 2v1 + 3v2 + 4v3 + 6v5 + 4v6 + 2v7 + 3v8) 1 0 0 0 0 = − 5 (α1 + 2α2 + 3α3 + 4α4 + 6α2 + 4α3 + 2α4 + 3α1) (L˜ − L˜ ) + 2(L˜ − L˜ ) + 3(L˜ − L˜ ) + 4(L˜ − L˜ ) (141) = − 1 2 1 3 2 4 3 5 4 5 ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 +6(L3 − L2) + 4(L4 − L3) + 2(L5 − L4) + 3(L2 − L1) 1 ˜ ˜ ˜ ˜ ˜ ˜0 ˜0 ˜0 ˜0 ˜0 = + 5 (L1 + L2 + L3 + L4 − 4L5 + 3L1 + 3L2 − 2L3 − 2L4, −2L5)

25 Which we can write out coordinate-wise as:

v0 = (−1, 1, 0, 0, 0 ) ⊕ ( 0, 0, 0, 0, 0 )

v1 = ( 0, 1, −1, 0, 0 ) ⊕ ( 0, 0, 0, 0, 0 )

v2 = ( 0, 0, 1, −1, 0 ) ⊕ ( 0, 0, 0, 0, 0 )

v3 = ( 0, 0, 0, 1, −1 ) ⊕ ( 0, 0, 0, 0, 0 ) 1 v4 = /5[( 1, 1, 1, 1, −4 ) ⊕ ( 3, 3, −2, −2, −2 )] (142)

v5 = ( 0, 0, 0, 0, 0 ) ⊕ ( 0, −1, 1, 0, 0 )

v6 = ( 0, 0, 0, 0, 0 ) ⊕ ( 0, 0, −1, 1, 0 )

v7 = ( 0, 0, 0, 0, 0 ) ⊕ ( 0, 0, 0, −1, 1 )

v8 = ( 0, 0, 0, 0, 0 ) ⊕ (−1, 1, 0, 0, 0 )

We recall that the Weyl group of WAn acts by coordinate-wise permutation. Moreover we would like to point out speciﬁcally that for any root system R:

∀α, β ∈ R s.t. α ⊥ β : Wα(β) = β (143)

For current purposes this means that the Weyl group of the one A4 acts trivially on the other. Furthermore we note that: h i h i S(A4 × A4) = {(0, 0, 0, 0, 0)} ⊕ S(A4) ∪ S(A4) ⊕ {(0, 0, 0, 0, 0)} (144) Bearing this in mind we observe that: R(A4 × A4) = WA4×A4 S(A4 × A4) h ~ i h ~ i = WA4 {0} ⊕ WA4 S(A4) ∪ WA4 S(A4) ⊕ WA4 {0} (145) h i h i = {~0} ⊕ R(A4) ∪ R(A4) ⊕ {~0}

From this we note that |R(A4 × A4) = 2|R(A4)| = 40. Futhermore we see that: 1 WA4×A4 v4 = 5 [P(1, 1, 1, 1, −4) ⊕ P(3, 3, −2, −2, −2)] (146)

By noting that the negative of v4 is also in R(E8) we can already observe that: R(A4 × A4) ∪ WA4×A4 ± v4 ⊂ R(E8) (147) However note that the cardinality of the left-hand side is lower than the right-hand side. Hence we need to find more roots by reflection. For this purpose, we consider the following root, obtained by applying reflections from W(A4 × A4) on the root v4: 1 β = 5 [(1, 1, 1, −4, 1) ⊕ (−2, −2, 3, 3, −2)] (148) where the permutation has been chosen in such a way that the coordinatewise multiplication of β and v4 is minimal. In fact, calculating the inner product between these vectors gives:

1 /5[( 1, 1, 1, −4, 1 ) ⊕ (−2, −2, 3, 3, −2 )] 1 hβ|v4i = = [3 − 8 − 24 + 4] = −1 (149) 1/5[( 1, 1, 1, 1, −4 ) ⊕ ( 3, 3, −2, −2, −2 )] 25 Thus we ﬁnd the following distinct root:

1/5[( 1, 1, 1, −4, 1 ) ⊕ (−2, −2, −2, 3, 3 )] 1 5 w := Wv4 (β) = β − hβ|v4iv4 = / [( 1, 1, 1, 1, −4 ) ⊕ ( 3, 3, −2, −2, −2 )] + (150) 1/5[( 2, 2, 2, −3, −3 ) ⊕ ( 1, 1, −4, 1, 1, )]

Including the coordinate-wise permutation of this root and its inverse we have the following subsets of R(E8): ~ ~ WA4×A4 v0 = P (−1, 1, 0, 0, 0) ⊕ {0} = R(A4) ⊕ {0} W v = {~0} ⊕ P(−1, 1, 0, 0, 0) = {~0} ⊕ R(A ) A4×A4 8 4 (151) 1 WA4×A4 ± v4 = ± 5 [P(1, 1, 1, 1, −4) ⊕ P(3, 3, −2, −2, −2)] 1 WA4×A4 ± w = ± 5 [P(3, 3, −2, −2, −2) ⊕ P(1, 1, 1, 1, −4)]

26 Hence we calculate the cardinality of the union of these disjoint sets: 5 5 5 5 R(A4 × A4) + WA4×A4 (±v4)| + |WA4×A4 (±w)| = 2|R(A4)| + 2 · + 2 · = 240 (152) 1 2 2 1 Thus we ﬁnd that: ∼ R(E8) = WA4×A4 S(A4 × A4) ∪ WA4×A4 (±v4) ∪ WA4×A4 (±w) (153) Or more speciﬁcally: ∼ 1 R(E8) = R(A4 × A4) ∪ ± 5 [P(1, 1, 1, 1, −4) ⊕ P(3, 3, −2, −2, −2)] 1 (154) ∪ ± 5 [P(3, 3, −2, −2, −2) ⊕ P(1, 1, 1, 1, −4)]

4.5 Describing E8 in the coordinates of A7 × A1

v7 v6 v5 v4 v3 v2 v1 v0 α7

0 α1 α6 α5 α4 α3 α2 α1

Figure 10: The subset of S(E8) generating a copy of R(A7 × A1)

Now we focus on the collection of roots (S(E8) ∪ {v0}) \{v6}. It can be shown that this subset generates a copy of R(A7 × A1) in R(E8) by the following identiﬁcation:  αk+1 for 0 ≤ k ≤ 5  0 vk = α1 for k = 7 (155)  α7 for k = 8

Then using Eq.108 we can write the remaining simple root of R(E8) as follows: 1 v6 = − 4 (v0 + 2v1 + 3v2 + 4v3 + 5v4 + 6v5 + 2v7 + 3v8) 1 0 = − 4 (α1 + 2α2 + 3α3 + 4α4 + 5α5 + 6α6 + 2α1 + 3α7) (L˜ − L˜ ) + 2(L˜ − L˜ ) + 3(L˜ − L˜ ) + 4(L˜ − L˜ ) (156) = − 1 2 1 3 2 4 3 5 4 4 ˜ ˜ ˜ ˜ ˜0 ˜0 ˜ ˜ +6(L6 − L5) + 4(L7 − L6) + 2(L2 − L1) + 3(L8 − L7) 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜0 ˜0 = + 4 (L1 + L2 + L3 + L4 + L5 + L6 − 3 + L7 − 3 + L8 + 2L1 − 2 + L2) Which we can write out coordinate-wise as:

v0 = (−1, 1, 0, 0, 0, 0, 0, 0 ) ⊕ ( 0, 0 )

v1 = ( 0, 1, −1, 0, 0, 0, 0, 0 ) ⊕ ( 0, 0 )

v2 = ( 0, 0, 1, −1, 0, 0, 0, 0 ) ⊕ ( 0, 0 )

v3 = ( 0, 0, 0, 1, −1, 0, 0, 0 ) ⊕ ( 0, 0 )

v4 = ( 0, 0, 0, 0, 1, −1, 0, 0 ) ⊕ ( 0, 0 ) (157)

v5 = ( 0, 0, 0, 0, 0, 1, −1, 0 ) ⊕ ( 0, 0 ) 1 v6 = /4[( 1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ ( 2, −2 )]

v7 = ( 0, 0, 0, 0, 0, 0, 0, 0 ) ⊕ (−1, 1 )

v8 = ( 0, 0, 0, 0, 0, 0, 1, −1 ) ⊕ ( 0, 0 )

27 We ﬁnd another distinct element by considering the root β ∈ WA7×A1 (v6): 1 β = 4 [(1, 1, 1, 1, −3, −3, 1, 1) ⊕ (−2, 2)] (158)

Calculating the inner product between this vector and v6 we ﬁnd:

1 /4[( 1, 1, 1, 1, −3, −3, 1, 1 ) ⊕ (−2, 2 )] 1 hβ|v6i = = [4 − 12 − 8] = −1 (159) 1/4[( 1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ ( 2, −2 )] 16 Thus we ﬁnd the following distinct root:

1/4[( 1, 1, 1, 1, −3, −3, 1, 1 ) ⊕ (−2, 2 )] 1 4 w := Wv6 (β) = β − hβ|v6iv6 = / [( 1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ ( 2, −2 )] + (160) 1/4[( 2, 2, 2, 2, −2, −2, −2, −2 ) ⊕ ( 0, 0 )] Mark, in particular, that the inverse of this root is contained in the set of its coordinate-wise permutations, i.e.

WA7×A1 (±w) = WA7×A1 (w). Hence we obtain the following subsets of R(E8): ~ ~ WA7×A1 v0 = P (−1, 1, 0, 0, 0, 0, 0, 0) ⊕ {0} = R(A7) ⊕ {0} W v = {~0} ⊕ P(−1, 1) = {~0} ⊕ R(A ) A7×A1 7 1 (161) 1 WA7×A1 ± v6 = ± 4 [P(1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ P(2, −2 )] 1 WA7×A1 w = 2 [P(1, 1, 1, 1, −1, −1, −1, −1 ) ⊕ P(0, 0 )] Hence we calculate the cardinality of the union of these disjoint sets: WA7×A1 v0 + WA7×A1 v7 + WA7×A1 (±v4)| + |WA7×A1 (±w)| 82 82 (162) = |R(A )| + |R(A )| + 2 · + = 240 7 1 3 1 4 0 Thus we ﬁnd that: ∼ R(E8) = WA7×A1 S(A7 × A1) ∪ WA7×A1 (±v6) ∪ WA7×A1 (w) (163) Or more speciﬁcally: ∼ 1 R(E8) = R(A7 × A1) ∪ ± 4 [P(1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ P(2, −2 )] 1 (164) ∪ 2 [P(1, 1, 1, 1, −1, −1, −1, −1 ) ⊕ P(0, 0 )]

4.6 Describing E8 in the coordinates of A5 × A2 × A1

v7 v6 v5 v4 v3 v2 v1 v0 00 α1

0 0 α2 α1 α5 α4 α3 α2 α1

Figure 11: The subset of S(E8) generating a copy of R(A5 × A2 × A1)

Now we focus on the collection of roots (S(E8) ∪ {v0}) \{v5}. It can be shown that this subset generates a copy of R(A5 × A2 × A1) in R(E8) by the following identiﬁcation:  αk+1 for 0 ≤ k ≤ 4  0 vk = αk−5 for 6 ≤ k ≤ 7 (165) 00  α1 for k = 8

28 Then using Eq.108 we can write the remaining simple root of R(E8) as follows:

1 v5 = − 6 (v0 + 2v1 + 3v2 + 4v3 + 5v4 + 4v6 + 2v7 + 3v8) 1 0 0 00 = − 6 (α1 + 2α2 + 3α3 + 4α4 + 5α5 + 4α1 + 2α2 + 3α1 ) (L˜ − L˜ ) + 2(L˜ − L˜ ) + 3(L˜ − L˜ ) + 4(L˜ − L˜ ) (166) = − 1 2 1 3 2 4 3 5 4 6 ˜ ˜ ˜0 ˜0 ˜0 ˜0 ˜00 ˜00 +5(L6 − L5) + 4(L2 − L1) + 2(L3 − L2) + 3(L2 − L1 ) 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜0 ˜0 ˜0 ˜00 ˜00 = + 6 (L1 + L2 + L3 + L4 + L5 − 5L6 + 4L1 − 2L2 − 2L3 + 3L1 − 3L2 ) Which we can write out coordinate-wise as:

v0 = (−1, 1, 0, 0, 0, 0 ) ⊕ ( 0, 0, 0 ) ⊕ ( 0, 0 )

v1 = ( 0, 1, −1, 0, 0, 0 ) ⊕ ( 0, 0, 0 ) ⊕ ( 0, 0 )

v2 = ( 0, 0, 1, −1, 0, 0 ) ⊕ ( 0, 0, 0 ) ⊕ ( 0, 0 )

v3 = ( 0, 0, 0, 1, −1, 0 ) ⊕ ( 0, 0, 0 ) ⊕ ( 0, 0 )

v4 = ( 0, 0, 0, 0, 1, −1 ) ⊕ ( 0, 0, 0 ) ⊕ ( 0, 0 ) (167) 1 v5 = /6[( 1, 1, 1, 1, 1, −5 ) ⊕ ( 4, −2, −2 ) ⊕ ( 3, −3 )]

v6 = ( 0, 0, 0, 0, 0, 0 ) ⊕ (−1, 1, 0 ) ⊕ ( 0, 0 )

v7 = ( 0, 0, 0, 0, 0, 0 ) ⊕ ( 0, −1, 1 ) ⊕ ( 0, 0 )

v8 = ( 0, 0, 0, 0, 0, 0 ) ⊕ ( 0, 0, 0 ) ⊕ (−1, 1 )

Note that the coordinate-wise permutation of these elements is not enough to reach all roots of E8. In fact, we need two coordinate-wise distinct elements. We will ﬁnd one of those by considering the root β ∈

WA5×A2×A1 (v5): 1 β = 6 [(1, 1, 1, 1, −5, 1) ⊕ (−2, 4, −2) ⊕ (−3, 3)] (168)

Calculating the inner product between this vector and v5 we ﬁnd:

1 /6[( 1, 1, 1, 1, −5, 1 ) ⊕ (−2, 4, −2 ) ⊕ (−3, 3 )] 1 hβ|v5i = = [4−10−8 + 4−18] = −1 (169) 1/6[( 1, 1, 1, 1, 1, −5 ) ⊕ ( 4, −2, −2 ) ⊕ ( 3, −3 )] 36

Thus we ﬁnd the following distinct root:

1/6[( 1, 1, 1, 1, −5, 1 ) ⊕ (−2, 4, −2 ) ⊕ (−3, 3 )] 1 6 w := Wv5 (β) = β − hβ|v5iv5 = / [( 1, 1, 1, 1, 1, −5 ) ⊕ ( 4, −2, −2 ) ⊕ ( 3, −3 )] + (170) 1/6[( 2, 2, 2, 2, −4, −4 ) ⊕ ( 2, 2, −4 ) ⊕ ( 0, 0 )]

Furthermore, we can ﬁnd another coordinate-wise distinct root by considering the following root:

1 γ = Wv8 ◦ Wv6 ◦ Wv7 ◦ Wv4 ◦ Wv3 (w) /6[(2, 2, 2, −4, −4, 2) ⊕ (−4, 2, 2) ⊕ (0, 0)] (171)

Again calculating the inner product between this vector and v5 we ﬁnd:

1 /6[( 2, 2, 2, −4, −4, 2 ) ⊕ (−4, 2, 2 ) ⊕ ( 0, 0 )] 1 hγ|v5i = = [6−8−10−16−8] = −1 (172) 1/6[( 1, 1, 1, 1, 1, −5 ) ⊕ ( 4, −2, −2 ) ⊕ ( 3, −3 )] 36

Thus the following root is an element of R(E8):

1/6[( 2, 2, 2, −4, −4, 2 ) ⊕ (−4, 2, 2 ) ⊕ ( 0, 0 )] 1 6 u := Wv5 (γ) = γ − hγ|v5iv5 = / [( 1, 1, 1, 1, 1, −5 ) ⊕ ( 4, −2, −2 ) ⊕ ( 3, −3 )] + (173) 1/6[( 3, 3, 3, −3, −3, −3 ) ⊕ ( 0, 0, 0 ) ⊕ ( 3, −3 )]

29 Considering the coordinate-wise permutations of these vectors we ﬁnd the following subsets of R(E8), denoting ˜ WA5×A2×A1 by W: ˜ W v0 = P(−1, 1, 0, 0, 0, 0) ⊕ {~0} ⊕ {~0} = R(A5) ⊕ {~0} ⊕ {~0} ˜ W v6 = {~0} ⊕ P(−1, 1, 0) ⊕ {~0} = {~0} ⊕ R(A2) ⊕ {~0} ˜ W v8 = {~0} ⊕ {~0} ⊕ P(−1, 1) = {~0} ⊕ {~0} ⊕ R(A1) (174) ˜ 1 W ± v5 = ± 6 [(1, 1, 1, 1, 1, −5 ) ⊕ (4, −2, −2 ) ⊕ (3, −3 )] ˜ 1 W ± w = ± 3 [(1, 1, 1, 1, −2, −2 ) ⊕ (1, 1, −2 ) ⊕ (0, 0 )] ˜ 1 W u = 2 [(1, 1, 1, −1, −1, −1 ) ⊕ (0, 0, 0 ) ⊕ (1, −1 )]

Where we note again that W˜ ± u = W˜ u. Hence we calculate the cardinality of the union of these disjoint sets: ˜ ˜ ˜ ˜ ˜ ˜ W(v0) + W(v6) + W(v8) + W(±v5) + W(±w) + W(u) 632 632 632 (175) = |R(A )| + |R(A )| + |R(A )| + 2 · + 2 · + = 240 5 2 1 1 1 1 2 1 0 3 0 1

Thus we ﬁnd that: ∼ ˜ ˜ ˜ ˜ R(E8) = W S(A5 × A2 × A1) ∪ W(±v5) ∪ W(±w) ∪ W(u) (176) Or more speciﬁcally:

∼ 1 R(E8) = R(A5 × A2 × A1) ∪ ± 6 [P(1, 1, 1, 1, 1, −5 ) ⊕ P(4, −2, −2 ) ⊕ P(3, −3 )] 1 ∪ ± 3 [P(1, 1, 1, 1, −2, −2 ) ⊕ P(1, 1, −2 ) ⊕ P(0, 0 )] (177) 1 ∪ 2 [P(1, 1, 1, −1, −1, −1 ) ⊕ P(0, 0, 0 ) ⊕ P(1, −1 )]

4.7 Describing E8 in the coordinates of D5 × A3

v7 v6 v5 v4 v3 v2 v1 v0

δ1

δ5 δ4 δ3 δ2 α3 α2 α1

Figure 12: The subset of S(E8) generating a copy of R(D5 × A3)

Now we focus on the collection of roots (S(E8) ∪ {v0}) \{v3}. It can be shown that this subset generates a copy of R(D5 × A3) in R(E8) by the following identiﬁcation:   αk+1 for 0 ≤ k ≤ 2 vk = δk−2 for 4 ≤ k ≤ 7 (178)  δ1 for k = 8

30 Then using Eq.108 we can write the remaining simple root of R(E8) as follows: 1 v3 = − 4 (v0 + 2v1 + 3v2 + 5v4 + 6v5 + 4v6 + 2v7 + 3v8) 1 = − 4 (α1 + 2α2 + 3α3 + 5δ2 + 6δ3 + 4δ4 + 2δ5 + 3δ1) ˜ ˜ ˜ ˜ ˜ ˜ (179) 1 (L2 − L1) + 2(L3 − L2) + 3(L4 − L3) + 5(L2 − L1) = − 4 +6(L3 − L2) + 4(L4 − L3) + 2(L5 − L3) + 3(L2 + L1)

1 ˜ ˜ ˜ ˜ = + 4 (2L1 − 2L2 − 2L3 − 2L4 − 2L5 + L1 + L2 + L3 − 3L4) Which we can write out coordinate-wise as:

v0 = ( 0, 0, 0, 0, 0 ) ⊕ (−1, 1, 0, 0 )

v1 = ( 0, 0, 0, 0, 0 ) ⊕ ( 0, 1, −1, 0 )

v2 = ( 0, 0, 0, 0, 0 ) ⊕ ( 0, 0, 1, −1 ) 1 v3 = /4[( 2, −2, −2, −2, −2 ) ⊕ ( 1, 1, 1, −3 )]

v4 = (−1, 1, 0, 0, 0 ) ⊕ ( 0, 0, 0 0 ) (180)

v5 = ( 0, −1, 1, 0, 0 ) ⊕ ( 0, 0, 0 0 )

v6 = ( 0, 0, −1, 1, 0 ) ⊕ ( 0, 0, 0 0 )

v7 = ( 0, 0, 0, −1, 1 ) ⊕ ( 0, 0, 0 0 )

v8 = ( 1, 1, 0, 0, 0 ) ⊕ ( 0, 0, 0 0 )

Recall that the action of Dn is such that:

WD5 [(2, −2, −2, −2, −2)] = P(2, −2, −2, −2, −2) ∪ P(2, 2, 2, −2, −2) ∪ P(2, 2, 2, 2, 2) (181)

In this sense we can ﬁnd an element β ∈ WD5×A3 (v3) such that: 1 β = 4 [(2, 2, 2, 2, 2) ⊕ (1, 1, −3, 1)] (182)

Then calculating inner products between this β and v3 gives:

1 /4[( 2, 2, 2, 2, 2 ) ⊕ ( 1, 1, −3, 1 )] 1 hβ|v3i = = [4 − 16 + 2 − 6] = −1 (183) 1/4[( 2, −2, −2, −2, −2 ) ⊕ ( 1, 1, 1, −3 )] 16 Thus we ﬁnd the following distinct root:

1/4[( 2, 2, 2, 2, 2 ) ⊕ ( 1, 1, −3, 1 )] 1 4 w := Wv3 (β) = β − hβ|v3iv3 = / [( 2, −2, −2, −2, −2 ) ⊕ ( 1, 1, 1, −3 )] + (184) 1/4[( 4, 0, 0, 0, 0 ) ⊕ ( 2, 2, −2, −2 )]

Considering the coordinate-wise permutations of these vectors we ﬁnd the following subsets of R(E8), denoting ˜ WD5×A3 by W: ˜ W v0 = {~0} ⊕ R(A3) ˜ W v4 = R(D5) ⊕ {~0} W˜ ± v = ± 1 [P( 2, −2, −2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 3 4 (185) 1 ∪ ± 4 [P( 2, 2, 2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ ± 4 [P( 2, 2, 2, 2, 2 ) ⊕ P(1, 1, 1, −3 )] ˜ 1 W w = 2 [P(±2, 0, 0, 0, 0 ) ⊕ P(1, 1, −1, −1 )] Where we note that W˜ ± w = W˜ w. Hence we calculate the cardinality of the union of these disjoint sets: ˜ ˜ ˜ ˜ W(v0) + W(v4) + W(±v3) + W(w) 54 54 54 5 4 (186) = |R(D )| + |R(A )| + 2 · + 2 · + 2 · + 2 · = 240 5 3 1 1 3 1 5 1 1 2

31 Thus we ﬁnd that: ∼ ˜ ˜ ˜ R(E8) = W S(D5 × A3) ∪ W(±v3) ∪ W(w) (187) Or more speciﬁcally:

∼ 1 R(E8) = R(D5 × A3) ∪ ± 4 [P( 2, −2, −2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ ± 4 [P( 2, 2, 2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 1 (188) ∪ ± 4 [P( 2, 2, 2, 2, 2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ 2 [P(±2, 0, 0, 0, 0 ) ⊕ P(1, 1, −1, −1 )]

4.8 Overview of the root system identities To summarize we have the following identities:

∼ 1 R(E8) = R(A8) ∪ ± 3 P(1, 1, 1, 1, 1, 1, −2, −2, −2 ) ∼ 1 R(E8) = R(D8) ∪ ± 2 P(−1, 1, 1, 1, 1, 1, 1, 1) 1 ∪ ± 2 P(−1, −1, −1, 1, 1, 1, 1, 1) ∼ 1 R(E8) = R(A4 × A4) ∪ ± 5 [P(1, 1, 1, 1, −4) ⊕ P(3, 3, −2, −2, −2)] 1 ∪ ± 5 [P(3, 3, −2, −2, −2) ⊕ P(1, 1, 1, 1, −4)] ∼ 1 R(E8) = R(A7 × A1) ∪ ± 4 [P(1, 1, 1, 1, 1, 1, −3, −3 ) ⊕ P(2, −2 )] 1 ∪ 2 [P(1, 1, 1, 1, −1, −1, −1, −1 ) ⊕ P(0, 0 )] ∼ 1 R(E8) = R(A5 × A2 × A1) ∪ ± 6 [P(1, 1, 1, 1, 1, −5 ) ⊕ P(4, −2, −2 ) ⊕ P(3, −3 )] 1 ∪ ± 3 [P(1, 1, 1, 1, −2, −2 ) ⊕ P(1, 1, −2 ) ⊕ P(0, 0 )] 1 ∪ 2 [P(1, 1, 1, −1, −1, −1 ) ⊕ P(0, 0, 0 ) ⊕ P(1, −1 )] ∼ 1 R(E8) = R(D5 × A3) ∪ ± 4 [P( 2, −2, −2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ ± 4 [P( 2, 2, 2, −2, −2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ ± 4 [P( 2, 2, 2, 2, 2 ) ⊕ P(1, 1, 1, −3 )] 1 ∪ 2 [P(±2, 0, 0, 0, 0 ) ⊕ P(1, 1, −1, −1 )]

32 5 Discussion

We have found several classical root systems embedded in the root system of E8. We used these subsystems to express the root system of E8 in the coordinates of the ambient space of those classical root systems. Further research can utilize these results to express the algebra of E8 as a product of these classical algebras and some of their representations. Another extension of this thesis can be made by ﬁnding further subsystems within the treated subsystems. Note that all the found subalgebras are simply laced. Hence we can apply our algorithm to ﬁnd the lowest root of these systems.

E8 An ···

E7 Dn ···

Figure 13: The lowest roots of all simply laced Lie algebras

Consequently, these lowest roots give rise to diﬀerent subsystems. First of all we note that An does not contain any non-trivial subalgebras that are reachable by the lowest root method. For the rest of the simply laced Lie algebras, we can summarized their subalgebras in Table 6. Thus we ﬁnd the following distinct algebras in E8 by iteratively applying the lowest root method (excluding the algebras already treated in this thesis):

2 2 2 • D6 × (A1) • (A3) × (A1)

2 4 • (D4) • (A2) 2 • D4 × (A2) 2 4 • (A2) × (A1) 2 • D4 × A2 × (A1) 6 • A2 × (A1) 4 • D4 × (A1) 8 • (A1)

Considering each of this algebras will give rise to diﬀerent coordinate representations of the root system of E8, which in turn will give rise to a diﬀerent way to express the algebra of E8 in terms of the found subalgebras and some of their representations.

Algebra Non-trivial subalgebras

2 Dn Dn−m × Dm, Dn−2 × (A1) 4 D4 (A1) 2 E8 A8, D8, A7 × A1, E7 × A1, E6 × A2, D5 × A3, A5 × A2 × A1,(A4) 2 E7 A7, D6 × A1, A5 × A2,(A3) × A1 2 E6 A5 × A2,(A2)

Table 6: List of non-trivial subalgebras of all simply laced Lie groups, reachable by applying the lowest root method once. Notice that in this table n ≥ 5 and m ≥ 3.

33 References

[1] Fulton, W. and Harris, J., Representation Theory: A First Course Springer, 2004. ISBN 0-387-97495-4.