Lie Algebras, Extremal Elements, and Geometries

Lie algebras, extremal elements, and geometries

Citation for published version (APA): Panhuis, in 't, J. C. H. W. (2009). Lie algebras, extremal elements, and geometries. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR643504

DOI: 10.6100/IR643504

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PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magniﬁcus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 12 oktober 2009 om 16.00 uur

door

Jozef Clemens Hubertus Wilhelmus in ’t panhuis

geboren te Roermond Dit proefschrift is goedgekeurd door de promotor: prof.dr. A.M. Cohen

Copromotor: dr. F.G.M.T. Cuypers

This research was ﬁnancially supported by NWO (Netherlands Organisation for Scien- tiﬁc Research) in the framework of the Free Competition, grant number 613.000.437. Preface

Lie algebras, extremal elements, and geometries

This thesis is about Lie algebras generated by extremal elements and geometries whose points correspond to extremal points, that is, projective points corresponding to extremal elements. Inside a Lie algebra g over a field F of characteristic not two, extremal elements are those nonzero elements x for which [x, [x, g]] ⊆ Fx. Extremal elements for which [x, [x, g]] = {0} are called sandwich elements. The definitions of extremal elements and sandwich elements in characteristic two are somewhat more involved. Sandwich elements were originally introduced in relation with the restricted Burn- side problem. An important insight for the resolution of this problem is the fact that a Lie algebra generated by finitely many sandwich elements is necessarily finite-dimensional. While this fact was first only proved under extra assumptions, later it was proved in full generality. Extremal elements play important roles in both classical and modern Lie algebra theory. In complex simple Lie algebras, or their analogues over other fields, extremal elements are precisely the elements that are long-root vectors relative to some maximal torus. In the classication of simple Lie algebras in small characteristics extremal elements are also useful: they occur in non-classical Lie algebras such as the Witt algebras. In the first chapter we give some definitions and basic results regarding Lie algebras, extremal elements, and the different geometries which are the subject of this thesis. Also we will already give a hint of how a Lie algebra can be related to a geometry using its extremal points: the points of the geometry are the extremal points in the Lie algebra and the lines are the projective lines all of whose points are extremal. Cohen and Ivanyos proved that the resulting geometry is a so-called root filtration space. Moreover, they showed that a root filtration space with a non-empty line set is the shadow space of a building. These buildings are geometrical and combinatorial structures introduced by Tits in order to obtain a better understanding of the semi-simple algebraic groups. If we are dealing with a Lie algebra for which no projective line consists entirely of extremal points, then the results of Cohen and Ivanyos are no longer applicable. There- fore, in that situation, the question is whether a non-trivial geometric structure can be associated to the extremal points in the Lie algebra. This is the subject of the second and third chapter. First, for Lie algebras generated by two or three extremal elements, we vi Preface

find the isomorphism type of the corresponding Lie algebra and give a description of the extremal elements. Then, for an arbitrary number of generators, we construct a geometry whose point set is the set of extremal points. As lines we take the hyperbolic lines: sets of extremal points corresponding to the extremal elements in a Lie subalgebra generated by two non-commuting extremal elements. If the field contains precisely two elements, then the resulting geometry is a connected Fischer space. This is a connected geometry in which each plane is isomorphic to a dual affine plane of order two or an affine plane of order three. Connected meaning that the collinearity graph of the geometry is connected. If the field contains more than two elements, then we take as lines the singular lines: sets of all extremal points commuting with all extremal points commuting with two distinct commuting extremal points. Using a result by Cuypers we prove that the resulting geometry is a polar space. This is a geometry in which each point not on a line is collinear with either one or all points of that line. In fact, the polar space we construct is non-degenerate, that is, no point is collinear with all other points. It was proven by Buekenhout and Shult that such a non-degenerate polar space is also the shadow space of a building. Then, in the fourth chapter, we consider the problem of describing all Lie algebras generated by a finite number of extremal elements over a field of characteristic not two. Cohen et al. proved that the Chevalley algebra of type A2 is the generic Lie algebra in case of three extremal generators. Moreover, in ’t panhuis et al. extended this result to more generators. There, starting from a graph, they constructed an affine variety whose points parametrize Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. In addition, for each Chevalley algebra of classical type they found a finite graph such that all points in some open dense subset of the corresponding variety parametrize Lie algebras isomorphic to this Chevalley algebra. We take a different view point. Starting from a connected simply laced Dynkin diagram of finite or affine type, we prove that the variety is an affine space and, assuming the Dynkin diagram is of affine type, we prove that the points in some open dense subset parametrize Lie algebras isomorphic to the Chevalley algebra corresponding to the associated Dynkin diagram of finite type. In the fifth chapter, we take a closer look at one type of geometry whose points correspond to extremal elements inside a Lie algebra: the class of finite irreducible cotriangular spaces. Each such cotriangular space is an example of a Fischer space in which each plane is isomorphic to a dual affine plane of order two. Hall and Shult proved that each irreducible cotriangular space is of three possible types, that is, triangular, symplectic, or orthogonal type. We use this fact to classify the polarized embeddings of a finite irreducible cotriangular space. Here, a polarized embedding is an injective map from the point set of the cotriangular space into the point set of a projective space satisfying certain properties. For instance, lines are mapped into lines and hyperplanes are mapped into hyperplanes. For the spaces of symplectic or orthogonal type we can describe, if the characteristic is not two, the polarized embeddings using the associated symplectic and Preface vii quadratic forms. For other characteristics the polarized embeddings can be described using the root systems of type E6, E7, and E8. For the spaces of triangular type the polarized embeddings can be described using the root systems of type An, n > 4. All this is an extension of the work by Hall who classified the polarized embeddings over the field with two elements. Finally, in the appendix, we give some of the basic terminology used throughout this thesis. viii Preface Contents

Prefacev

Contents ix

1 Preliminaries1 1.1 Lie algebras...... 1 1.1.1 Linear Lie algebras...... 2 1.1.2 Chevalley algebras...... 4 1.1.3 Kac-Moody algebras...... 5 1.2 Extremal elements...... 6 1.3 Geometries...... 11 1.3.1 Planes...... 11 1.3.2 Polar spaces...... 12 1.3.3 Fischer spaces...... 13 1.3.4 Cotriangular spaces...... 16 1.3.5 Root ﬁltration spaces...... 20

2 Lie subalgebras of Lie algebras without strongly commuting pairs 25 2.1 Introduction...... 25 2.2 The multiplication table...... 25 2.3 Lie subalgebras generated by hyperbolic pairs...... 27 2.3.1 Isomorphism type...... 27 2.3.2 Extremal elements...... 27 2.4 Lie subalgebras generated by symplectic triples...... 28 2.4.1 Isomorphism type...... 29 2.4.2 Extremal elements...... 33 2.5 Lie subalgebras generated by unitary triples...... 39 2.5.1 Isomorphism type...... 40 2.5.2 Extremal elements...... 47

3 Constructing geometries from extremal elements 49 3.1 Introduction...... 49 x Contents

3.2 From Lie algebra to polar space...... 51 3.3 From Lie algebra to Fischer space...... 61 3.4 From Fischer space to Lie algebra...... 62 3.4.1 Proof of the main theorem...... 63 3.4.2 Some examples...... 67

4 Constructing simply laced Lie algebras from extremal elements 69 4.1 Introduction and main results...... 69 4.2 The variety structure of the parameter space...... 71 4.3 The sandwich algebra...... 74 4.3.1 Weight grading...... 74 4.3.2 Relation with the root system of the Kac-Moody algebra.... 77 4.3.3 Simply laced Dynkin diagrams of finite type...... 78 4.3.4 Simply laced Dynkin diagrams of affine type...... 79 4.4 The parameter space and generic Lie algebras...... 81 4.4.1 Scaling...... 81 4.4.2 The Premet relations...... 82 4.4.3 The parameters...... 82 4.4.4 Simply laced Dynkin diagrams of finite type...... 83 4.4.5 Simply laced Dynkin diagrams of affine type...... 85 4.5 Notes...... 89 4.5.1 Recognising the simple Lie algebras...... 89 4.5.2 Other graphs...... 90 4.5.3 Geometries with extremal point set...... 90

5 Classifying the polarized embeddings of a cotriangular space 93 5.1 Introduction...... 93 5.2 Polarized embeddings...... 94 5.2.1 Notation...... 94 5.2.2 Deﬁnition...... 94 5.2.3 Equivalence...... 95 5.2.4 Quotient embeddings...... 96 5.2.5 Natural embedding...... 96 5.3 The dimension of a polarized embedding...... 97 5.3.1 Triangular type...... 98 5.3.2 Symplectic type...... 98 5.3.3 Orthogonal type...... 99 5.4 Polarized quotient embeddings...... 99 5.4.1 Polarizing criteria...... 100 5.4.2 Equivalence...... 101 5.5 Equivalence of polarized embeddings: triangular type...... 104 5.5.1 Characterizing the polarized embeddings...... 104 Contents xi

5.5.2 The universal embedding...... 105 5.5.3 Quotient embeddings...... 106 5.5.4 The equivalence classes...... 107 5.6 Equivalence of polarized embeddings: X7 ...... 111 5.6.1 Characterizing the polarized embeddings...... 111 5.6.2 The universal embedding...... 117 5.6.3 Quotient embeddings...... 117 5.6.4 The equivalence classes...... 119 5.7 Equivalence of polarized embeddings: symplectic type...... 120 5.7.1 Field characteristic...... 120 5.7.2 Dimensionality...... 121 5.7.3 Embedding lines...... 122 5.7.4 Quotient embeddings...... 124 5.7.5 The universal embedding...... 124 5.7.6 The equivalence classes...... 126 5.8 Equivalence of polarized embeddings: orthogonal type...... 127 5.8.1 Field characteristic...... 127 5.8.2 Dimensionality...... 127 5.8.3 Embedding lines in characteristic two...... 128 5.8.4 The equivalence classes and the universal embedding: characteristic not two...... 131 5.8.5 The equivalence classes and the universal embedding: characteristic two...... 134

A Basic terminology 137 A.1 Afﬁne varieties and polynomial maps...... 137 A.2 Generalized Cartan matrices and Dynkin diagrams...... 137 A.3 Root systems...... 139 A.4 Algebras and modules...... 142 A.5 Gradings...... 143 A.6 Symplectic, orthogonal and Hermitian spaces...... 144

Bibliography 147

Index 151

Acknowledgements 157

Curriculum Vitae 159 xii Contents Chapter 1

Preliminaries

In this chapter we introduce some of the notation, basic terminology, and results used throughout this thesis regarding Lie algebras, extremal elements, and geometries. Some concepts not deﬁned here can be found in AppendixA.

1.1 Lie algebras

A Lie algebra over a ﬁeld F is an algebra g over F whose multiplication [·, ·]: g×g → g satisﬁes the anti-commutativity identities and the Jacobi identities, that is,

∀x,y,z∈g :[x, x] = 0 ∧ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

Lie algebras were introduced by Lie to study the concept of inﬁnitesimal transformations. Independently, they were also introduced by Killing (1884) in an effort to study non-Euclidean geometry. For an introduction into Lie algebras over characteristic 0 we recommend Humphreys (1978). In the remainder of this thesis we will omit the brackets: we write xyz instead of [x, [y, z]] and (xy)z instead of [[x, y], z]. Moreover, for x an element of a Lie algebra g we write adx to indicate left multiplication by x. In other words,

adx : g → g, y 7→ xy.

Example 1.1 For any associative algebra A with multiplication ∗ : A × A → A another algebra ALie can be constructed. As a vector space ALie is A, but the multiplication on ALie is different from the multiplication on A. For x, y ∈ ALie we deﬁne xy := x ∗ y − y ∗ x. This ensures ALie is a Lie algebra. It is the Lie algebra associated to A.

Now, let g(1) = g1 = g be a Lie algebra. Then, for all integers n > 1, we can deﬁne

gn := [g, gn−1] and g(n) := [g(n−1), g(n−1)]. 2 Chapter 1. Preliminaries

n If there is an n ∈ N with g = {0}, then g is called nilpotent. Moreover, if there is an (n) n ∈ N with g = {0}, then g is called solvable. A non-abelian Lie algebra without any proper solvable ideals is called semi-simple and a nilpotent subalgebra of a Lie algebra is called a Cartan subalgebra if it equals its normalizer.

Example 1.2 For an integer n > 1 and F a ﬁeld we deﬁne tn(F) as the Lie algebra associated to the matrix algebra consisting of all upper triangular matrices with entries in F. The subalgebra of tn(F) consisting of all matrices with zeroes on the diagonal is denoted by nn(F). Both tn(F) and nn(F) are solvable. However, only nn(F) is also nilpotent.

2n+1 Example 1.3 For n ∈ N and F a ﬁeld deﬁne hn(F) as the vector space F together with the multiplication induced by

(j − i) if |j − i| = 1 and i, j ∈ [2n], = 2n+1 i j 0 otherwise.

Here, (and in the remainder of this thesis) [m] and [k, m] are deﬁned such that

[1, m] = [m] if k = 1, and ∀ : {k, . . . , m} = k

This makes hn(F) a nilpotent Lie algebra called the Heisenberg Lie algebra of dimension 2n + 1 over F. In the special case that n = 1 we write h(F) instead of hn(F).

1.1.1 Linear Lie algebras

An important example of a Lie algebra over a ﬁeld F is the general linear Lie algebra

gl(V ) := End(V )Lie of V . Here, End(V ) is the set of endomorphisms of a vector space V over F with the usual composition as multiplication. Any subalgebra of gl(V ) is called a linear Lie algebra and theorems by Ado and Iwasawa (Jacobson 1962, Chapter 6) prove that every (ﬁnite-dimensional) Lie algebra is isomorphic to some linear Lie algebra. n 2 If V = F , for a certain n ∈ N, then dim gl(V ) = n and we write gln(F) instead of gl(V ). In this setting, we identify End(V ) with the algebra of all n × n-matrices with entries in F. The Lie algebras tn(F) and nn(F) from Example 1.2 are examples of linear Lie algebras. Other linear Lie algebras are the Lie algebras of classical type. They are depicted in Examples 1.4–1.7. 1.1. Lie algebras 3

Example 1.4 Let F be a ﬁeld and let n ∈ N. Then the traceless matrices in gln+1(F) form a subalgebra denoted by sln+1(F). As a vector space it is spanned by the matrices Ei,j (i, j ∈ [n + 1] and i 6= j) and Ei,i − Ei+1,i+1 (i ∈ [n]). Here Ei,j is the (n + 1) × (n + 1)-matrix having 1 at position (i, j) and 0 elsewhere. The Lie algebra sln+1(F) is said to be of type An and is referred to as the special linear Lie algebra of dimension 2 n + 2n over F.

Example 1.5 Let F be a field, let n ∈ N, let In ∈ gln(F) be the identity matrix, and 2n+1 define f to be the bilinear form on F defined by 1 0 0  0 0 In . 0 In 0

We deﬁne o2n+1(F) as the subalgebra of gl2n+1(F) consisting of those matrices A sat- n isfying f(Ax, y) = −f(x, Ay) for all x, y ∈ F . The following matrices form a basis:

• Ei+1,j+1 − En+j+1,n+i+1 for i, j ∈ [n],

• Ei+1,n+j+1 − Ej+1,n+i+1 for i, j ∈ [n] with i < j, and

• En+i+1,j+1 − En+j+1,i+1 for i, j ∈ [n] with i < j.

The Lie algebra o2n+1(F) is said to be of type Bn and is referred to as the (odd) orthog- 2 onal Lie algebra of dimension 2n + n over F.

2n Example 1.6 Let F be a field, let n ∈ N, and define f to be the bilinear form on F defined by 0 I n . −In 0

We deﬁne sp2n(F) as the subalgebra of gl2n(F) consisting of those matrices A which n satisfy f(Ax, y) = −f(x, Ay) for all x, y ∈ F . The following matrices form a basis:

• Ei,n+i for i ∈ [n],

• En+i,i for i ∈ [n],

• Ei,j − En+j,n+i for i, j ∈ [n],

• Ei,n+j + Ej,n+i for i, j ∈ [n] with i < j, and

• En+i,j + En+j,i for i, j ∈ [n] with i < j.

The Lie algebra sp2n(F) is said to be of type Cn and is referred to as the symplectic Lie 2 algebra of dimension 2n + n over F. 4 Chapter 1. Preliminaries

2n Example 1.7 Let F be a field, let n ∈ N, and define f to be the bilinear form on F defined by 0 I n . In 0

We deﬁne o2n(F) as the subalgebra of gl2n(F) consisting of those matrices A which n satisfy f(Ax, y) = −f(x, Ay) for all x, y ∈ F . The following matrices form a basis:

• Ei,j − En+j,n+i for i, j ∈ [n],

• Ei,n+j − Ej,n+i for i, j ∈ [n] with i < j, and

• En+i,j − En+j,i for i, j ∈ [n] with i < j.

The Lie algebra o2n(F) is said to be of type Dn and is referred to as the (even) orthogonal 2 Lie algebra of dimension 2n − n over F. The linear Lie algebra in the next example will return in Chapter2.

Example 1.8 Let n ∈ and let be a field which is the fixpoint set of an involution σ N nF of a field F. This makes V = F an n-dimensional vector space over F and, if F 6= F, a 2n-dimensional vector space over F. Next, let f : V × V → F be a Hermitian form relative to σ. Then we define un(F, f) to be the subalgebra of gl(V ) over F consisting of those matrices A satisfying f(Ax, y) + f(x, Ay) = 0, for all x, y ∈ V . This is a Lie algebra over F, but not over F in the case that F 6= F (because f is linear in the first, but not in the second variable). It is called the unitary Lie algebra 2 of dimension n over F. Intersecting un(F, f) with sln(F) gives another Lie algebra 2 sun(F, f) called the special unitary Lie algebra of dimension n − 1 over F.

1.1.2 Chevalley algebras The finite-dimensional simple complex Lie algebras are classified using the irreducible root systems and the Dynkin diagrams of finite type (Killing 1884, Cartan 1894). For the relevant definitions regarding root systems and Dynkin diagrams we refer to Appendix A. Here, we show how the semi-simple complex Lie algebras give rise to Lie algebras over other fields. Therefore, let g be a semi-simple complex Lie agebra. Then Humphreys (1978) says that g contains a Cartan subalgebra g0. Now, a root system Φ can be associated to g: the roots relative to g0 are the linear functionals α on g0 satisfying

g := {x ∈ g | ∀ : hx = α(h)x}= 6 {0}. α h∈g0 This makes M g = g0 ⊕ gα α∈Φ 1.1. Lie algebras 5 a root space decomposition of g. In addition, Humphreys says that g has a so-called Chevalley basis relative to Φ. By deﬁnition, this basis contains one nonzero element eα ∈ gα for each root α ∈ Φ, and hα := eαe−α for each root α ∈ Φ with

∀α,β∈Φ : α + β ∈ Φ ⇒ eαeβ = −e−αe−β ∈ gα+β.

An important property of this Chevalley basis is

∀ : g ⊗ is a Lie algebra over . F a ﬁeld Z F F If Γ is the Dynkin diagram corresponding to Φ, then this Lie algebra is called the Cheval- ley algebra over F of type Γ. Moreover, if Γ is simply laced, then also the corresponding Lie algebra is called simply laced. If g is a simple complex Lie algebra, then for a ﬁeld the Chevalley algebra g ⊗ F Z F is often simple, but not always (Seligman 1967, Strade 2004). Examples of Chevalley algebras are the Lie algebras introduced in Examples 1.4– 1.7. They are of classical type, that is, An, Bn, Cn, Dn, respectively.

1.1.3 Kac-Moody algebras The Chevalley algebras were generalized by Kac (1990) to Kac-Moody algebras and their equivalents over other fields. These Kac-Moody algebras are complex Lie algebras constructed from a Dynkin diagram. Here, we give the construction in case Γ is a Dynkin diagram of finite type and we point at a Chevalley basis giving rise to a Chevalley algebra of type Γ. So, let Γ = (Π, ∼) be a finite type Dynkin diagram and let (Ax,y)x,y∈Π be its generalized Cartan matrix. Then the Kac-Moody algebra gKM over C of type Γ is the free Lie algebra generated by 3 · |Π| generators, denoted ex, fx, hx for x ∈ Π, modulo the relations  hxhy = 0,   exfx = hx, ∀x,y∈Π :  hxey = Axyey,  hxfy = −Axyfy, and  e f = 0,  x y  1−Axy ∀x6=y∈Π : adex ey = 0, 1−A  ad xy f = 0. fx y Π For x ∈ Π, assign to ex, fx, hx the weights αx, −αx, 0 ∈ Z , respectively. Here, αx is the element with a 1 on position x and zeroes elsewhere. This induces a weight for each word over the 3 · |Π| generators of gKM. If we speak of the weight of a monomial in the generators, then we mean the weight of the corresponding word. Now, we have a 6 Chapter 1. Preliminaries grading of g by weight: KM M gKM = (gKM)β. Π β∈Z Π Here, for each β ∈ Z , the summand (gKM)β is the weight space consisting of all monomials of weight β. In fact, the root system Φ of gKM of type Γ satisfies

Π Φ = {β ∈ Z \{0} | (gKM)β 6= {0}}.

It contains the simple roots αx for x ∈ Π. A Chevalley basis of gKM consists of the images of hx, x ∈ Π, and one vector e ∈ (g ) for every root α ∈ Φ, where e and e may be taken as the images of α KM α αx −αx ex and fx (Carter 1972, Section 4.2). It gives rise to the Chevalley algebra of type Γ.

1.2 Extremal elements

Here, we consider extremal elements inside Lie algebras. Most of the results and the definitions in this section come from Cohen and Ivanyos (2006) and, assuming the characteristic is not two, Cohen, Steinbach, Ushirobira, and Wales (2001). Let g be a Lie algebra over a field F. Then a non-zero element x ∈ g is called an extremal element if there exists a map gx : g → F, which is by definition linear, satisfying the extremal identities:

∀y∈g : xxy = 2gx(y)x, (1.1)

∀y,z∈g : xyxz = gx(yz)x − gx(z)xy − gx(y)xz. (1.2)

Note that identities (1.2) go back to Premet and were ﬁrst used by Chernousov (1989). Therefore, they are also referred to as the Premet identities.

Lemma 1.9 (Cohen and Ivanyos 2006) If char(F) 6= 2, then the Premet identities follow from the remaining extremal identities.

Lemma 1.10 (Cohen and Ivanyos 2006) A Lie algebra generated by extremal elements is linearly spanned by extremal elements.

We denote the set of extremal elements in a Lie algebra g over F by E(g) and the corresponding set of extremal points {Fx | x ∈ E(g)} by E(g). Usually, it is clear which Lie algebra g is meant. Then we write E and E instead of E(g) and E(g), respectively.

Example 1.11 Let g be the Lie subalgebra of sl2(F) generated by 0 1 0 0 x := and y := . 0 0 1 0 1.2. Extremal elements 7

Then, g = sl2(F) and either all nonzero matrices in g are extremal or only the matrices of rank 1. To be more speciﬁc,

 S −1  Fx ∪ Fy ∪ F(δx + δ y + xy) if char(F) 6= 2, and ∗ E ∪ {0} = δ∈F  g otherwise.

For yet another example we need the concept of inﬁnitesimal (Siegel) transvections. Therefore, let V be a vector space over F containing an element x and let h be a linear functional on V . Then

V → V, y 7→ h(y)x is called an inﬁnitesimal transvection if h(x) = 0. If V admits a non-degenerate symmetric bilinear form f, and if V contains two elements x, y ∈ V with f(x, x) = f(x, y) = f(y, y) = 0, then

V → V, z 7→ f(x, z)y − f(y, z)x is called an inﬁnitesimal Siegel transvection.

Example 1.12 Let g be a classical Chevalley algebra over a field of characteristic not two. If g is a special linear Lie algebra or a symplectic Lie algebra, then all infinitesimal transvections on g are extremal and generate g. Otherwise, all infinitesimal Siegel transvections on g are extremal and generate g. See for instance Postma (2007).

For x, y ∈ E we write

 E ⇐⇒ x = y,  −2 F F   E−1 ⇐⇒ xy = 0, (x, y) ∈/ E−2, and Fx + Fy ⊆ E ∪ {0}, (x, y) ∈ E0 ⇐⇒ xy = 0 and (x, y) ∈/ E−2 ∪ E−1,   E1 ⇐⇒ xy 6= 0 and gxy = 0,  E2 ⇐⇒ gxy 6= 0.

In addition, if (x, y) ∈ ∪j∈[−2,i]Ej, for some i ∈ [−2, 2], then we write (x, y) ∈ E≤i. Analogously, for x, y ∈ E and i ∈ [−2, 2], we say that

(Fx, Fy) ∈ E(≤)i ⇐⇒ (x, y) ∈ E(≤)i.

By deﬁnition,

E × E = E−2 ]E−1 ]E0 ]E1 ]E2.

Note that to ensure the validity of the results of Cohen and Ivanyos (2006) for charac- 8 Chapter 1. Preliminaries

teristic two the deﬁnition of E−1 above is slightly different from the one used by Cohen and Ivanyos. Given two linearly independent commuting extremal elements x, y they used as deﬁning criterium

(x, y) ∈ E−1 ⇐⇒ ∀z∈g : xyz = gy(z)x + gx(z)y.

Though this does not make any difference over characteristic not two, it might make a difference over characteristic two. Next, let X,Y be two distinct extremal points. Then there is an i ∈ [−2, 2] such that (X,Y ) ∈ Ei. Now, the pair (X,Y ) is said to be hyperbolic if i = 2, special if i = 1, polar if i = 0, strongly commuting if i = −1, and commuting if i ≤ 0. Let (X,Y ) be a hyperbolic pair. Then the set of extremal points corresponding to the extremal elements of g in the Lie algebra hX,Y i generated by X and Y is called the hyperbolic line on X and Y . If Z ∈ E makes (X,Y,Z) a hyperbolic path of length two, that is, (Y,Z) ∈ E2, then (X,Y,Z) is called a symplectic triple if (Y,Z) ∈ E0 and a unitary triple if (Y,Z) ∈ E2. Here, a hyperbolic path is simply a path in (E, E2).

Example 1.13 The Lie algebra of Example 1.11 satisﬁes

E ⊕ E ⊕ E if char( ) = 2, and E × E = −2 −1 1 F E−2 ⊕ E2 otherwise.

Moreover, if char(F) = 2 and X1,X2 ∈ E, then

(X1,X2) ∈ E1 ⇐⇒ X1X2 ⊆ X1 + X2.

If x ∈ E and gx = 0, then we call x a sandwich element. The corresponding extremal point we call a sandwich point. We write S(g) and S(g) for the sets of sandwich elements and sandwich points, respectively. Again, if it clear which Lie algebra g is meant, we omit g.

Example 1.14 The Lie algebra of Examples 1.11 and 1.13 satisﬁes

xy if char( ) = 2, and S ∪ {0} = F F {0} otherwise.

Lemma 1.15 (Cohen and Ivanyos 2006) If g is a Lie algebra generated by extremal elements, then the Lie subalgebra hSi generated by the sandwich elements is an ideal of g.

If gx can be chosen to be identically zero for an extremal element x, then we insist that it is chosen to be identically zero. In this way, we ensure that gx is uniquely determined for each extremal element x. Moreover, we obtain that an extremal element x is a sandwich element if and only if gx = 0. 1.2. Extremal elements 9

Lemma 1.16 (Cohen and Ivanyos 2006) Let x ∈ E. With the restriction that gx is chosen to be identically zero if x is a sandwich element, the map gx is a uniquely deﬁned functional on g.

For x ∈ E we call gx the extremal functional on x. As the following proposition points out, it gives rise to a unique bilinear form on g which we call the extremal form.

Proposition 1.17 (Cohen and Ivanyos 2006) Suppose that g is a Lie algebra over F generated by E. Then g is linearly spanned by E and there is a unique bilinear form g : g × g → F such that

∀x∈E∀y∈g : g(x, y) = gx(y).

The form g is symmetric and associative, that is,

∀x,y,z∈g : g(x, y) = g(y, x) ∧ g(x, yz) = g(xy, z).

For a Lie algebra g = hEi with extremal form g, we write gxy and gxyz instead of g(x, y) and g(x, yz) for all x, y, z ∈ g. Because of the fact that g is both symmetric and associative this is well deﬁned. However, it may cause confusion with the extremal functional in the case that xy or xyz is extremal. Therefore, we will make sure that it is clear from the context what is meant. The following lemma describes the possible isomorphism types of a Lie subalgebra generated by two extremal elements.

Lemma 1.18 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F, and let L be a Lie subalgebra of g generated by two linearly independent extremal elements x and y. Then,

(i) L = Fx + Fy is abelian, if (x, y) ∈ E−2 ∪ E−1 ∪ E0, ∼ (ii) L = h(F), if (x, y) ∈ E1, and ∼ (iii) L = sl2(F), if (x, y) ∈ E2.

Moreover, xy ∈ E if and only if (x, y) ∈ E1 if and only if (x, xy) ∈ E−1. If there are no strongly commuting or special pairs, then the following lemma shows that no non-extremal element becomes extremal after restricting to a component of (E, E2).

Lemma 1.19 Let g be a Lie algebra over F generated by extremal elements, let L be a Lie subalgebra generated by the points in a component of (E, E2), and assume E±1 = ∅. Then E(L) ⊆ E(g). 10 Chapter 1. Preliminaries

Proof. Let x ∈ E(L) and y, z ∈ g. We need to prove

xxz = 0 and xyxz = g(x, yz)x − g(x, z)xy − g(x, y)xz.

Therefore, let M be the direct sum of the extremal points not in the connected component of (E, E2) containing Fx. This ensures g = L⊕M. Now, there is a v ∈ L and a collection E0 of extremal elements commuting with x such that X z = v + w. w∈E0

Hence, since x ∈ E(L) commutes with E0, X xyxz = xyxv + xyxw = xyxv = g(x, yv)x − g(x, v)xy − g(x, y)xv w∈E0 = g(x, yv)x − g(x, v)xy − g(x, y)xv X + (−g(y, xw)x − g(x, w)xy − g(x, y)xw) w∈E0 = g(x, yv)x − g(x, v)xy − g(x, y)xv X + (g(x, yw)x − g(x, w)xy − g(x, y)xw) w∈E0 X X X = g(x, y(v + w)x − g(x, v + w)xy − g(x, y)x(v + w) w∈E0 w∈E0 w∈E0 = g(x, yz)x − g(x, z)xy − g(x, y)xz.

Thus, indeed, x ∈ E(g).

Finally, let g again be a Lie algebra. Then we deﬁne for each extremal element x ∈ g and each scalar α the exponential map exp(x, α): g → g by

2 exp(x, α)y = y + αxy + α gxyx.

If x ∈ E, then we often write exp(x) instead of exp(x, 1). Note that

∞ X 1 ∀ : char( ) 6= 2 ⇒ exp(x) = adn. x∈E F n! x n=0

Lemma 1.20 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F containing an extremal element x. Then

∀ ∀ : exp(x, α) ∈ Aut(g) ∧ exp(x, α)y ∈ E. α∈F y∈E 1.3. Geometries 11

1.3 Geometries

First some basic terminology. Let (P, L) be a pair consisting of a set P of points and a set L of lines. Moreover, suppose each line in L is a subset of P of size at least two. Now, (P, L) is called a point-line space. If any two distinct points are on at most one line, then (P, L) is called a partial linear space. If any two distinct points are on exactly one line, then (P, L) is called a linear space. Let X be a subset of P. Then it is a subspace of (P, L) if any line intersecting X in at least two points is completely contained in X . Moreover, if X is a proper subspace, that is, ∅= 6 X 6= P, then X is called a hyperplane of (P, L) if and only if each line in L intersects X . For a projective space this is in accordance with the classical notion of a hyperplane as the kernel of a non-trivial linear functional. If we deﬁne K to be the set of lines in L completely contained in X , then, assuming X is a subspace, (X , K) is a point-line space. Note that in this situation, X will also be called a point-line space and (X , K) will also be called a subspace. Next, consider the intersection of all subspaces of (P, L) containing X . This is again a subspace and we denote it by hX i. The elements of X are called the generators of hX i and hX i is said to be generated by X . Suppose n is the minimal cardinality of a generating set of (P, L), then n is said to be the generating rank of (P, L). Moreover, if the cardinality of X equals the generating rank, then X is said to be a basis of (P, L). The collinearity graph of a point-line space (P, L) is the graph where two (possibly coinciding) points in P are adjacent if and only if there is a line in L containing both of them. The complement is called the co-collinearity graph. If the collinearity graph or its complement is connected, then (P, L) is called connected or co-connected, respectively. Two points in (P, L) are called collinear if they are adjacent in the collinearity graph. Finally, two point-line spaces are said to be isomorphic if there exists a bijection of the point sets that is simultaneously a bijection of the line sets. In the remainder of this section we will take a closer look at the different point-line spaces which will be the subject of this thesis.

1.3.1 Planes A plane is a subspace of a point-line space generated by two distinct intersecting lines. A projective plane is a point-line space such that, • given any two distinct points, there is exactly one line containing both of them,

• given any two distinct lines, there is exactly one intersection point, and

• there are four distinct points such that no line contains more than two of them. If all lines of a projective plane have the same number r of points, then it is said to be of order r. 12 Chapter 1. Preliminaries

Figure 1.1: Dual afﬁne plane of order two Figure 1.2: Afﬁne plane of order three

An affine plane is a projective plane from which a single line and all points on that line are removed. A dual affine plane is a projective plane from which a single point and all lines through that point are removed. A transversal coclique in a dual affine plane is the set of points of a dual affine plane incident with a removed line. A (dual) affine plane corresponding to a projective plane of order r is also said to be of order r. The dual affine plane of order two and the affine plane of order three (also known as Young’s geometry) are depicted in Figures 1.1 and 1.2. There the lines are coloured in such a way that two lines intersect if and only if they have different colours. Note that in Figure 1.1 each pair of non-collinear points is an example of a transversal coclique.

1.3.2 Polar spaces

A polar space is a partial linear space in which any point not on a line is connected to either one or all points of that line. This axiom was introduced by Buekenhout and Shult (1974). Polar spaces are the subject of Chapter3. Given two points x and y we write x ⊥ y to denote that they are collinear and we write x⊥ to denote the set of points collinear with x. If no two points x and y in a polar space satisfy x⊥ = y⊥, then the polar space is called non-degenerate. Moreover, if a polar space (P, L) is non-degenerate, then the polar graph, the collinearity graph of (P, L), determines (P, L) uniquely. See for example Johnson (1990). The rank of a non-degenerate polar space (P, L) is the largest non-negative integer n for which there exists a chain X1 ⊆ ... ⊆ Xn of length n, where the Xi are singular subspaces. Here, a subspace is called singular if all points in the subspace are collinear. 1.3. Geometries 13

The non-degenerate polar spaces have been classiﬁed by Veldkamp (1959,1960) and Tits (1974) under the assumption that their rank is at least three. A non-degenerate polar space of rank at least four having at least three points per line can be proven to be isomorphic to a so-called classical polar space. See for instance Cuypers, Johnson, and Pasini (1993). These classical polar spaces can be constructed starting from projective spaces. For an introduction into projective spaces and polar spaces we refer to Taylor (1992) and Cameron (1991).

Example 1.21 Let V be a vector space carrying a symplectic form f, then the partial linear space Sp(V, f) = (P, L) with P the set of projective points on which f vanishes identically and L the set of projective lines completely contained in P is a polar space.

Example 1.22 Let V be a vector space carrying a Hermitian form f, then the partial linear space U(V, f) = (P, L) with P the set of projective points on which f vanishes identically and L the set of projective lines completely contained in P is a polar space.

Example 1.23 Let V be a vector space carrying a quadratic form Q, then the partial linear space O(V,Q) = (P, L) with P the set of projective points on which Q vanishes identically and L the set of projective lines completely contained in P is a polar space. The polar spaces of Examples 1.21–1.23 are the classical polar spaces of symplectic, unitary, or orthogonal type, respectively.

1.3.3 Fischer spaces A Fischer space is a partial linear space in which each plane is isomorphic to either a dual afﬁne plane of order two or an afﬁne plane of order three. We denote the intersection of collinearity and non-equality in a Fischer space by ∼ and the union of non-collinearity and equality by ⊥. Moreover, for a point x in a ∼ ⊥ Fischer space (P, L) we write x , x , and ∆x to denote the sets {y ∈ P | x ∼ y}, {y ∈ P | x ⊥ y}, and x⊥ \{x}, respectively. Now, a connected and co-connected Fischer space in which no two points x and y satisfy x∼ ∪ {x} = y∼ ∪ {y} or x∼ = y∼ is called irreducible. Important examples of Fischer spaces can be constructed using so-called 3-transpositions. A conjugacy class D of 3-tranpositions in a group G is a class of elements of order two, that is, transpositions, such that for all d, e ∈ D, the order of the product de is 1, 2, or 3. If in addition G is generated by D, then G is called a 3-transposition group The basic example of a 3-transposition group is the symmetric group. There, the class of transpositions is a class of 3- transpositions. Given a 3-transposition group, one can construct a point-line space whose points are the 3- transpositions and whose lines are those triples of 3-transpositions contained in a subgroup generated by two non-commuting 3-transpositions. This point-line space will then be a Fischer space. 14 Chapter 1. Preliminaries

Theorem 1.24 (Buekenhout 1974) Each connected Fischer space is isomorphic to a Fischer space coming from a 3-transposition group.

The finite 3-transposition groups containing no non-trivial normal solvable subgroups were classified by Fischer. This result was reproved by Cuypers and Hall after having removed the assumption of finiteness and having restricted his prohibition of solvable normal subgroups to those which are central. This induces a classification of the irreducible Fischer spaces. However, before giving this classification we first introduce the relevant Fischer spaces.

Example 1.25 If Ω is a set, then the partial linear space (P, L) with

P = {{i, j} | i, j ∈ Ω} and L = {{x, y, z} | x, y, z ∈ P ∧ |x ∪ y ∪ z| = 3} is denoted by T (Ω). We write Tn instead of T (Ω) if Ω = [n] for a certain n ∈ N.

Example 1.26 Suppose (V, f) is a symplectic space over the ﬁeld F2. Then the partial linear space (P, L) with P = V \{0} and L = {{x, y, x + y} | x, y ∈ P ∧ f(x, y) = 1} is denoted by HSp(V, f). 2n If V = F2 (n ∈ N), then we can take f as the symplectic form with

n X ((x1, . . . , x2n), (y1, . . . , y2n)) 7→ (x2i−1y2i + y2i−1x2i), i=1 and we write HSp2n(2) instead of HSp(V, f).

Example 1.27 Suppose (V,Q) is an orthogonal space over the ﬁeld F2. Moreover, let f be the symplectic form associated to Q. Then the partial linear space (P, L) with

P = {x | x ∈ V \ Rad(f) ∧ Q(x) = 1} and L = {{x, y, x + y} | x, y, x + y ∈ P} is denoted by NO(V,Q). 2n+1 If V = F2 (n ∈ N), then we write NO2n+1(2) instead of NO(V,Q) and we 1.3. Geometries 15 can assume Q is the quadratic form with

n X 2 (x1, . . . , x2n+1) 7→ x2i−1x2i + x2n+1. i=1

2n If V = F2 (n ∈ N), then there are two possibilities. Either we can take Q as the quadratic form with n X (x1, . . . , x2n) 7→ x2i−1x2i, i=1 + and we write NO2n(2) instead of NO(V,Q), or we can take Q as the quadratic form with n X 2 2 (x1, . . . , x2n) 7→ x2i−1x2i + x2n−1 + x2n, i=1 − and we write NO2n(2) instead of NO(V,Q).

Example 1.28 Suppose (V,Q) is an orthogonal space over the ﬁeld F3 and let ∈ {+, −}. Then the partial linear space (P, L) with

P = {Fx | x ∈ V ∧ Q(x) = 1} and L = {hX,Y i ∩ P | X,Y ∈ P ∧ |hX,Y i ∩ P| = 3} is denoted by N O(V,Q).

Example 1.29 Suppose (V, f) is a Hermitian space over the ﬁeld F4. Then the partial linear space (P, L) with

P = {X ∈ P(V ) | f(X,X) = 0} and

L = {hX,Y i ∩ P | X,Y ∈ P ∧ f(X,Y ) = F ∧ |hX,Y i ∩ P| = 3} is denoted by HU(V, f). n If V = F4 , then we write HU n(2) instead of HU(V, f).

Example 1.30 The Fischer spaces corresponding to the 3-transposition groups

Fi22, Fi23, Fi24, Ω(8, 2) : Sym3, Ω(8, 3) : Sym3 are called the sporadic Fischer spaces. 16 Chapter 1. Preliminaries

Theorem 1.31 (Fischer 1971, Cuypers and Hall 1995) If Π is a Fischer space such that each component of the co-collinearity graph generates an irreducible Fischer space, then Φ is isomophic to

•T (Ω) for a set Ω,

•HS p(V, f) for a symplectic space (V, f) over the ﬁeld F2,

•NO (V,Q) for an orthogonal space (V,Q) over the ﬁeld F2, ± •N O(V,Q) for an orthogonal space (V,Q) over the ﬁeld F3,

•HU (V, f) for a Hermitian space (V, f) over the ﬁeld F4, or

• a sporadic Fischer space corresponding to one of the sporadic groups Fi22, Fi23, Fi24, Ω(8, 2) : Sym3, or Ω(8, 3) : Sym3

1.3.4 Cotriangular spaces A cotriangular space is a partial linear space in which any line contains exactly three points and any point not on a line is connected to either no or all but one of the points of that line. A connected cotriangular space is called irreducible if no two non-collinear points have the same set of non-collinear points. It was proven by Shult (1974) and Hall (1989) that each irreducible cotriangular space is an example of a Fischer space containing no affine planes of order three. There- fore, for cotriangular spaces ∼ and ⊥ are defined in the same way as for Fischer spaces. In fact, the Fischer spaces from Examples 1.25–1.27 are all that is needed to give a complete classification of the irreducible cotriangular spaces. They are the subject of Chapter5. In that chapter also the cotriangular space as defined in Example 1.32 will be considered. This cotriangular space will turn out to be a convenient description of NO7(2).

Example 1.32 Deﬁne

P = {{0}} ∪ {{i, j} | i, j ∈ [8] ∧ i < j} ∪ {{0, i, j} | (i, j) ∈ [4] × [5, 8]} ∪ {{0, i, j, k, l} | (i, j, k, l) ∈ [4]2 × [5, 7]2 ∧ i < j ∧ k < l}.

Moreover, for x, y ∈ P deﬁne

x ÷ y = (x ∪ y) \ (x ∩ y), and x ÷c y = {0} ÷ ([0, 8] \ (x ÷ y)). 1.3. Geometries 17

This enables us to deﬁne [ L = {{x, y, x ÷ y}, {x, y, x ÷c y}} ∩ 2P . x,y∈P

A straightforward check shows that X7 := (P, L) is a cotriangular space generated by

B := {{0}} ∪ {{i, i + 1} | i ∈ [6]}.

Now, deﬁne

(x{0}, x{1,2}, x{2,3}, . . . , x{6,7})

:= (3 + 5 + 7, 2 + 5 + 7, 1 + 3 + 5 + 7, 4 + 5 + 7,

1 + 4 + 6 + 7, 5 + 7, 1 + 3 + 4 + 5 + 6 + 7).

Then the map sending each y ∈ B to xy induces the isomorphism ∼ X7 = NO7(2).

Theorem 1.33 (Shult 1974, Hall 1989) Each irreducible cotriangular space is isomophic to

•T (Ω) for a set Ω of size at least 5,

•HS p(V, f) for a symplectic space (V, f) of dimension at least 6 over the field F2, •NO (V,Q) for an orthogonal space (V,Q) of dimension at least 6 over the field F2. Moreover, each plane in an irreducible cotriangular space is isomorphic to a dual affine plane of order two.

Amongst the different cotriangular spaces occurring in this theorem we can prove the following isomorphisms.

Lemma 1.34 + ∼ NO6 (2) = T8, and

∀ : HSp (2) ∼ NO (2). n∈N 2n = 2n+1 Proof. The ﬁrst isomorphism is readily checked. The last isomorphism follows from the fact that modulo 2n+1 the point sets coincide. 18 Chapter 1. Preliminaries

Hence, for a ﬁnite irreducible cotriangular space (P, L) it makes sense to say that (P, L) is of ∼ • triangular type if there is an n ≥ 5 such that (P, L) = Tn, ∼ ∼ • symplectic type if there is an n ≥ 3 such that (P, L) = HSp2n(2) = NO2n+1(2), • orthogonal type if there is an n ≥ 3 and an ∈ {±} with (, n) 6= (+, 3) such ∼ that (P, L) = NO2n(2).

Thus, if we restrict ourselves to the ﬁnite case, as in Chapter5, then Theorem 1.33 translates to the following theorem.

Theorem 1.35 A ﬁnite irreducible cotriangular space is of triangular, symplectic or orthogonal type.

Now, the following proposition gives the generating rank of each ﬁnite irreducible cotriangular space.

Proposition 1.36 (Hall 1983) Let n ≥ 4 be an integer. Then Tn+1 has generating rank ± n, HSp2n−2(2) has generating rank 2n − 1, NO2n(2) has generating rank 2n, and − NO6 (2) has generating rank 6. Another way to obtain cotriangular spaces is starting from the simply laced root systems of types A and E. We refer to AppendixA for the relevant deﬁnitions regarding root systems.

Example 1.37 Let Xm be one of the root systems Em with m ∈ [6, 8] or Am with m ≥ 4 an integer. Moreover, let Φ be the root system of type Xn with simple system {ai | i ∈ [n]} and assume char(F) 6= 2 if X = E. Then the partial linear space (P, L) with P = {Fx | x ∈ Φ} and L = {{Fx, Fy, Fz} | x, y, z ∈ Φ ∧ z ∈ Fx + Fy} is denoted by R(Xm). Cotriangular spaces isomorphic to R(Xm) are said to be of type Xm.

The following lemma gives useful isomorphisms.

Lemma 1.38 Set

− + (M6, M7, M8) := (NO6 (2), NO7(2), NO8 (2)).

Then Tn+1 is of type An for all integers n ≥ 4 and Mn is of type En for all n ∈ [6, 8]. 1.3. Geometries 19

Proof. The map which sends F(i − j) (i, j ∈ [n + 1] with i < j) to {i, j} induces the isomorphism involving Tn+1. The other isomorphisms are readily checked. We end with giving some lemmas which will be of particular use in Chapter5.

Lemma 1.39 Let Π be a connected cotriangular space. Then the diameter of the collinearity graph of Π is at most two. Proof. Suppose (v, x, y, z) is a path of length three in the collinearity graph of Π. Then, by deﬁnition, both v and z are collinear to at least two of the three points on the line through x and y. Hence, there must exist at least one point w on the line through x and y which is collinear to both v and z Thus, the diameter of Π is at most two.

∼ Lemma 1.40 For all positive integers n there are subspaces M2n−1 = NO2n+1(2), ∼ ∓ ∼ ∼ ± M2n = NO2n(2), M2n+1 = NO2n+1(2), M2n+2 = NO2n+2(2) of NO2n+3(2) such that M2n−1 ⊆ M2n ⊆ M2n+1 ⊆ M2n+2 ⊆ N O2n+3(2).

Proof. For each point x in a cotriangular space of orthogonal type generated by m ∈ N points, ∆x is a cotriangular space of symplectic type generated by m − 1 points. There- ∼ ∓ ∼ fore, it is sufﬁcient to prove that there are subspaces M2n = NO2n(2) and M2n+1 = ∓ NO2n+1(2) of NO2n+2(2) with

± M2n ⊆ M2n+1 ⊆ N O2n+2(2).

+ Now, for NO2n+2(2) deﬁning

+ M2n+1 := h(x1, . . . , x2n+2) ∈ N O2n+2(2) \{(0,..., 0, 1, 1)} |

(x2n+1, x2n+2) ∈ {(0, 0), (1, 1)}i, + M2n := h(x1, . . . , x2n+2) ∈ N O2n+2(2) |

(x2n−1, . . . , x2n+2) ∈ {(0, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 1), (0, 1, 1, 1)}i,

− does the job. For N2n+2(2) deﬁning

− M2n+1 := h(x1, . . . , x2n+2) ∈ N O2n+2(2) \{(0,..., 0, 1, 0)} | x2n+2 = 0i, − M2n := h(x1, . . . , x2n+2) ∈ N O2n+2(2) | (x2n+1, x2n+2) = (0, 0)i. does the job.

± Lemma 1.41 Let n ≥ 3 and let M be a subspace of N2n(2) isomorphic to a dual afﬁne plane of order two. Then, hMi⊥⊥ = hMi. 20 Chapter 1. Preliminaries

Proof. Clearly, hMi⊥⊥ ⊇ hMi. So, it sufﬁces to prove hMi⊥⊥ ⊆ hMi. Let x, y, z be three pairwise collinear points generating M and let p, q ∈ z∼ ∩ x⊥ ∩ y⊥ with p 6= q. Then, it readily follows that ∼ ∼ − hM, pi = hM, qi = NO4 (2).

In other words, we can identify hM, pi and hM, qi with

hn−3 + n−1 + n, n−2 + n−1 + n, n−1, ni if δ = −, and with

hn−5 + n−1 + n, n−4 + n−1 + n, n−3 + n−2 + n−1, n−3 + n−2 + ni otherwise. Using this, it is easily checked that hM, pi⊥⊥ = hM, pi and hM, qi⊥⊥ = hM, qi. Consequently,

hMi⊥⊥ ⊆ hM, pi⊥⊥ ∩ hM, qi⊥⊥ = hM, pi ∩ hM, qi.

Hence, it is sufﬁcient to prove that hM, pi ∩ hM, qi = hMi. − In NO4 (2) the span of a subspace isomorphic to a dual afﬁne plane of order two − and a point outside this subspace is NO4 (2) itself. Moreover, p is the only point in hM, pi connected to z but not to x and y. In other words, q cannot be a point of hM, pi. Thus, indeed, ⊥⊥ M ⊆ hM, pi ∩ hM, qi = hMi.

1.3.5 Root ﬁltration spaces

Let (P, L) be a partial linear space equipped with a quintuple (Pi)i∈[−2,2] of disjoint symmetric relations with

P × P = P−2 ]P−1 ]P0 ]P1 ]P2.

Moreover, deﬁne

∀i∈[−2,2] : P≤i := ∪j∈[−2,i]Pj, and

∀i∈[−2,2]∀x∈P : Pi(x) := Pi ∩ {(x, y) | y ∈ P}. 1.3. Geometries 21

Then (P, L) is called a root ﬁltration space with ﬁltration (Pi)i∈[−2,2] if

•P −2 is equality on P,

•P −1 is collinearity of distinct points of P,

• there is a map P1 → P, denoted by (y, z) 7→ yz such that, if (y, z) ∈ P1 and x ∈ Pi(y) ∩ Pj(z), then yz ∈ P≤i+j(x),

•P ≤0(x) ∩ P≤−1(y) = ∅ for each (x, y) ∈ P2,

•P ≤−1(x) and P≤0(x) are subspaces of (P, L) for each x ∈ P, and

•P ≤1 is a hyperplane of (P, L) for each x ∈ P. A root ﬁltration space is called non-degenerate if in addition to the previous properties also

•P 2 6= ∅ for each x ∈ P, and

• the graph (P, P−1) is connected. For a thorough introduction into root ﬁltration spaces we refer to Cohen and Ivanyos (2006). Now, in the same way as for a polar space we deﬁne the rank as the largest non- negative integer n for which there exists a chain

X1 ⊆ ... ⊆ Xn of length n, where the Xi are singular subspaces. Again by singular we mean that all points in the subspace are collinear. Examples 1.42–1.47 give some examples of root ﬁltration spaces coming from Co- hen and Ivanyos (2006).

Example 1.42 Let (P, L) be a linear space and deﬁne P−1 as the set consisting of all pairs of distinct collinear points. Then (P, L) is a root ﬁltration space with Pi = ∅ for all i ∈ [0, 2].

Example 1.43 Let (P, L) be a partial linear space without lines and define P2 as the set consisting of all pairs of distinct points. Then (P, L) is a root filtration space with Pi = ∅ for all i ∈ [−1, 1]. Even, if we keep P±1 = ∅ and allow for P0 6= ∅, then (P, L) is a root filtration space.

Example 1.44 Let (P, L) be a polar space, define P2 as the set consisting of all pairs of non-collinear points, and define P0 as the complement in P × P of P−2 ]P2. Then (P, L) is a root filtration space with P±1 = ∅= 6 P0. 22 Chapter 1. Preliminaries

Example 1.45 Let (P, L) be a generalized hexagon, that is, a point-line space whose collinearity graph has diameter 6 and girth 12, define P−1 as the set consisting of all pairs of collinear points, and define Pi (i ∈ [1, 2]) as the set consisting of all pairs of points at mutual distance i + 1. Moreover, for each pair (x, y) ∈ P1 define xy as the unique point collinear with x and y. This results in a root filtration space with P0 = ∅.

Example 1.46 Let P be a projective space, let H be a collection of hyperplanes such that the intersection of all hyperplanes is empty, and take P as the set of all point- hyperplane pairs where the point is contained in the hyperplane. Now, for the line set L take those sets consisting of all (x, H) with H a fixed hyperplane and x running through the points of a line in H, and those sets consisting of all (x, H) with x a fixed point and H running through the hyperplanes in H containing a fixed co-dimension 2 subspace of P containing X. This makes (P, L) is a root filtration space with  P−2 ⇐⇒ x = y ∧ H = K,  P ⇐⇒ x = y ∨ H = K,  ≤−1 ∀(x,H),(y,K)∈P : ((x, H), (y, K)) ∈ P≤0 ⇐⇒ x ∈ K ∧ y ∈ H,  P≤1 ⇐⇒ x ∈ K ∨ y ∈ H,  P2 ⇐⇒ x∈ / K ∧ y∈ / H.

Example 1.47 Let (M, P) be a non-degenerate polar space and deﬁne L as the set of pencils of lines on a point which sits in a singular plane. Singular meaning that all points are collinear. This in contrast to non-singular which we use to denote the existence of non-collinear points. This makes (P, L) a root ﬁltration space with  P−2 ⇐⇒ l = m,  P ⇐⇒ hl, mi is a singular plane,  −1 ∀l,m∈P :(l, m) ∈ P0 ⇐⇒ hl, mi is a non-singular plane or the union l ∪ m,  P1 ⇐⇒ ∃!n∈P : hl, ni and hm, ni are singular planes,  P2 ⇐⇒ (l, m) ∈/ P≤1.

Theorem 1.48 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F containing no sandwich elements and generated by E. Moreover, define F as the set of projective lines all of whose points belong to E. Then (E, F) is a root filtration space with filtration (Ei)i∈[−2,2]. Furthermore, each connected component of (E, E2) is either a non-degenerate root filtration space or a root filtration space with an empty set of lines. The non-degenerate root filtration spaces have been classified by Cohen and Ivanyos (2007). 1.3. Geometries 23

Theorem 1.49 (Cohen and Ivanyos 2007) Each non-degenerate root filtration space with finite rank is the shadow space of a building. Here, a building is a combinatorial and geometrical structure introduced by Tits as a means to understand the structure of groups of Lie type. For the theory of buildings we refer to Tits (1974), Ronan (1989), and Cohen (1995). In Chapter3 we will consider root filtration spaces having an empty set of lines. 24 Chapter 1. Preliminaries Chapter 2

Lie subalgebras of Lie algebras without strongly commuting pairs

2.1 Introduction

We consider an arbitrary Lie algebra g over a field F generated by a set of extremal elements but not containing any strongly commuting pairs. Then, in addition, because of Lemma 1.18, there are no special pairs. In g we consider a Lie subalgebra L generated by a hyperbolic pair, a symplectic triple, or a unitary triple. This implies that L is generated by a hyperbolic path in (E, E2) of length at most two. We find the possible isomorphism types of L and in some interesting cases we find an explicit description of the extremal elements of g in L. The latter will be of use in Chapter3. For that reason the assumption that no strongly commuting pairs exist was made. Note that Cohen et al. (2001) gave a description of L assuming char(F) = 2 but without assuming the non-existence of strongly commuting pairs.

2.2 The multiplication table

If L is a Lie subalgebra of g over a ﬁeld F generated by no more than three extremal elements, then the extremal identities can be used to determine the multiplication table of L.

Proposition 2.1 If g is a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by three (possibly coinciding) extremal elements x, y, z then Table 2.1 determines the multiplication on L.

Note that the entries below the diagonal in Table 2.1 are simply the negatives of the corresponding entries above the diagonal. Therefore, the lower diagonal part of the table is left empty. 26 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs yz z y xy xz xz xz yz xy yz g g xyz g g xy xy xyz xyz xz xy xz xz g g xyz g g g y g xz yz g 2 2 2 2 g yz g xz − − yxz yxz yxz − − − xyz − 0 y y − xz xy − − g yz xz xy x x yxz g g 2 z g g g yz xz z x xz xz − − g yz yz g − − + + xy g g yz g xyz xz xyz xy g xyz g g xz g g g xyz g 2 xy 2 xyz g 2 g g − − +2 − − − +2 z yz xz g xy xyz xy xz xy xyz yz xyz g g yz yz g g 2 g g x yz yz yxz + xyz xyz + + − + − 0 xz xy x yz xyz xz xy x y z xyz g g y g g yz g g L yz 2 yz − − yz g − − g g xyz xyz +2 xyz xz g g g xy xz g 2 − − g g 2 2 2 − − − xz xy z yz yz y yz yz g g yz yz 0 g xz xy − + yz g 2 xyz g g 2 z y − + + xyz xyz g g The multiplication table of xy z xz x g xz xz xz 0 g xy + xz g 2 yxz g Table 2.1: 2 x − − xyz g y yxz x xy + xy 0 g xy g 2 2 − xyz − 0 z yz xz 0 y xy 0 x z y x yz xz xy xyz yxz 2.3. Lie subalgebras generated by hyperbolic pairs 27

2.3 Lie subalgebras generated by hyperbolic pairs

We determine both the isomorphism type and the extremal elements in a Lie subalgebra generated by a hyperbolic pair.

2.3.1 Isomorphism type Because of Lemma 1.18, the following result is obvious.

Proposition 2.2 If g is a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a hyperbolic pair (Fx, Fy), then ∼ • L = Fx + Fy + Fxy = sl2(F), and

xy if char( ) = 2, and • C(L) = F F {0} otherwise.

Note, in this proposition, xy is extremal relative to L whereas it is not extremal relative to g. Hence, if char(F) = 2, then L is a proper Lie subalgebra of g.

2.3.2 Extremal elements

In Example 1.11 we gave a description of the extremal elements in sl2(F). However, this description was dependent on the characteristic. The following proposition shows that this dependence can be eliminated if we are dealing with a Lie subalgebra isomorphic to sl2(F) inside another Lie algebra that does not contain any strongly commuting or special pairs.

Proposition 2.3 Let g be a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a hyperbolic pair (Fx, Fy). Moreover, suppose E±1 = ∅. Then

[ 2 2 (E ∩ L) ∪ {0} = F(λ x + gxyµ y + λµxy). λ,µ∈F The extremal elements described in this proposition can be identiﬁed with the traceless 2 × 2-matrices of rank 1.

Proof of Proposition 2.3. Because of Proposition 2.2, ∼ L = Fx + Fy + Fxy = sl2(F) and 2 2 2 −1 ∀ ∗ : λ x + xy + g µ y = λ exp(y, −λ µ)x ∈ E. λ,µ∈F xy 28 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

Hence, [ 2 2 F(λ x + gxyµ y + λµxy) ⊆ (E ∩ L) ∪ {0}. λ,µ∈F If char(F) 6= 2, then we are done. See also Example 1.11. Therefore, assume char(F) = 2, let α, β, γ ∈ F, and suppose z := αx + βy + γxy ∈ E. First, suppose γ = 0. Then α and β cannot both be zero. Moreover, if one of them is zero, then we are done. Therefore, we assume αβ 6= 0. Then

∀u∈E : αβ(xy)u = αβxyu + αβyxu = (αx + βy)(αx + βy)u = zzu = 0.

Hence,

(xy)(xy)u = 0, and ∀u,v∈E : (xy)u(xy)v + g(xy)uvxy + g(xy)u(xy)v + g(xy)v(xy)u = 0.

This implies (x, y) ∈ E1 = ∅. This is a contradiction. Thus, the sum of two non- commuting extremal elements cannot be extremal and we can assume γ = 1. In addition since xy∈ / E, we can assume α 6= 0 or β 6= 0. Suppose β = 0 and α 6= 0. Then z = αx + xy and gyz = αgxy 6= 0. As a consequence,

2 αz + gxyy = α(αx + xy) + gxyy = α x + gxyy + αxy ∈ E.

This is in contradiction with the fact that the sum of two non-commuting extremal elements is not extremal. In the same way we ﬁnd a contradiction if α = 0. Hence, we can assume αβ 6= 0. Now, z = αx + βy + xy and

2 αz + (gxy − αβ)y = α(αx + βy + xy) + (gxy − αβ)y = α x + gxyy + αxy ∈ E.

If gxy 6= αβ, then we obtain a contradiction with the fact that the sum of two non- 2 commuting extremal elements is not extremal. Hence, gxy = αβ and αz = α y + gxyy + αxy. We conclude

[ 2 2 F(λ x + gxyµ y + λµxy) ⊇ (E ∩ L) ∪ {0}. λ,µ∈F

2.4 Lie subalgebras generated by symplectic triples

First we determine the possible isomorphism types of Lie subalgebras generated by symplectic triples. Then we use this to give an explicit description of the extremal elements in these Lie subalgebras. This will be of use in Chapter3. 2.4. Lie subalgebras generated by symplectic triples 29

2.4.1 Isomorphism type We determine the isomorphism type of a Lie subalgebra generated by a symplectic triple assuming the non-existence of strongly commuting or special pairs.

Proposition 2.4 Let g be a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a symplectic triple (X,Y,Z). Moreover, assume E±1 = ∅. Then

∃(x,y,z)∈X×Y ×Z :(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

Moreover, if • (a, b, c) = (xy + yz, 2x − xyz, x + z − xyz),

• T = Fx + Fy + Fxy, and

• R = Fa + Fb + Fc, then

• L = R o T is 6-dimensional, ∼ • T = sl2(F), and

• C(L) = Fc. In particular, if    α β η −η   γ −α θ −θ  M =   α,β,γ,η,θ,ζ ∈ F θ −η ζ −ζ    θ −η ζ −ζ  is the Lie subalgebra of the symplectic Lie algebra over F deﬁned by the symplectic form  0 1 0 0 −1 0 0 0 f =   ,  0 0 0 1 0 0 −1 0 then the map induced by

0 1 0 0 0 0 0 0 0 1 −1 1 0 0 0 0 1 0 0 0 0 0 0 0 (x, y, z) 7→   ,   ,   0 0 0 0 0 0 0 0 0 1 −1 1 0 0 0 0 0 0 0 0 0 1 −1 1 induces an isomorphism between L and M. 30 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

In the remainder of this section we assume g is a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a symplectic triple (X,Y,Z). Moreover, we assume there are no strongly commuting or special pairs in E. Now, it is obvious that there are x, y, z ∈ E generating L with (x, y), (y, z) ∈ E2 and (y, z) ∈ E0. Scaling the extremal generators makes that we can assume

(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

In addition, we will assume

• (a, b, c) = (xy + yz, 2x − xyz, x + z − xyz),

• T = Fx + Fy + Fxy, and

• R = Fa + Fb + Fc, A ﬁrst step towars the proof of Proposition 2.4 is the following lemma.

Lemma 2.5 ∼ L = R + T and T = sl2(F). Proof. Clearly, R + T ⊆ L. Moreover, substituting xz = 0 in Table 2.1 gives the following multiplication table for R + T .

x y xy a b c x 0 xy −2x −b 0 0 y −xy 0 2y 0 −a 0 xy 2x −2y 0 −a b 0 a b 0 a 0 2c 0 b 0 a −b −2c 0 0 c 0 0 0 0 0 0

Table 2.2: The multiplication table of R + T

This table shows that R + T is closed under multiplication. Hence, R + T is a Lie subalgebra of L containing x, y and z = x − b + c. Since L is generated by x, y and z, we obtain L ⊆ R + T. ∼ Finally, T = sl2(F) follows from the fact that (x, y) ∈ E2. 2.4. Lie subalgebras generated by symplectic triples 31

Next, we want to prove the linear independence of the elements in

{x, y, z, a, b, c}.

However, ﬁrst we prove that a, b, and c are non-zero. Lemma 2.6 The elements a, b, and c are all non-zero. Proof. Since a = by and b = ax either a = b = 0 or a 6= 0 6= b. 1 Suppose a = b = 0 and suppose char(F) 6= 2. Then c = 2 ab = 0 and L is 3- dimensional. Consequently, z is an extremal element in Fx + Fy + Fxy commuting with x. However, the only extremal elements in there commuting with x are the nonzero multiples of x. Hence, x and z are linearly dependent. This is a contradiction. Therefore, suppose char(F) = 2. Then

2 −1 ∀ ∗ : ω x + z = exp(y, (ω + 1) )exp(x, ω)exp(y, 1)z ∈ E. ω∈F \{1}

Hence, (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, a 6= 0 6= b. Next, suppose |F| > 2 and c = 0. Then,

−1 ∀ ∗ : (1 + ω)x + ω(ω + 1)z = exp(y, (ω + 1) )exp(z, ω)exp(y, 1)x ∈ E. ω∈F \{1}

Hence, (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, c 6= 0 if |F| > 2. Finally, suppose |F| = 2, take an arbitrary extension F over characteristic two, and consider the Lie subalgebra L generated by x, y, and z over F and deﬁne a, b, and c as before. Now, again, since |F| > 2, we know c = 0 implies that (Fx, Fz) is strongly commuting. Hence, x + z ⊆ E(g ⊗ ). F F F F

In particular x + z is extremal in g. We conclude (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, c 6= 0 also if |F| = 2.

Lemma 2.7 The elements x, y, xy, a, b, and c are linearly independent.

Proof. Since (x, y) ∈ E2, the elements x, y and xy are linearly independent. char(F) 6= 2. Because of Lemma 2.6, a, b, c 6= 0. First, suppose a ∈ T and let α, β, γ be scalars such that a = αx + βy + γxy. This implies

0 = ya = y(αx + βy + γxy) = −αxy + 2γy.

Hence,

α = γ = 0, a = βy, and 0 = xx(a − βy) = −xb + 2βx = 2βx.

In particular, β = 0. This is in contradiction with a 6= 0. We conclude, a∈ / T . 32 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

Next, suppose b ∈ T + Fa and let α, β, γ, δ be scalars such that b = αx + βy + γxy + δa. This implies

0 = xb = x(αx + βy + γxy + δa) = βxy − 2γx − δb, 0 = x(βxy − 2γx − δb) = −2βx, 0 = yy(βxy)y = yy(2γx + δb)y = −4γy, 1 1 0 = − (−2βx)y + xx(4γy) = βxy − 2γx = −δb, and 2 4 0 = yyb = yy(αx + βy + γxy + δa) = −2αy.

This gives α, β, γ, δ = 0. This is in contradiction with b 6= 0. Consequently, b∈ / T +Fa. Finally, suppose c ∈ T + Fa + Fb, and let α, β, γ, δ, be scalars such that c = αx + βy + γxy + δa + b. Then

0 = xc = x(αx + βy + γxy + δa + b) = βxy − 2γx − δb, and 0 = yc = y(αx + βy + γxy + δa + b) = −αxy + 2γy − a.

Since x, y, xy, a and b are linearly independent, all scalars must be zero. This is in contradiction with c 6= 0. Consequently, c∈ / T + Fa + Fb. Thus, if char(F) 6= 2, then x, y, xy, a, b, and c are linearly independent. char(F) = 2. Because of Lemma 2.6, a, b, c 6= 0. First, suppose a ∈ T and let α, β, γ be scalars such that a = αx + βy + γxy. This implies

0 = ya = y(αx + βy + γxy) = αxy, and 0 = za = z(αx + βy + γxy) = αxz + γa. xy and a are nonzero. Hence, α = γ = 0. Consequently, a = βy and

0 = (βy + a)xz = βyxz + axz = βb + (ax)z = βb + bz = βb.

In other words, β = 0. This is in contradiction with a 6= 0. Consequently, a∈ / T . Next, suppose b ∈ T + Fa and let α, β, γ, δ be scalars such that b = αx + βy + γxy + δa. This implies

0 = yxb = yx(αx + βy + γxy + δa) = δa, 0 = xb = x(αx + βy + γxy + δa) = βxy + δb, 0 = yzb = y(αx + βy + γxy + δa) = αb + δa, and 0 = zb = z(αx + βy + γxy + δa) = αxz + γa + δb.

Consequently, α = β = γ = δ = 0. This is in contradiction with b 6= 0. Hence, b∈ / T + Fa. 2.4. Lie subalgebras generated by symplectic triples 33

Finally, suppose c ∈ T + Fa + Fb, and let α, β, γ, δ, be scalars such that c = αx + βy + γxy + δa + b. Then 0 = xc = x(αx + βy + γxy + δa + b) = βxy + δb, 0 = yc = y(αx + βy + γxy + δa + b) = αxy + a, and 0 = zc = z(αx + βy + γxy + δa + b) = βb + αxz + γb + δb. Since x, y, xy, a and b are linearly independent, all scalars must be zero. This is in contradiction with c 6= 0. Consequently, c∈ / T + Fa + Fb. Thus, if char(F) 6= 2, then x, y, xy, a, b, and c are linearly independent.

Now it remains to ﬁnd C(L) and to prove that L is indeed a semi-direct product.

Lemma 2.8 L = R o T and C(L) = Fc. Proof. It follows from Table 2.2 that R is an ideal of L and that T is a Lie subalgebra of L. Consequently, since the nonzero elements of R are linearly independent from the elements of T , we obtain that T ∩ R = {0}. In particular, L = R o T. Table 2.2 says Fc ⊆ C(L). Moreover, because of Lemma 2.6, a 6= 0 6= b. Therefore, let w = αx + βy + γxy + δa + b + ηc ∈ C(L) for certain α, β, γ, δ, , η ∈ F. Then 0 = xw = βxy − 2γx − δb, 0 = yw = −αxy + 2γy − a, and 0 = aw = γa + αb + 2c. Consequently, since x, y, xy, a and b are linearly independent, all scalars are zero and C(L) ⊆ Fc.

Proof of Proposition 2.4. If we identify (x, y, z) with

0 1 0 0 0 0 0 0 0 1 −1 1 0 0 0 0 1 0 0 0 0 0 0 0   ,   ,   , 0 0 0 0 0 0 0 0 0 1 −1 1 0 0 0 0 0 0 0 0 0 1 −1 1 then it is readily checked that L and M have the same multiplication table. Thus, the proposition follows from Lemmas 2.5–2.8.

2.4.2 Extremal elements Here, we will ﬁnd an explicit description of the extremal elements in the Lie subalgebra L that is described in Proposition 2.4. Note that through the isomorphism with the Lie algebra M as described in Proposition 2.4 they correspond to rank-1 matrices. 34 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

Proposition 2.9 Let g be a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a symplectic triple (X,Y,Z). Moreover, assume E±1 = ∅. Then

∃(x,y,z)∈X×Y ×Z :(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

Now, deﬁne (a, b, c) := (xy + xz, 2x − xyz, x + z − xyz), and  2 2 2  u(κ, λ, µ) := κ x − λ y + κλxy + λµa + κµb + µ c, ∀ : κ,λ,µ∈F  U(κ, λ, µ) := Fu(κ, λ, µ). Then [ [ (E ∩ L) ∪ {0} ⊆ U(0, 0, 1) ∪ U(0, 1, µ) ∪ U(1, λ, µ), µ∈F λ,µ∈F [ [ (E ∩ L) ∪ {0} ⊇ U(0, 1, µ) ∪ U(1, λ, µ), µ∈F λ,µ∈F and, provided char(F) 6= 2,

(E ∩ L) ∪ {0} ⊇ U(0, 0, 1).

Moreover,   u(κ, λ, µ) ∈ E \ U(0, 0, 1)  ∀ : κ,λ,µ∈F =⇒  S 0  C (u(κ, λ, µ)) = 0 U(κ, λ, µ ) ∩ E. E∩L µ ∈F Note that this proposition gives a complete description of the extremal elements if the characteristic is not two. In characteristic two the question remains whether the central elements in the Lie subalgebra are extremal. Now, because of Proposition 2.4 we know that there are x, y, and z with

(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

Therefore, the ﬁrst step in proving Proposition 2.9 consists of showing that the non- central candidate extremal elements are indeed extremal.

Lemma 2.10 [ U(κ, λ, µ) \ U(0, 0, 1) ⊆ E ∩ L. κ,λ,µ∈F 2.4. Lie subalgebras generated by symplectic triples 35

Proof. The lemma follows from

x, y, z ∈ E, 2 u(κ, 0, 0) = κ x for κ ∈ F, 2 u(0, λ, 0) = −λ y for λ ∈ F, 2 −1 ∗ u(κ, λ, 0) = κ exp(y, −κ λ)x for κ ∈ F and λ ∈ F, 2 −1 ∗ u(κ, λ, −κ) = κ exp(y, −κ λ)z for κ ∈ F and λ ∈ F, −1 ∗ u(κ, λ, µ) = exp(x, −λ (µ + κ))u(−µ, λ, µ) for κ ∈ F and λ, µ ∈ F , and −1 ∗ u(κ, 0, µ) = exp(y, −κ )u(κ, 1, µ) for κ ∈ F and µ ∈ F.

Next, we prove that the elements in the center of L are also extremal if char(F) 6= 2.

Lemma 2.11 Suppose char(F) 6= 2. Then

∅= 6 U(0, 0, 1) \{0} ⊆ E ∩ L.

Proof. Since the center of L is 1-dimensional,

∗ U(0, 0, 1) \{0} = F c 6= ∅.

Therefore, it is sufﬁcient to prove

∀v∈g : ccv = 2gvcc.

We prove this using the extremality of x, z, w := x + z + xyz = −u(2, 0, −1), and the fact that xw = zw = 0. Let v ∈ g. Then, using the extremal identities, we ﬁnd

−xvxyz = −gx(vyz)x + gxvxyz + gxyzxv = gv(xyz)x + gvxxyz, and

−zvzyx = gv(zyx)z + gvzzyx = gv(xyz)z + gvzxyz.

Adding both equations gives

−(xvxyz + zvzyx) = gv(xyz)x + gvxxyz + gv(xyz)z + gvzxyz

= gv(xyz)(x + z) + gv(x+z)xyz.

Consequently,

(x + z)wv = (x + z)(x + z + xyz)v

= 2gvxx + 2gvzz + 2xzv − (xvxyz + zvzyx)

= 2gvxx + 2gvzz + 2xzv + gv(xyz)(x + z) + gv(x+z)xyz. 36 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

Hence,

gv(xyz)(x + z) + gv(x+z)xyz = −2gvxx − 2gvzz − 2xzv + (x + z)wv.

Moreover,

gvcc = gv(x+z−xyz)(x + z − xyz)

= gv(x+z+xyz)(x + z + xyz) − 2(gv(x+z)xyz + gv(xyz)(x + z))

= gvww − 2(gv(x+z)xyz + gv(xyz)(x + z)).

Combining all this we indeed obtain ccv = 4xxv + 4xzv − 2xwv + 4zxv + 4zzv − 2zwv − 2wxv − 2wzv + wwv

= 2gvww + 8gvxx + 8xzv − 4xwv + 8gvzz − 4zwv

= 2gvww − 4(gv(xyz)(x + z) + gv(x+z)xyz)

= 2gvcc.

So, indeed ccv = 2gvcc for all v ∈ g.

Lemmas 2.12–2.14 prove the extremality of the remaining non-central elements.

Lemma 2.12 Suppose char(F) 6= 2. Then [ E ∩ L ⊆ U(κ, λ, µ). κ,λ,µ∈F Proof. Let v ∈ E ∩ L. Because of Proposition 2.4, we know a 6= 0 6= b. Moreover, 2 there are β, γ, δ, , η, κ ∈ F such that v = κ x + βy + γxy + δa + b + ηc. This implies 1 1 −4(κ2γx + βγy − κ2βxy + (γδ − 3β)a + (γ + 3κ2δ)b + δc) = vvxy ∈ v. 4 4 F Suppose γ = 0, then 3 3 κ2βxy − βa + κ2δb + δc ∈ v. 4 4 F In other words, ∈ / {κ, β, δ} = {0}, δ∈ / {κ, β, } = {0}, {κ, β, δ, } = {0}, β∈ / {κ, } = {0}, or κ∈ / {β, δ} = {0}. Hence,

2 v ∈ (a + Fc) ∪ (b + Fc) ∪ Fc ∪ F(βy + δa + ηc) + F(κ x + b + ηc).

If v ∈ (a + Fc) ∪ (b + Fc), then vx 6= 0 or vy 6= 0 which is in contradiction with gvx = gvy = 0 and the fact that E1 = ∅. If v ∈ F(βy + δa + ηc), then the relation 2 vvx ∈ Fv gives v ∈ U(0, β, βδ). If v ∈ F(κ x + b + ηc), then the relation vvy ∈ Fv 2.4. Lie subalgebras generated by symplectic triples 37 gives v ∈ U(κ, 0, ). Consequently, we can assume γ 6= 0. As a consequence, 1 1 κ2x + βy − βγ−1κ2xy + (δ − 3βγ−1)a + ( + 3γ−1δκ2)b + γ−1δc ∈ v. 4 4 F If κ = β = 0, then also δ = = 0 and v ∈ U(0, 0, 1). Therefore, we assume κ 6= 0 or 2 2 2 β 6= 0. But then, γ = −κ β. This proves the existence of a λ ∈ F such that β = −λ and γ = κλ. Substitution of these identities gives 1 1 κ2x − λ2y + κλxy + (δ − 3κ−1λ)a + ( + 3δκλ−1)b + δκ−1λ−1c ∈ v. 4 4 F In particular,

4δ = δ − 3κ−1λ ∧ 4 = + 3δκλ−1 ∧ η = δκ−1λ−1.

2 −1 Consequently, (δ, , η) = (λµ, −κµ, −µ ) with µ = −κ . Thus, v ∈ U(κ, λ, µ).

Lemma 2.13 Suppose char(F) = 2. Then [ E ∩ L ⊆ F(u(κ, λ, µ) + νc). κ,λ,µ,ν∈F Proof. Let v ∈ E ∩ L. Because of Proposition 2.4 we know a, b, and c are all non-zero. Moreover, we know there are β, γ, δ, , η, κ ∈ F such that

v = κ2x + βy + γxy + δa + b + ηc.

This implies (γδ + β)a + (γ + κ2δ)b = vvxy = 0. Suppose γ 6= 0 = δ. Then = 0 and

(βκ2 + γ2)a = vva = 0.

2 Hence, there is a λ ∈ F such that β = λ and γ = κλ. This implies that both κ and λ are non-zero. Moreover, v ∈ F(u(κ, λ, 0) + ηc). Suppose γ 6= 0 6= δ. Then γ = βδ−1 and

0 = δ0 = δ(γ + κ2δ) = δ(βδ−12 + κ2δ) = β2 + κ2δ2.

2 −1 This implies there is a λ ∈ F such that β = λ , κδ = λ, and γ = βδ = κλ. In particular, both κ and λ are non-zero. Therefore, we can deﬁne µ := κ−1. Then −1 −1 2 δ = κ λ = λµ and = κ κ = κµ. This implies v ∈ F(u(κ, λ, µ) + (ν + µ )c). 38 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

Thus, we can assume γ = 0 and β = κ2δ = 0. Consequently,

γ = 0 and δ = = 0 ∨ β = κ = 0 ∨ δ = κ = 0 ∨ β = = 0.

Suppose δ = = 0. This implies βκ2a = vva = 0. Hence, β = 0 or κ2 = 0. Consequently, v ∈ F(u(κ, 0, 0) + ηc) ∪ F(u(0, β, 0) + βηc).

Next, suppose β = κ = 0. This implies vx 6= 0 or vy 6= 0 whereas gv(x) = gv(y) = 0. This is in contradiction with the fact that E1 = ∅. Then, suppose δ = κ = 0. Both δ = = 0 and κ = β = 0 we already treated. Therefore, we can assume and β are non-zero. Since a 6= 0, this is in contradiction with βa = vvxy = 0. Finally, suppose β = = 0. Both β = κ = 0 and = δ = 0 we already treated. Therefore, we can assume δ and κ are non-zero. Since b 6= 0, this is in contradiction 2 with δκ b = vvxy = 0.

Lemma 2.14 Suppose char(F) = 2. Then,

∀ ∀ ∗ : u(κ, λ, µ) + νc ∈ (L \ E) ∪ U(0, 0, 0, 1). κ,λ,µ∈F ν∈F Proof. First, observe that a, b, and c are non-zero because of Proposition 2.4. Next, let κ, λ, µ, ν ∈ F with ν non-zero, deﬁne v := u(κ, λ, µ) + νc, and suppose v ∈ E \ U(0, 0, 1). Now, it is sufﬁcient to derive a contradiction. If κ = λ = 0, then v ∈ U(0, 0, 1). This is in contradiction with v ∈ E \ U(0, 0, 1). Therefore, because of symmetry, we can assume κ = 1. Moreover, since exp(y, λ)u(1, λ, µ) = u(1, 0, µ), we can assume λ = 0. Now,

x + νc = exp(u(0, 1, µ))exp(y)v ∈ E and y + νc = exp(x)exp(y)(x + νc) ∈ E.

As a consequence,

( 2 v1(ρ) := x + ρ y + ρxy + νc = exp(y, ρ)(x + νc) ∈ E, and ∀ ∗ : ρ∈F 2 2 2 −1 v2(ρ) := x + ρ y + ρxy + ρ νc = ρ exp(x, ρ )(y + νc) ∈ E.

Hence, since E−1 = ∅,

−2 2 ∀ ∗ : u(1, ρ, 0, 0) = (1 + ρ) (ρ v (ρ) + v (ρ)) ∈/ E. ρ∈F \{1} 1 2 For F a ﬁeld with |F| > 2 this gives the required contradiction. Therefore, we assume |F| = 2. Now, to obtain a contradiction we have to extend our ﬁeld. Therefore, let F be non-trivial extension of F containing a non-zero element ρ 6= 1. Then

L := L ⊗ = x + y + z + a + b + c F F F F F F F F 2.5. Lie subalgebras generated by unitary triples 39 is the Lie subalgebra of g := g ⊗ generated by x, y, and z. Moreover, since the F F set E of extremal elements in g is a subset of the set E of extremal elements in g, we know v1(ρ), v2(ρ) ∈ E ⊆ E. However, since E−1 = ∅, the sum v1 + v2 is not in E. Hence, also v1 + v2 ∈/ E. In particular, (v1, v2) ∈ E0 and u(1, ρ, 0, 0) = −2 2 (1 + ρ) (ρ v1(ρ) + v2(ρ)) ∈/ E. This is the required contradiction if F = F2.

Now, it remains to determine the centralizer CE∩L(w) of a non-central w ∈ E ∩ L.

Lemma 2.15 [ ∀ : u(κ, λ, µ) ∈ E \ U(0, 0, 1) ⇒ C (u(κ, λ, µ)) = U(κ, λ, µ0) ∩ E. κ,λ,µ∈F E∩L 0 µ ∈F Proof. Let κ, λ, µ ∈ F with u(κ, λ, µ) ∈ E \ U(0, 0, 1). Then κ and λ cannot both be zero. Hence, because of symmetry we can assume κ = 1. Moreover, since exp(y, λ)u(1, λ, µ) = u(1, 0, µ), we can assume λ = 0. Now, it is readily checked that

[ 0 CE∩L(u(κ, λ, µ)) \ U(0, 0, 1) ⊇ U(κ, λ, µ ) ∩ E. 0 µ ∈F

So, let v ∈ CE∩L(u(κ, λ, µ)) \ U(0, 0, 1) and let α, β, γ ∈ F with v = u(α, β, γ). Then

0 = −u(1, 0, µ)v = 2αβx + β2xy + β2µa + (βγ + αβµ)b + βγµc.

In other words, β = 0 and, since v∈ / U(0, 0, 1), α 6= 0. Now deﬁne µ0 := α−1γ. Then v ∈ U(1, 0, µ0) = U(κ, λ, µ0). Thus,

[ 0 CE∩L(u(κ, λ, µ)) \ U(0, 0, 1) ⊆ U(κ, λ, µ ) ∩ E. 0 µ ∈F

Proof of Proposition 2.9. First, Lemmas 2.10 and 2.11 prove that the extremal elements which we expect to be extremal are indeed extremal. Next, Lemmas 2.12–2.14 show that the extremal elements which we expect not to be extremal are indeed not extremal. Finally, Lemma 2.15 gives a description of the extremal elements in the centralizer CE∩L(w) of a non-central extremal element w ∈ E ∩ L.

2.5 Lie subalgebras generated by unitary triples

First we determine the possible isomorphism types of Lie subalgebras generated by unitary triples over an arbitrary ﬁeld. Then we use this to give an explicit description of the extremal elements in these Lie subalgebras provided that the ﬁeld in question contains exactly two elements. This will be of use in Chapter3. 40 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

2.5.1 Isomorphism type Here, we determine the isomorphism type of a Lie subalgebra generated by a unitary triple assuming the non-existence of strongly commuting or special pairs.

Proposition 2.16 Let g be a Lie algebra over a ﬁeld F containing a Lie subalgebra L generated by a unitary triple (X,Y,Z) but not by a symplectic triple. Moreover, assume E±1 = ∅. 2 Then, there is an irreducible polynomial t − δt − ∈ F[t] with δ ∈ {0, 1} and there are linearly independent extremal elements x, y, z generating L such that • δ = 0 if char(F) 6= 2,

• (gxy, gyz, gxz, gxyz) = −(1, 1, , δ), {8} if char( ) 6= 3, • dim(L) ∈ F {7, 8} otherwise,

{{0}} if char( ) 6= 3, • C(L) ∈ F {{0}, F(x + y + z − xyz − yxz)} otherwise. Moreover, if char(F) = 3, then dim(L) = 7 if and only if C(L) = {0}. 2 Furthermore, if δ 6= 0 or char(F) 6= 2, then F := F[t]/(t − δt − ) is a quadratic 3 3 extension of F and there is a hermitian form f : F × F → F such that L is isomorphic to su3(F, f) or su3(F, f)/C(su3(F, f)). In the remainder of this section we assume g is a Lie algebra over a field F containing a Lie subalgebra L generated by unitary triple (X,Y,Z) but not by a symplectic triple. Moreover, we assume the non-existence of strongly commuting or special pairs in E. The first step towards proving Proposition 2.16 is finding the irreducible polynomial and the extremal elements x, y, z satisfying the right parameters.

2 Lemma 2.17 There is an irreducible polynomial t −δt− ∈ F[t] and there are extremal elements x, y and z generating L such that

(gxy, gyz, gxz, gxyz) = −(1, 1, , δ) with δ = 0 if char(F) 6= 2 and δ ∈ {0, 1} if char(F) = 2. Proof. Let x, y and z be extremal elements generating L. Then,

∀α∈ : g(exp(x,α)y)xz = g 2 = gxyz − 2αgxzgyz. F (y+αxy+gxyα x)xz

1 −1 −1 Hence, if char(F) 6= 2 we can assume gxyz = 0 by taking α = 2 gxyzgxz gyz . Further scaling makes that we can assume

(gxy, gyz, gxz, gxyz) = −(1, 1, , δ). 2.5. Lie subalgebras generated by unitary triples 41

Here, ∈ F, δ = 0 if char(F) 6= 2, and δ ∈ {0, 1} if char(F) = 2. 2 Therefore, suppose α ∈ F such that α − δα − = 0. Then L is generated by the symplectic triple (Fx, Fy, Fexp(y, α)z). This is in contradiction with our assumptions. 2 Consequently, t − δt − ∈ F[t] is irreducible.

In the remainder of this section we assume δ, , x, y, and z are as described in Lemma 2.17. If char(F) = 2 and δ = 0, then there is nothing much we can say. However, we do have the following lemma.

Lemma 2.18 Suppose char(F) = 2 and δ = 0. Then dim(L) = 8 and C(L) = {0}.

Proof. First we prove that L is 8-dimensional. Therefore, deﬁne

(a1, . . . , a8) := (x, y, z, xy, xz, yz, xyz, yxz). P Then, L = i∈[8] Fai. Thus, it is sufﬁcient to prove X ∀j∈[8] : aj ∈/ ai. i∈[j−1] This is trivial for j < 4. The other cases we check one by one. j = 4. Suppose xy = αx + βy + γz for certain α, β, γ ∈ F. Then

0 = gxxy = β + γ, 0 = gyxy = α + γ, and 0 = gzxy = α + β.

Consequently, xy = γ(x + y + z). Additionally, since xy 6= 0, also γ 6= 0. As a consequence, 0 = xxy = γ(xy + xz) and 0 = yxy = γ(xy + yz).

In other words, xz = xy = yz and xyz = xxy = 0. Deﬁne ! γ + 1 1 + γ2 (κ, λ, µ) := , 1 + , . γ(1 + ) γ2(1 + )

Since is not a square we obtain that λµ 6= 0 and

λy + µz = exp(κz)exp(x)y ∈ E.

This is in contradiction with Proposition 2.3. Thus, a1, . . . , a4 are linearly independent. In the same way we can prove, for j ∈ [5, 6], that aj is linearly independent from a1, a2, and a3. j = 5. Suppose xz = αx + βy + γz + κxy for certain α, β, γ, κ ∈ F. Then κ 6= 0 42 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs and in the same way as before we can prove (α, β, γ) = γ(1, , 1). Hence,

xz = γ(x + y + z) + κxy.

Suppose γ 6= 0. Then,

0 = γ−1xxz = xy + xz = γ(ηx + y + z) + (κ + )xy.

This is in contradiction with the linear independence of a1, . . . , a4. Consequently, γ = 0 and since xz 6= 0 also κ 6= 0. This implies

xz = κxy, yxz = κyxy = 0, zxy = κ−1zxz = 0, and xyz = yxz + zxy = 0.

Moreover, 0 = κ2xyxy = κxyxz = κxy + κxz = xz + κxz. In other words, κ = , xz = xy, and yz = xy + yzyx = xy. Now, deﬁne (v, w) := (exp(y)z, 2exp(x)z). Then it is readily checked that

gvw = 1 and v + w = exp((1 + )z)exp(x)y ∈ E.

This is in contradiction with Proposition 2.3. Thus, a1, . . . , a5 are linearly independent. In the same way we can prove that a6 is linearly independent with a1, . . . , a4 and also with a1, a2, a3, a5. j = 6. Suppose yz = αx + βy + γz + κxy + λxz for certain α, β, γ, κ, λ ∈ F. Then, in the same way as before, we can prove (α, β, γ) = γ(1, , 1). Hence,

yz = γ(x + y + z) + κxy + λxz.

If γ = 0, then xyz = 0 and xy + yz = yxyz = 0. The latter is in contradiction with the linear independence of a4 and a6. Consequently, γ 6= 0 and 0 = xyz = γ(xy + xz). This is in contradiction with the linear independence of a4 and a5. Thus, a1, . . . , a6 are linearly independent. j = 7. Suppose xyz = αx+βy +γz +κxy +λxz +µyz for certain α, β, γ, κ, λ, µ ∈ F. Then, in the same way as before, we can prove (α, β, γ) = γ(1, , 1). Hence,

xyz = γ(x + y + z) + κxy + λxz + µyz.

Suppose γ = 0. Then

0 = yxxyz = µyxyz = µxy + µyz, and 0 = x(xy + yz + yxyz) = xyz + λxyxz = (κ + λ)xy + µyz. 2.5. Lie subalgebras generated by unitary triples 43

Since a4, a5, a6 are linearly independent, we obtain κ = λ, µ = 0 and xyz = λ(xy + xz). In particular, κ 6= 0. Otherwise, we ﬁnd a linear dependence between a4 and a6. Hence,

yxz = λ−1λy(xy + xz) = λ−1yxyz = λ−1(xy + yz), zxy = λλ−1z(xy + yz) = λzyxz = λ(xz + yz), and zxy = −1λ−1λz(xy + xz) = −1λ−1zxyz = −1λ−1(xz + yz).

We conclude λ−1−1 = λ and = λ−2. This is in contradiction with the fact that is not a square. Thus γ 6= 0 and

0 = xxyz = γ(xy + xz) + µxyz = γ(xy + xz) + µγ(x + y + z) + κµxy + λµxz + µ2yz = µγ(x + y + z) + (κµ + γ)xy + (λµ + γ)xz + µ2yz.

Hence, the linear independence of a1, . . . , a6 implies µ = γ = 0. This is in contradiction with γ 6= 0. Thus, a1, . . . , a7 are linearly independent. j = 8. Suppose there are α, β, γ, κ, λ, µ, ν ∈ F such that yxz = αx + βy + γz + κxy + λxz + µyz + νxyz. Then

0 = gxw = β + γ,

0 = gyw = α + γ, and

0 = gzw = α + β.

Hence, (α, β, γ) = γ(1, , 1, 1). Moreover,

0 = xy + xz + xyxz = µxyz + (γ + 1)(xy + xz), 0 = xyyxz = (γ + ν)xyz + λ(xy + xz), and 0 = xzyxz = (γ + ν)xyz + κ(xy + xz).

Using the linear independence of xy, xz and xyz we obtain γ = ν = 1 and κ = λ = µ = 0. In particular,

0 = z(x + y + z + xyz + yxz) = xz + yz + xz + yz + xz + yz = xz + yz.

This is in contradiction with the linear independence of xz and yz. Thus, a1, . . . , a8 are linearly independent and L is 8-dimensional.

C(L) = {0}. Let w := αx + βy + γz + κxy + λxz + µyz + νxyz + ξyxz ∈ C(L) for 44 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs certain α, β, γ, κ, λ, µ, ν, ξ ∈ F. Then

0 = gxw = β + γ,

0 = gyw = α + γ, and

0 = gzw = α + β.

Hence, (α, β, γ) = γ(1, , 1). Moreover,

0 = xw = γxy + γxz + µxyz + ξxyxz = (γ + ξ)(xy + xz) + µxyz, 0 = yw = γxy + γyz + λyxz + νyxyz = (γ + ν)(xy + yz) + λyxz, and 0 = zw = γxz + γyz + κzxy + νzxzy + ξzyzx = (γ + ν + ξ)(xz + yz) + κzxy.

Now, the linear independence of xy, xz, yz, xyz and yxz proves that all scalars are zero. In other words, w = 0 and C(L) = {0}.

This proves Proposition 2.16 assuming that char(F) = 2 and δ = 0. Therefore, assume char(F) 6= 2 or δ = 1 in the remainder of this section. This implies that we can deﬁne

2 F := F[t]/(t − δt − ).

Because of Lemma 2.17 this is a quadratic extension of F. Now, let ζ1 and ζ2 be the two 2 (distinct) zeroes of t − δt − in F, that is, ζ1ζ2 = − and ζ1 + ζ2 = −δ, and deﬁne

• σ : F → F as the map interchanging ζ1 and ζ2, and ﬁxing F,

• M := diag(−1, 1, −1), and

3 3 3 3 T σ • f : F × F → F as the map sending (u, v) ∈ F × F to u Mv .

It is readily checked that f is a Hermitian form relative to the involution σ. First we analyze the generators and the ideals of the 8-dimensional special unitary Lie algebra su3(F, f). Then we show that L is isomorphic to su3(F, f)/C(su3(F, f)) or su3(F, f).

Lemma 2.19 su3(F, f) is generated by three extremal elements a, b, and c with

(gab, gbc, gac, gabc) = −(1, 1, , δ).

T T T Proof. If char(F) = 2, then deﬁne (a, b, c) := (a1 a2, b1 b2, c1 c2) with

−1 a1 = ζ ( ζ1 ζ2 0 ), b1 = ( 0 ζ1 ζ2 ), c1 = ( ζ1 0 ζ2 ), −1 a2 = ζ ( ζ2 ζ1 0 ), b2 = ( 0 ζ2 ζ1 ), c2 = ( ζ2 0 ζ1 ). 2.5. Lie subalgebras generated by unitary triples 45

T T T Otherwise, deﬁne (a, b, c) := (a1 a2, b1 b2, c1 c2) with

a1 = ( 1 1 0 ), b1 = ( 0 1 1 ), c1 = ( 1 −1 0 ),

a2 = ( 1 −1 0 ), b2 = ( 0 −1 1 ), c2 = ( 1 1 0 ).

This makes a, b and c extremal (Postma 2007, Section 2.6.1). Now, consider the Lie algebra M generated by a, b and c over F. Since a, b, c ∈ su ( , f), also M ⊆ su ( , f). We prove that dim(M ⊗ ) = 8. This implies 3 F 3 F F F dim(M) = 8 and M = su3(F, f). Deﬁne I = diag(1, 1, 1) and consider the group G generated by

{I + αa | α ∈ F} ∪ {I + αb | α ∈ F} ∪ {I + αc | α ∈ F}.

By deﬁnition, M ⊗ F is the Lie algebra corresponding to G. F 3 Next, deﬁne V := F . Then

V = Fa1 + Fb1 + Fc1 = Fa2 + Fb2 + Fc2.

Moreover, it is readily checked that G acts transitively on V . Therefore, G is SL(V ) or Sp(V ). See McLaughlin (1967, Lemma 3) or Cameron and Hall (1991, Theorem 1). However, Sp(V ) does not act transitively on V because V is 3-dimensional. In other words, M ⊗ = sl ( ) and, indeed, F F 3 F dim(M ⊗ ) = 8. F F

Hence, su3(F, f) is generated by a, b, and c. Finally, if char(F) = 2, we have to replace b by the extremal element exp(a, )b to enforce (gab, gbc, gac, gabc) = −(1, 1, , δ).

Lemma 2.20 If char(F) 6= 3, then ∼ • L = su3(F, f) and C(L) = {0}. Otherwise,

• C(L) = F(x + y + z − xyz − yxz), and ∼ • L = su3(F, f) and C(L) is 1-dimensional, or ∼ • L = su3(F, f)/C(su3(F, f)) and C(L) is 0-dimensional. Proof. Let a, b and c be as deﬁned in the proof of Lemma 2.19. Then the map from su3(F, f) to L induced by sending a to x, b to y and c to z is a Lie algebra homomor- 46 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs phism. This follows from applying the extremal identities and

(gxy, gxz, gyz, gxyz) = (gab, gac, gbc, gabc).

Consequently, L is isomorphic to a quotient of su3(F, f). Suppose L is not isomorphic to su3(F, f). Then su3(F, f) contains a proper ideal i such that ∼ L = su3(F, f)/i.

Obviously, i ⊗ is a proper ideal in sl ( ). However, sl ( ) is simple if char( ) n. F F 3 F n F F - Moreover, if char(F) | n, then each proper ideal of sln(F) is 1-dimensional. Hence, cha(F) = 3 and dim(i) = dim(i ⊗ ) = 1. F F Now, because of the Lie algebra homomorphism, it is sufﬁcient to prove that

C(su3(F, f)) = F(a + b + c − abc − bac).

So, consider C(su3(F, f)). Using the extremal identities, it is readily checked that

F(a + b + c − abc − bac) ⊆ C(su3(F, f)).

Therefore, let

d := αa + βb + γc + κab + λac + µbc + νabc + ξbac ∈ C(su3(F, f)) for certain α, β, γ, κ, λ, µ, ν, ξ ∈ F. Then

0 = ad = βab + γac − 2κa − 2λa + µabc + ξab + ξac = −2(κ + λ)a + (β + ξ)ab + (γ + ξ)ac + µabc, and 0 = bd = −αab + γbc + 2κb + λbac − 2µb − νab + νbc = 2(κ − µ)b − (α + ν)ab + (γ + ν)bc + λbac.

Consequently, using the linear independence of the basis elements we obtain

(α, β, γ, κ, λ, µ, ν, ξ) = α(1, , 1, 0, 0, 0, −1, −1).

Thus, indeed, C(su3(F, f)) = F(a + b + c − abc − bac).

Proof of Proposition 2.4. Take δ, , x, y, and z as described in Lemma 2.17. If char(F) = 2 and δ = 0, then the proof follows from 2.18. Otherwise, the proof follows from Lemma 2.20. 2.5. Lie subalgebras generated by unitary triples 47

2.5.2 Extremal elements Here, we will ﬁnd an explicit description of the extremal elements in the Lie subalgebra that is described in Proposition 2.16 assuming F = F2. This will be of use in Chapter3. Note that the extremal elements we ﬁnd correspond to either the rank 1 matrices inside su3(F, f) for a Hermitian form f or the rank 1 matrices in the Lie subalgebra M of the symplectic Lie algebra as described in Proposition 2.9.

Proposition 2.21 Let g be a Lie algebra over F2 containing a Lie subalgebra L generated by a unitary triple (X,Y,Z) = (F2x, F2y, F2z), assume the non-existence of strongly commuting or special pairs in E, deﬁne w := x + y + z + xy + xz + yz, and deﬁne

X := {x, y, z, x + y + xy, x + z + xz, y + z + yz}, and Y := {w + xyz, w + yxz, w + zxy}.

Then,

gxyz = 0 ⇒ X ⊆ E ∩ L ⊆ X ∪ {w + xy + yz + yxz},

gxyz = 1 ⇒ E ∩ L = X ∪ Y.

Proof. If gxyz = 0, then applying Proposition 2.9 with Z replaced by F(y + z + yz) gives X ⊆ E ∩ L ⊆ X ∪ {w + xy + yz + yxz}.

Therefore, we assume gxyz = 1. Then, X ∪ Y ⊆ E ∩ L follows from

x + y + xy = exp(x)y, x + z + xz = exp(x)z, y + z + yz = exp(y)z, and

w + zxy = exp(z)(x + y + xy), w + yxz = exp(y)(x + z + xz), w + xyz = exp(x)(y + z + yz).

Moreover, using Proposition 2.16 we obtain that x, y, xy, xz, yz, xyz and yxz are linearly independent. It is readily checked that wwxyz 6= 0 for each linear combination w outside X ∪ Y . Thus, indeed, X ∪ Y = E ∩ L. 48 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs Chapter 3

Constructing geometries from extremal elements

3.1 Introduction

In this chapter we consider an arbitrary Lie algebra bg over a field F generated by a set of extremal elements and we find an ideal i of bg such that the extremal point set E of g := bg/i is the point set of a geometry which relates g to a so-called building: a combinatorial and geometrical structure introduced by Tits as a means to understand the structure of groups of Lie type. For the theory of buildings we refer to Tits (1974). For i an arbitrary ideal containing the ideal generated by the sandwich elements a natural way to construct a geometry from g is described by Cohen and Ivanyos (2006). There, the line set consists of those projective lines all of whose points are extremal and each E2-connected subspace of the resulting geometry is either a non-degenerate root filtration space or a root filtration space with no lines. In the latter case, E−1 = ∅. The non-degenerate root filtration spaces have been classified by Cohen and Ivanyos (2007). They proved that a non-degenerate root filtration space is a so-called shadow space of a building. This relates our Lie algebra g to a building provided that E−1 6= ∅. Therefore, we assume in the remainder of this chapter that E−1 = ∅. Because of Lemma 1.18 this implies E1 = ∅. Hence, we can assume there are no strongly commuting or special pairs in E. Because of Lemma 1.19 we know that no non-extremal element becomes extremal after restricting to a component of (E, E2). Consequently, since the extremal elements outside a component of (E, E2) linearly span an ideal of g, it is no restriction to assume that (E, E2) is connected. We will do so throughout this chapter. Note that this also implies the non-existence of sandwich elements provided that the component in question contains at least two elements. Now, the root filtration space corresponding to g has an empty set of (projective) lines and there are two natural ways to equip E with a non-empty set of lines: one can 50 Chapter 3. Constructing geometries from extremal elements

take as line set the set H consisting of all hyperbolic lines or, assuming E0 6= ∅, one can take the set F consisting of all isotropic lines, that is, lines of the form {X,Y }⊥⊥ with (X,Y ) ∈ E0. Here, ⊥= E≤0.

Example 3.1 Let bg = g be a simple Lie algebra over F = F2 generated by x, y and z with (x, y), (x, z) ∈ E2 and (y, z) ∈ E0. Then, because of Proposition 2.9,

E = {x, y, z, x + y + xy, x + z + xz, x + y + z + xy + xz + yxz}.

Moreover, since F contains only one nonzero element we can identify E with E. Hence,

H ={{x, y, x + y + xy}, {x + y + xy, z, x + y + z + xy + xz + yxz}, {x, z, x + z + xz}, {x + z + xz, y, x + y + z + xy + xz + yxz}}.

In a picture (E, H) looks as follows:

It is the dual affine plane of order two which is, by definition, an example of a Fischer space. Now, we are ready to state the main results of this chapter. They depend on the field in question.

Theorem 3.2 Let bg be a Lie algebra over a ﬁeld F 6= F2 generated by its set Eb of extremal elements and deﬁne  P (x + z) if char( ) = 2,  F F i := (x,z)∈Eb: gx= gz  {0} otherwise.

Then i is an ideal and the quotient Lie algebra g := bg/i is generated by extremal elements. Moreover, if we assume connectedness of (E, E2), the non-existence of strongly commuting or special pairs, and the existence of a polar pair, then (E, F) is a non- degenerate polar space if F is the set of all isotropic lines in E.

Theorem 3.3 Let g be a Lie algebra over the ﬁeld F = F2 generated by extremal elements. If we assume connectedness of (E, E2) and the non-existence of strongly commuting pairs or special pairs, then (E, H) is a connected Fischer space if H is the set of all hyperbolic lines in E. 3.2. From Lie algebra to polar space 51

Note that the non-degenerate polar spaces of rank at least three have been classiﬁed by Tits (1974) and the connected Fischer spaces have been classiﬁed by Buekenhout (1974). See also Sections 1.3.2 and 1.3.3. Finally, we also show how to go back in case of a connected Fischer space. As- suming the characteristic is two, we construct a (possibly trivial) Lie algebra over F generated by extremal elements corresponding to the points of the Fischer space. In this Lie algebra the extremal generators commute if and only if the corresponding points are non-collinear in the Fischer space.

3.2 From Lie algebra to polar space

In this section we prove Theorem 3.2. Therefore, let bg be a Lie algebra over a field F 6= F2 generated by extremal elements. Moreover, define g := bg/i with i as defined in Theorem 3.2. The first thing we need to prove is that g is a Lie algebra. This follows from the following lemma.

Lemma 3.4 The linear subspace i is an ideal of bg. Proof. If char(F) 6= 2, then obviously i = {0} is an ideal. Therefore, suppose char(F) = 2 and take x, y, z ∈ E with x + z ∈ i. Then, scaling y makes that we can assume gxy = gyz = 1. In particular, with exp : g → Aut(g) the exponential map as deﬁned in Section 1.2,

y(x + z) = (exp(y)x + x + y) + (exp(y)z + y + z) = (x + z) + (exp(y)x + exp(y)z).

Since exp(y) is an automorphism sending perpendicular points to perpendicular points, we obtain (exp(y)x, exp(y)z) ∈ E0. Moreover, if w ∈ E, then, as exp(y) is an involution,

gw(exp(y)x) = g(w, exp(y)x) = g(exp(y)w, exp(y)exp(y)x) = g(exp(y)w, x) = g(exp(y)w, z) = g(exp(y)exp(y)w, exp(y)z) = g(w, exp(y)z)

= gw(exp(y)z).

Hence, y(x + z) ∈ i. Thus, i is indeed an ideal. Consequently, g is a Lie algebra over F. This proves the ﬁrst part of Theorem 3.2. For the second part we need to assume the connectedness of (E, E2), the non-existence of strongly commuting pairs or special pairs, and the existence of a polar pair. In particular, E±1 = ∅= 6 E0. 52 Chapter 3. Constructing geometries from extremal elements

We will make use of a theorem by Cuypers.

Theorem 3.5 (Cuypers 2006) Let (E, H) be a point-line space, denote the union of equality and non-collinearity in (E, H) by ⊥, write ∼ instead of 6⊥, and deﬁne F as the set of isotropic lines, that is, the set of lines of the form {X,Z}⊥⊥ with X ⊥ Z and X 6= Z. Then (E, F) is a non-degenerate polar space provided (E, H) satisﬁes the following properties.

1. (E, H) is a connected but non-linear partial linear space. 2. Each line contains at least four points. 3. Each three distinct points X,Y,Z with X ∼ Y ∼ Z ⊥ X generate a subspace isomorphic to a dual affine plane. 4. If a point is not collinear with two points of a transversal coclique, then it is not collinear with any point of that coclique. 5. If X and Z are points satisfying X⊥ ⊆ Z⊥, then X⊥ = Z⊥. 6. If X and Z are points satisfying X⊥ = Z⊥, then X = Z. 7. (E, ⊥) is connected. Recall that a “transversal coclique”, as mentioned in property 4, is the set of points inside a dual affine plane incident with a line removed from the corresponding projective plane. Now, let (E, H) and ⊥ as defined in Theorem 3.2 and write ∼ instead of 6⊥. Then proving this theorem is equivalent to proving that (E, H) satisfies the seven properties as formulated in Theorem 3.5. We check these properties one by one.

Property 1 Lemma 3.6 (E, H) is a connected non-linear partial linear space.

Proof. Since (E, E2) is connected, also (E, H) is connected. Moreover, since a hyperbolic line is uniquely determined by any two of its points, (E, H) is a partial linear space. However, E0 6= ∅. As a consequence, (E, H) is non-linear.

Property 2 Lemma 3.7 Each line of (E, H) contains at least four points. Proof. For each hyperbolic line H there are extremal elements x, y such that

2 2 2 H = {F(λ x + µ y + λµxy) | λ, µ ∈ F} = {F(x + µ y + µxy) | µ ∈ F} ∪ {Fy}.

The cardinality of this set is |F| + 1 ≥ 4. 3.2. From Lie algebra to polar space 53

Property 3 Lemma 3.8 Each three distinct points X,Y,Z of (E, H) with X ∼ Y ∼ Z ⊥ X generate a subspace isomorphic to a dual afﬁne plane. Proof. Let X,Y,Z be three points of (E, H) with X ∼ Y ∼ Z ⊥ X and denote the point-line space generated by these three points by (P, L). Then, X,Y,Z are extremal points of g with (X,Y ), (X,Z) ∈ E2 and (Y,Z) ∈ E0. In particular, (X,Y,Z) is a symplectic triple and there are x, y, z ∈ E with

(x, y, z) ∈ X × Y × Z and (gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

Now, deﬁne (a, b, c) := (xy + xz, 2x − xyz, x + z − xyz), and

u(κ, λ, µ) := κ2x − λ2y + κλxy + λµa + κµb + µ2c, and ∀κ,λ,µ∈ : F U(κ, λ, µ) := Fu(κ, λ, µ). Then it follows from Proposition 2.9 that

P = {U(0, 1, µ) | µ ∈ F} ∪ {U(1, λ, µ) | λ, µ ∈ F}, L = {{U(0, 1, µ)} ∪ {U(1, λ, κ + λµ) | λ ∈ F} | κ, µ ∈ F}, and M = {{U(0, 1, µ) | µ ∈ F}} ∪ {{U(1, λ, µ) | µ ∈ F} | λ ∈ F}, where M is used to denote the set of maximal cocliques in (P, L). Next, deﬁne P0 := P ∪ {C}, with C the center of the Lie algebra generated by {x, y, z}, and deﬁne

M0 := {M ∪ {C} | M ∈ M}.

Removing C and all lines through C from (P0, L ∪ M0) gives (P, L). Consequently, it is sufﬁcient to prove that (P0, L ∪ M0) is isomorphic to a projective plane. This induces the following proof obligations with respect to (P0, L ∪ M0). 1. Any line is uniquely determined by any two of its points. 2. Given any two distinct points, there is at least one line containing both of them. 3. Given any two distinct lines, there is at least one intersection point. 4. There are four points such that no line contains more than two of them.

These are readily checked. 54 Chapter 3. Constructing geometries from extremal elements

Property 4 Lemma 3.9 If a point of (E, H) is not collinear with two points of a transversal coclique, then it is not collinear with any point of that coclique.

Proof. Let V = Fv be a point not collinear with two points X and Z of a transversal coclique T . Since E1 = ∅, it is sufﬁcient to prove

∀w∈W ∈T : gvw = 0.

Note that there is a point Y such that (X,Y,Z) is a symplectic triple in (E, H). Conse- quently, there are x, y, z ∈ E with

(x, y, z) ∈ X × Y × Z and (gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0).

Following the proof of Lemma 3.8, we obtain

T ⊆ {{U(0, 1, µ, ν) | µ, ν ∈ F}} ⊆ P(Fx + Fz + Fxyz).

Since, V ⊥ X,Z, we obtain vx = vz = 0 and gvx = gvz = 0. Consequently, it is sufﬁcient to prove gv(xyz) = 0. Indeed,

gv(xyz) = g(vx)(yz) = g0(xz) = 0.

Property 5 Lemma 3.10 If X and Z are points of (E, H) with X⊥ ⊆ Z⊥, then X⊥ = Z⊥.

⊥ ⊥ Proof. Let X = Fx and Z = Fz be two extremal points in E with X ⊆ Z . Then ⊥ ⊥ ⊥ X ∈ X ⊆ Z . In other words, X ⊥ Z. Moreover, since E= 6 Z (otherwise (E, E2) would be disconnected), there is an extremal point Y = Fy on a hyperbolic line with Z. If X ⊥ Y , then also Y ⊥ Z. So, Y is also on a hyperbolic line with X. After applying a suitable scaling we can assume

(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0). Next, deﬁne

h := exp(y, −1)exp(z, 1)exp(x, −1)exp(y, 1) ∈ Aut(g).

Then h2 = 1 and h(X) = h(Fx) = Fh(x) = Fz = Z. As a consequence

⊥ ⊥ ⊥ ⊥ ⊥ ⊥ Z = h(X) = h(X ) ⊆ h(Z ) = h(Z) = X . 3.2. From Lie algebra to polar space 55

Property 6 (characteristic not two)

Lemma 3.11 Suppose char(F) 6= 2. Then X = Z for all points X,Z of (E, H) with X⊥ = Z⊥. Proof. Let X,Z be points in (E, H) with X⊥ = Z⊥. If X = Z, then we are done. Therefore, we assume X 6= Z and we consider a point Y ∈ E collinear with X. This ⊥ ⊥ point exists because of the connectedness of (E, E2). Moreover, since X = Z , also Y ∼ Z. In other words, (X,Y,Z) is a symplectic triple and, because of Proposition 2.9, there exists a triple (x, y, z) ∈ X × Y × Z with

(gxy, gyz, gxz, gxyz) = (−1, −1, 0, 0) such that the center C = Fc = F(x + z − xyz) ⊥ of the Lie algebra generated by X, Y , and Z is a point in {X,Y,Z} . However, (E, E2) is connected. Therefore, there must be an extremal point W = Fw∈ / {X,Y,Z} collinear to C. If W is not collinear to X, then it is also not collinear with Z. But then, in the same way as in the proof of Lemma 3.9, we can prove that W is not collinear with C. Hence, W is collinear to both X and Z. This proves that (X, W, C) is a symplectic triple. Moreover, after a suitable scaling, we can assume gwx = gwc = −1. This makes

h := exp(w, −1)exp(c, 1)exp(x, −1)exp(w, 1) an involution on g interchanging x and c. In particular,

h(Y ) ∼ h(X) = C and h(Y ) ⊥ h(C) = X.

Consequently, also h(Y ) ⊥ Z. However, we already saw in the proof of Lemma 3.9 that this implies h(Y ) ⊥ C. This is in contradiction with h(Y ) ∼ C. Thus, X = Z.

Property 6 (characteristic two)

Lemma 3.12 Suppose char(F) = 2. Then X = Z for all points X,Z of (E, H) with X⊥ = Z⊥. Inspired by Cuypers (2006) we introduce some more notation. First, deﬁne an equivalence relation ≈ on E such that

X ≈ Y ⇔ X⊥ = Y ⊥ and deﬁne

I := ≈-equivalence classes. 56 Chapter 3. Constructing geometries from extremal elements

Moreover, for all (X,Y ) ∈ E2 deﬁne

IY := unique I ∈ I such that Y ∈ I and

IX,Y := {IZ | Z a point on the hyperbolic line on X and Y }. Now, proving Lemma 3.12 is equivalent to proving that |I| = 1 for all I ∈ I. Since the proof is a bit more involved than the proof of the other properties, we split it up in several lemmas. Moreover, we assume char(F) = 2. Furthermore, we note that two points X,Z with X⊥ = Z⊥ are by deﬁnition in the same connected component of (E, ⊥). Hence, in addition we can assume that (E, H) is connected.

Lemma 3.13 I is hexp(y) | y ∈ Ei-invariant.

Proof. E0 and E2 are hexp(y) | y ∈ Ei-invariant. Hence, the same holds for I.

Lemma 3.14 Suppose (X,Y ) ∈ E2. Then IX,Y is hexp(x) | x ∈ Xi-invariant. h Proof. Let I ∈ IX,Y and let h ∈ hexp(x) | x ∈ Xi. Then there are Z and Z on the h hyperbolic line on X and Y with I = IZ and h(Z) = Z . Consequently,

h I = I h ∈ I . Z X,Y

Lemma 3.15 Let (X,Y ) ∈ E2, let Z ∈ IX , and let (P, L) be the subspace of (E, H) generated by X, Y , and Z. Then

∀V,W ∈P : V ⊥ W ⇒ IV = IW .

Proof. If X = Z, then the lemma trivially follows. Therefore, assume X 6= Z. This implies, (X,Y,Z) is a symplectic triple and (P, L) is isomorphic to a dual afﬁne plane. Next, let (V,W ) ∈ E≤0 be a pair of points in (P, L), let M be the maximal coclique of (P, L) containing V and W , and let L be the Lie subalgebra of g generated by X, Y , and Z. Since hexp(u) | u ∈ E ∩Li acts transitively on the maximal cocliques of (P, L), we can assume that X,Z ∈ M. Consequently, because of symmetry and because of Lemma 3.10 it is sufﬁcient to prove that X⊥ ⊆ W ⊥. Suppose

⊥ ⊥ (W, X, Y, Z) = (Fw, Fx, Fy, Fz) and U = Fu ∈ X = Z .

Then, combining Proposition 2.9 with W, X, Z ∈ M gives

w ∈ X + Z + Fxyz and guw ∈ Fgux + Fguz + Fgu(xyz) = Fg(ux)(yz) = {0}.

⊥ ⊥ ⊥ Thus, U ∈ W and X ⊆ W . 3.2. From Lie algebra to polar space 57

Lemma 3.16 Let (X,Y ) ∈ E and let Z ∈ I . Then I is hexp(z) | z ∈ ∪ Zi- 2 X X,Y Z∈IX invariant.

Proof. Let Z ∈ IX . It is sufﬁcient to prove that IX,Y is hexp(z) | z ∈ Zi-invariant. If X = Z, then the lemma follows from Lemma 3.14. Therefore, assume X 6= Z. This implies that (P, L) is isomorphic to a dual afﬁne plane of order two. Let I ∈ IX,Y and let h ∈ hexp(z) | z ∈ Zi. Then there is a point W in the h hyperbolic line on X and Y such that I = IW and there is a point W in the hyperbolic line on W and Z such that h(W ) = W h. Moreover, there is exactly one point V in the hyperbolic line on X and Y which is perpendicular to W h. Consequently, because of Lemma 3.15,

h(I) = h(I ) = I h = I ∈ I . W W V X,Y

Lemma 3.17 Suppose (X,Y ) ∈ E2, let Z ∈ IX , and let (P, L) be the subspace of (E, H) generated by X, Y , and Z. Then

∃(x,z)∈X×Z ∀W ∈P : exp(x)IW = exp(z)IW .

Proof. If X = Z, then the lemma trivially follows. Therefore, suppose X 6= Z. In particular, (X,Y,Z) is a symplectic triple and (P, L) is isomorphic to a dual afﬁne plane of order two. Now, clearly,

∃(x,y,z)∈X×Y ×Z :(gxy, gyz, gxz) = (1, 1, 0).

Therefore, let W = Fw ∈ P. Then

w ∈ Fx + Fy + Fz + Fxy + Fxz + Fxyz.

In particular, gxw = gwz and

g = g = (g + g )2 = 0. (exp(x)w)(exp(z)w) (w+gwxx+xw)(w+gwzz+zw) xw wz Consequently, exp(x)W ⊥ exp(z)W and

exp(x)IW = Iexp(x)W = Iexp(z)W = exp(z)IW .

Lemma 3.18 Suppose (X,Y,Z) is a symplectic triple with Z ∈ IX .Then

∼ ⊥ ∼ ∃(x,z)∈X×Z ∀W ∈E : exp(x)IW 6= exp(z)IW ⇒ X ∩ Y ⊆ W . 58 Chapter 3. Constructing geometries from extremal elements

Proof. Let W = Fw ∈ E and let (x, y, z) ∈ X × Y × Z such that gxy = gxz. First we derive some of the properties which have to be satisﬁed to ensure exp(x)IW 6= exp(z)IW . If X ⊥ W , then also W ⊥ Z and gxw = gwz = 0. This would imply

exp(x)IW = Iexp(x)W = IW = Iexp(z)W = exp(z)IW .

Therefore, we can assume (X, W, Z) is a symplectic triple. In addition, after applying a suitable scaling, we can assume that that gxw = 1. However, if gwz = gxw = 1, then

g = g = (g + g )2 = 0. (exp(x)w)(exp(z)w) (w+gwxx+xw)(w+gwzz+zw) xw wz

−1 and exp(x)IW = exp(z)IW . Therefore, we assume ω := gwz 6= 1. Next, we want to describe the center of the Lie subalgebra L generated by x, w, and z. Therefore, deﬁne ( v(λ, µ) := λ2x + µ2ωz − λµωxwz, and ∀λ,µ∈ : F V (λ, µ) := Fv(λ, µ).

Then, because of Proposition 2.9, ⊥ ∼ ⊥ {V (0, 1)} ∪ {V (1, µ) | µ ∈ F \{1}} ⊆ X ∩ W ∩ Z Moreover, V (1, 1) ⊆ C(L). −1 −1 −1 Now, deﬁne V := Fv with v := (ω + 1) v(1, ω ). Then, V ∈ IX = IZ . Hence, Y ∼ V and gvy 6= 0. Moreover, x + z + v = x + z + (ω−1 + 1)−1(x + ω−1z + xwz) = x + z + (ω−1 + 1)−1ω−1(ωx + z + ωxwz) = x + z + (1 + ω)−1(ωx + z + ωxwz) = (1 + ω)−1((1 + ω + ω)x + (1 + ω + 1)z + ωxwz) = (1 + ω)−1(x + ωz + ωxwz) ∈ V (1, 1) ⊆ C(L).

Using this we can prove that W ∼ Y . 0 00 0 00 0 00 Deﬁne (w , w ) = exp(x)exp(w)(z, v) and (W ,W ) = (Fw , Fw ). Then,

w + w0 + w00 = exp(x)exp(w)(x + z + v) = x + z + v, and (W ⊥, (W 0)⊥, (W 00)⊥) = exp(x)exp(w)(X⊥,Z⊥,V ⊥).

In particular, W ⊥ = (W 0)⊥ = (W 00)⊥ 3.2. From Lie algebra to polar space 59

Suppose W ⊥ Y . Then, Y ⊥ W 0,W 00 and

0 = gwy = g(x+z+v+w0+w00)y = gvy 6= 0. This is a contradiction. Consequently, indeed, W ∼ Y . It remains to show that W ∼ Y 0 for all other Y 0 ∈ X∼ ∩ Y ⊥. Therefore, let Y 0 ∈ X∼ ∩ Y ⊥. Then (X,Y 0,Z) is a symplectic triple and, since Y ⊥ Y 0, we have 0 exp(x)IY 0 = exp(z)IY 0 . But this can only be the case if gxy0 = gy0z for all y ∈ Y . Now 0 0 repeating the proof with Y replaced by Y proves that indeed W ∼ Y .

Lemma 3.19 Suppose (X,Z) ∈ E2 with Z ∈ IX . Then

∃(x,z)∈X×Z ∀W ∈E : exp(x)IW = exp(z)IW .

Proof. Since (E, H) is connected, there is an Y making (X,Y,Z) a symplectic triple. Consequently, because of Lemma 3.18 there are (x, z) ∈ X × Z such that

∼ ⊥ ∀W ∈E : exp(x)IW 6= exp(z)IW ⇒ W ∼ X ∩ Y .

∼ ⊥ Let W ∈ E. If W ⊥ Y , then W 6∼ X ∩ Y . Hence, if W ⊥ Y , then exp(x)IW = exp(z)IW . Moreover, if W ⊥ X, then also W ⊥ Z and exp(x)IW = IW = exp(z)IW . Therefore, we can assume X,Y,Z ∈ W ∼. If W is a point of the hyperbolic line on X and Y , then set X0 := Z. Otherwise, X0 := X. This ensures that W is not a point of the hyperbolic line on X0 and Y . Next, we turn to another point-line space having I as point set and

⊥⊥ FI := {{IU ,IV } | (U, V ) ∈ E0} as line set. This is well-deﬁned since

∀U,V ∈ E : U ⊥ V ⇔ IU ⊥ IV .

By definition, (I, FI ) satisfies all the properties of Theorem 3.5. Hence, it is a non- degenerate polar space. If we remove a hyperplane from a non-degenerate polar space whose lines contain at least three points, then what we get is called an affine polar space. ⊥ See for example Cohen and Shult (1990). In particular, (I\ IX0 , ⊥) is the polar graph of an affine polar space. ⊥ The results by Cohen and Shult (1990) prove that two points in (I\ IX , ⊥) have distance at most three. Moreover, their distance is three if and only if IX0 is on the hyperbolic line through the two points. Consequently, we can assume that there is a point

V ∈ E such that IY ⊥ IV ⊥ IW and IV ∼ IX0 . Now, suppose exp(x)IW 6= exp(z)IW . Then W ∼ V because V ∈ (X0)∼ ∩ Y ⊥ = X∼ ∩ Y ⊥. This is in contradiction with IV ⊥ IW . Thus, exp(x)IW = exp(z)IW . 60 Chapter 3. Constructing geometries from extremal elements

Lemma 3.20 Suppose there exists X 6= Z with IX = IZ . Then there is a pair (x, z) ∈ X × Z with gxy = gyz for all y ∈ E. Proof. Because of Lemma 3.19 there is a pair (x, z) ∈ X × Z such that

∀W ∈E : exp(x)IW = exp(z)IW .

Therefore, let w ∈ E and deﬁne W := Fw. Then,

exp(x)IW = exp(z)IW .

Consequently, exp(x)W ⊥ exp(z)W. Hence,

0 = g = g = (g + g )2. (exp(x)w)(exp(z)w) (w+gwxx+wx)(w+gwzz+wz) wx wz

Thus, gwx = gwz.

Now we are ready to prove the sixth property in case the ﬁeld characteristic is two.

Proof of Lemma 3.12. To obtain g we divided out an ideal which makes sure that there are no two distinct commuting extremal elements x and z such that gxy = gyz for all extremal elements y. Consequently, because of Lemma 3.20 there cannot exist ⊥ ⊥ distinct extremal points X and Z with X = Z .

Property 7 (characteristic not two)

Lemma 3.21 Suppose char(F) 6= 2. Then (E, ⊥) is connected.

Proof. Since E0 6= ∅, there are two non-collinear points X and Z in E. Necessarily, they are in the same component of (E, ⊥). Suppose Y is an extremal point in another component. Then (X,Y,Z) is a symplectic triple. Because of Proposition 2.9 we know that the center C of the Lie algebra generated by X, Y , and Z is extremal but not collinear with X, Y , and Z. This can only be the case if X, Y , and Z are in the same component as C. This is a contradiction with the fact that X and Y are in different components. Consequently, (E, ⊥) has only one component. Thus, (E, ⊥) is indeed connected.

Propery 7 (characteristic 2)

Lemma 3.22 Suppose char(F) = 2. Then (E, ⊥) is connected. Proof. We prove this lemma using so-called transvection (sub)groups (Cuypers 2006). 3.3. From Lie algebra to Fischer space 61

A conjugacy class P of abelian subgroups of a group G is called a class of F- transvection (sub)groups if G = hPi and if for all A, B ∈ P we have AB = BA or A and B are full unipotent subgroups of the group hA, Bi which is isomorphic to (P )SL2(F). For P a class of F-transvections in a group G the point-line space (P, L) with L = {{C ∈ P | C ⊆ hA, Bi} | A, B ∈ P ∧ AB 6= BA} is called the geometry of F-transvection groups in G. We deﬁne G := hPi with P := {[x] | x ∈ E}. Here, for each x ∈ E the point [x] denotes the 1-parameter subgroup {exp(x, α) | α ∈ F}. It is readily checked that P is indeed a class of F-transvection (sub)groups in G. Now, in the same way as for (E, H) deﬁne ⊥ in the geometry (P, L) of F-transvection groups in G as the union of equality and non-collinearity. Then

∀x,y∈E : Fx ⊥ Fy ⇔ [x] ⊥ [y].

Therefore, suppose (E, ⊥) is not connected. Then the same holds for (P, ⊥) and there exist extremal elements x and y such that [x]⊥ = [y]⊥ (Cuypers 2006, Lemma 2.9). As ⊥ ⊥ a consequence (Fx) = (Fy) . This contradicts the sixth property. Thus, (E, ⊥) is connected.

3.3 From Lie algebra to Fischer space

The results from the previous section are not applicable if the ﬁeld has exactly two elements. However, in this situation we can prove that the hyperbolic lines give rise to a connected Fischer space.

Theorem 3.23 Let g be a Lie algebra over the field F = F2 generated by extremal elements. Moreover, assume the non-existence of strongly commuting pairs or special pairs. Then (E, H) is a Fischer space if H is the set of all hyperbolic lines in E. If in addition (E, E2) is connected, then also (E, H) is connected. Proof. Take H as the set of all hyperbolic lines in E and consider an arbitrary plane in (E, H). By definition, the points in this plane correspond to the extremal elements in the Lie subalgebra L of g generated by three extremal elements x, y, and z with (x, y), (x, z) ∈ E2. Using Propositions 2.9 and 2.21 an explicit description of the extremal elements of g in L is readily obtained. Next, a straightforward check shows that our plane is isomorphic to either the dual affine plane of order two or the affine plane of order three. In other words, (E, H) is a Fischer space. Moreover, obviously, (E, H) is connected if (E, E2) is connected. 62 Chapter 3. Constructing geometries from extremal elements

3.4 From Fischer space to Lie algebra

Let (P, L) be a connected Fischer space and deﬁne ⊥ and ∼ such that ( x ∼ y if x and y are distinct collinear points, and ∀x,y∈P : x ⊥ y otherwise.

Next, identify (P, L) with the connected Fischer space coming from a 3-transposition group G as described in Section 1.3.3 and use · for the action of the group by conjugation. In particular, the group action of a point x ∈ P on a point y ∈ P is denoted by x · y and ( z if there is a line {x, y, z} ∈ L, ∀x,y∈P : x · y = y otherwise.

Now, deﬁne gP as the formal vector space over F2 linearly spanned by the points of our Fischer space. We can turn this vector space into an algebra over F2 by deﬁning the multiplication as the bilinear map determined by sending a pair (x, y) of basis elements to x + y + x · y = x + y + z if there is a line {x, y, z} ∈ L, xy : = 0 otherwise.

Note, gP is not necessarily a Lie algebra. Therefore, we will ﬁnd a set A of subsets of P, which we call a vanishing set, and divide out the ideal iP,A generated by the vector space X X A := F2A with ∀A∈A : A = x. A∈A x∈A

This makes gP,A := gP /iP,A a quotient algebra of gP for each vanishing set A. It turns out that choosing the right vanishing set results in a Lie algebra.

Example 3.24 Take (P, L) the afﬁne plane of order three:

Then gP,{P} is an 8-dimensional Lie algebra and gP,L = {0}. 3.4. From Fischer space to Lie algebra 63

Given a vanishing set A the ideal iP,A might be much bigger than the vector space A. Therefore, we introduce the notion of an admissible vanishing set: a vanishing set A with ∀x∈P ∀A∈A : x.A ∈ A.

Given an admissible vanishing set A we will prove that iP,A equals A or gP . To describe the vanishing sets giving rise to Lie algebras we introduce the notion of an afﬁne vanishing set: a vanishing set A with

∀afﬁne plane A in (P,L) : A ∈ A.

Given a vanishing set A we will prove that gP,A is a Lie algebra if and only if A is an affine vanishing set. The vanishing sets as described in Example 3.24 are both admissible and affine. Also, we will prove for each affine vanishing set A that the nonzero images of the points of (P, L) in gP,A are extremal. This will prove the following theorem.

Theorem 3.25 Let (P, L) be a connected Fischer space and let A be a vanishing set. Then ( A, or A admissible ⇒ iP,A = gP . Moreover, gP,A is a Lie algebra ⇐⇒ A is afﬁne.

Furthermore, if A is afﬁne and gP,A is non-trivial, then the images of the points of (P, L) in gP,A are extremal and gP,A is a Lie algebra generated by these images.

Note that the construction described in this section gives Lie algebras over the field F2. Of course, tensoring with the appropriate field will give us Lie algebras over arbitrary fields of characteristic two. At the end of this chapter we address the question of which Lie algebras can be obtained via the construction described here. However, first we prove Theorem 3.25.

3.4.1 Proof of the main theorem We assume

• (P, L) is a connected Fischer space, and

•A is a vanishing set.

Moreover, we identify the points of (P, L) with their images in gP,A. The ﬁrst step towards the proof of Theorem 3.25 is the following lemma.

Lemma 3.26 Suppose A is admissible. Then, iP,A = A or iP,A = gP . 64 Chapter 3. Constructing geometries from extremal elements

Proof. Let A ∈ A and let x ∈ P. Then

xA = |{y ∈ A | x ∼ y}|x + A + x · A = |{y ∈ A | x ∼ y}|x + A + x · A.

Since A is admissible, xA, A, and x · A are all elements of iP,A. Consequently, also

|{y ∈ A | x ∼ y}|x ∈ iP,A.

If |{y ∈ A | x ∼ y}| is not even, then x ∈ iP,A and, since P acts transitively on x, P ⊆ iP,A. This can only be the case if iP,A = gP . Consequently, either

∀x∈P ∀A∈A : xA = A + x · A ∈ A, in which case iP,A = A, or iP,A = gP .

Lemma 3.27

gP,A is a Lie algebra ⇐⇒ A is affine. Proof. First, suppose A is an affine vanishing set. Then, it is sufficient to check the anti- commutativity identities and the Jacobi identities for arbitrary points in P assuming that the points in an affine plane add up to zero. Therefore, let x, y, z ∈ P be three distinct points. Then, xx = 0 because each line contains three distinct points. Consequently, it is sufficient to prove xyz + yxz + xzy = 0.

If x ⊥ y ⊥ z ⊥ x, then obviously xyz + yxz + xzy = 0. Suppose x ∼ y ⊥ z ⊥ x and let u ∈ P such that {x, u, y} ∈ L. Since z is perpendicular to x and y, it is also perpendicular to w. Hence,

xyz + yzx + zxy = zxy = z(x + u + y) = zx + zu + zy = 0.

Suppose x ∼ y ⊥ z ∼ x. Then x, y, and z generate a dual afﬁne plane of order two inside (P, L). Consequently, there are u, v, w ∈ P such that {x, u, y}, {x, v, z}, {u, w, z}, {v, w, y} ∈ L. Moreover, u ⊥ v and x ⊥ w. Hence,

xyz + yzx + zxy = y(x + v + z) + z(x + u + y) = xy + yv + xz + uz = (x + u + y) + (v + w + y) + (x + v + z) + (u + w + z) = 0.

Suppose x ∼ y ∼ z ∼ x. If {x, y, z} ∈ L, then xyz = yxz = zxy = 0. Consequently, we can assume there are u, v, w ∈ P such that {x, u, y}, {x, v, z}, {y, w, z}, {u, v, w} ∈ L. If x, y, and z generate a dual afﬁne plane of order two, then u ⊥ z, v ⊥ y, w ⊥ x, 3.4. From Fischer space to Lie algebra 65 and xyz + yzx + zxy = x(y + w + z) + y(x + v + z) + z(x + u + y) = (xy + xz) + (xy + yz) + (xz + yz) = 0.

In other words, we can assume that x, y, and z generate an afﬁne plane of order three. In particular, this implies there are a, b, c ∈ P such that x+y+z+u+v+w+a+b+c = 0, {a, u, z}, {b, v, y}, {c, w, x} ∈ L, and

xyz + yzx + zxy = x(y + w + z) + y(x + v + z) + z(x + u + y) = (xy + wx + xz) + (xy + vy + yz) + (xz + uz + yz) = (c + w + x) + (b + v + y) + (a + u + z) = 0.

So, indeed gP,A is a Lie algebra generated by P. From the last equality it also follows that gP,A can only be a Lie algebra if A = 0 for all afﬁne planes A, that is, A is afﬁne. This concludes the proof.

Lemma 3.28 Suppose A is an afﬁne vanishing set. Then the nonzero images of the points of (P, L) are extremal.

Proof. Let x ∈ P and deﬁne gx : gP → F2 as the map induced by sending y ∈ P to 1 if x ∼ y, g (y) := x 0 otherwise.

Let y, z ∈ P. Now, it is sufﬁcient to prove that

xxy = 0 and xyxz = gx(yz)x + gx(z)xy + gx(y)xz assuming that for each afﬁne plane the points add up to zero. If there is a u ∈ P such that {x, y, u} ∈ L, then

xxy = x(x + y + u) = xy + xu = (x + y + u) + (x + u + y) = 0.

Otherwise, xxy = x0 = 0. Thus, indeed, xxy = 0. Let M be the subspace of (P, L) generated by x, y, and z. If M contains at most one line, then the identity

xyxz = gx(yz)x + gx(z)xy + gx(y)xz is readily checked. Therefore, we will assume M contains at least two lines. As a consequence, M is either a dual afﬁne plane of order two or an afﬁne plane of order 66 Chapter 3. Constructing geometries from extremal elements three. We consider these two cases separately.

Dual afﬁne plane of order two. If y ⊥ z, then gx(yz) = gx(0) = 0. Otherwise, there is a u ∈ P such that {y, z, u} ∈ L. Since each point of M not on a line of M is collinear with exactly two points of that line, this would imply

gx(yz) = gx(y + z + u) = 2 + 0 = 0.

We conclude gx(yz) = 0. Consequently, it is sufﬁcient to prove xyxz = gx(z)xy + gx(y)xz. If x ⊥ z, then gx(z) = 0, xz = 0, and

xyxz = xy0 = 0 = 0xy + gx(y)0 = gx(z)xy + gx(y)xz.

Therefore, we can assume x ∼ z. Moreover, since xxyz = 0 and xzxy = xyxz + xxyz = xyxz, we can also assume y ∼ z. Suppose y ⊥ z. Then there are u, v, w ∈ P such that {x, u, y}, {x, v, z}, {u, w, z}, {v, w, y} ∈ L. Moreover, u ⊥ v and x ⊥ w. Hence,

xyxz = xy(x + v + z) = x(x + u + y) + x(v + w + y) = (x + u + y) + (x + u + y) + (x + v + z) + (x + u + y) = (x + u + y) + (x + v + z) = xy + xz

= gx(z)xy + gx(y)xz.

Consequently, we can assume y ∼ z. This implies there are u, v, w ∈ P such that {x, u, y}, {x, v, z}, {y, w, z} ∈ L. Moreover, u ⊥ z, v ⊥ y, and w ⊥ x. Hence,

xyxz = xy(x + v + z) = x(x + u + y) + x(y + w + z) = (x + u + y) + (x + u + y) + (x + u + y) + (x + v + z) = (x + u + y) + (x + v + z) = xy + xz

= gx(z)xy + gx(y)xz.

Afﬁne plane of order three. There are u, v, w, a, b, c ∈ P such that

x + y + z + u + v + w + a + b + c = 0, {x, u, y}, {x, v, z}, {y, w, z}, {x, a, w}, {y, b, v}, {x, b, c} ∈ L, and {x, y, z, u, v, w, a, b, c} = M. 3.4. From Fischer space to Lie algebra 67

Hence, gx(yz) = gx(y + w + z) = 1, and

xyxz = xy(x + v + z) = x(x + u + y) + x(y + b + v) + x(y + w + z) = (x + u + y) + (x + b + c) + (x + v + z) + xy + (x + a + w) + xz = u + y + xy + xz + b + c + v + z + a + w = x + xy + xz

= gx(yz)x + gx(z)xy + gx(y)xz.

Proof of Theorem 3.25. Lemmas 3.26–3.28.

3.4.2 Some examples If we are given a Fischer space (P, L) not containing any afﬁne planes, then each vanishing set A will give us a quotient Lie algebra gP,A of the non-zero Lie algebra gP,∅. In general, gP,∅ will be of classical type.

Example 3.29 Let n ∈ N and suppose (P, L) is the cotriangular Fischer space Tn. Then Tn does not contain any affine planes of order three. As a consequence, gP,∅ is a n Lie algebra over F2 of dimension 2 . This Lie algebra is isomorphic to on(F2) through the linear map gP,∅ → on(F2) induced by the map sending a point {i, j} of Tn to the rank-1 matrix Ei,i + Ei,j + Ej,i + Ej,j in on. If we are given a Fischer space (P, L) that does contain an affine plane, then the empty vanishing set will not give us a Lie algebra. However, taking the vanishing set containing all affine planes will give us a Lie algebra. This Lie algebra might be zero.

Example 3.30 Let n ∈ N, suppose (P, L) is the Fischer space HUn(2), and let A be the collection of all afﬁne planes. Max Horn computed the dimension of gP,A for small values of n.

n 2 3 4 5 6 7 8 9 dim(gP,A(F2)) 3 8 30 45 78 119 176 249 Next, deﬁne

⊥ B := {B ⊆ P | |B| = 6 ∧ ∃x∈P : B ⊆ x ∧ ∀l∈L : |B ∩ l| ≤ 2}, that is, each element of B consists of six points all of which are perpendicular to a single point and no three of which are one a single line. For small values of n Max Horn 2 veriﬁed that gP,B is a Lie algebra of dimension n − 1 isomorphic to a special unitary Lie algebra. 68 Chapter 3. Constructing geometries from extremal elements

Example 3.31 Suppose (P, L) is the Fischer space coming from the Fischer group Fii, for some i ∈ [22, 24], and let A be the collection of all affine planes. If i ∈ [23, 24], then there is a point outside an affine plane that is collinear to all points of that affine plane. Following the proof of Lemma 3.26 we then find that gP,A = {0}. However, if i = 22, then there is no such point. In fact, Max Horn and Dan Roozemond verified that 2 for this case gP,A is isomorphic to the Lie algebra of type E6 over F2. This leads to a geometric proof that the Fischer group Fi22 embeds in the group of exceptional Lie type 2 2 E6 also known as E6(2). Chapter 4

Constructing simply laced Lie algebras from extremal elements

4.1 Introduction and main results

In Chapter2 we considered the problem of describing Lie algebras generated by at most three extremal elements under the assumption that strongly commuting or special pairs do not exist but without putting any restrictions on the field characteristic. Cohen et al. (2001), in ’t panhuis, Postma, and Roozemond (2009), Postma (2007), and Roozemond (2005) considered similar problems but now with more extremal generators in a slightly different setting: no restrictions were made regarding the existence of strongly commuting or special pairs and characteristic two was excluded. In this chapter we will do the same. See also Draisma and in ’t panhuis (2008). Because of Proposition 1.18, all 3-dimensional Lie algebras over a field F with a pair (x, y) of extremal generators are parameterised by gxy: all algebras with gxy 6= 0 are isomorphic to sl2(F), the Chevalley algebra of type A1, while the other algebras are nilpotent and isomorphic to h(F). This is a prototypical example of our results. The next smallest case of three generators is treated by Cohen et al. (2001), Zel0manov and Kostrikin (1990), and also by the results in this chapter. There the generic Lie algebra is the Chevalley algebra of type A2 and more interesting degenerations exist. We now generalize and formalize this example to the case of more generators, where we also allow for the flexibility of prescribing that certain generators commute. Thus, let Γ = (Π, ∼) be a finite simple graph without loops or multiple edges and denote the set of edges by Σ. Fixing a field F of characteristic not two, we denote by F the quotient of the free Lie algebra over F generated by Π modulo the relations

∀x6∼y∈Π : xy = 0.

So, F depends both on Γ and on F, but we will not make this dependence explicit in the 70 Chapter 4. Constructing simply laced Lie algebras from extremal elements

∗ ∗ Π notation. We write F for the space of all functionals F → F and for every g ∈ (F ) , also written (gx)x∈Π, we denote by L(g) the quotient of F by the ideal generated by the set {xxy − 2gx(y)x | x ∈ Π ∧ y ∈ F }. By construction, L(g) is a Lie algebra generated by extremal elements, corresponding to the vertices of Γ, which commute whenever they are not connected in Γ. For x ∈ Π ∗ Π the element gx is needed to express the extremality. Hence, for g ∈ (F ) the element g can be identified with the extremal form of L(g). If Γ is not connected, then both F and L(g) naturally split into direct sums over all connected components of Γ. So, it is no restriction to assume that Γ is connected. We will do so throughout this chapter. In the Lie algebra L(0) the elements of Π map to sandwich elements. Zel0manov and Kostrikin (1990) proved that this Lie algebra is finite-dimensional. For general g ∈ (F ∗)Π Cohen et al. (2001) proved that dim(L(g)) ≤ dim(L(0)). This also follows from the proof of Theorem 4.1. It is therefore natural to focus on the Lie algebras L(g) of the maximal possible dimension dim(L(0)). This leads us to define X as the set containing all g ∈ (F ∗)Π for which dim(L(g)) = dim(L(0)). It is the parameter space for all maximal-dimensional Lie algebras of the form L(g). In the two-generator case above Γ is the graph with two vertices joined by an edge. The sandwich algebra L(0) is the three-dimensional Heisenberg algebra, and X is the affine line. All Lie algebras corresponding to non-zero points of X are isomorphic to the Chevalley algebra of type A1. The first main result of this chapter is that X carries a natural structure of an affine variety. To specify this structure we note that I(0) is a homogeneous ideal relative to the natural N-grading that F inherits from the free Lie algebra generated by Π.

Theorem 4.1 Let F be a ﬁeld of characteristic not two and let Γ = (Π, ∼) be a connected ﬁnite simple graph without loops or multiple edges. Then the parameter space

X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))} is naturally the set of F-rational points of an affine variety defined over F. This variety can be described as follows. Fix any finite-dimensional homogeneous subspace V of F such that V + I(0) = F . Then the restriction map

∗ Π X → (V ) , g 7→ (gx|V )x∈Π

∗ Π maps X injectively onto the set of F-rational points of a closed subvariety of (V ) . This yields an F-variety structure on X which is independent of the choice of V . We prove this theorem in Section 4.2. In Section 4.3 we first derive some relations between the sandwich algebra L(0) and the positive part of the Kac-Moody algebra of type Γ. Then we determine L(0) explicitly in the case where Γ is a simply laced Dynkin diagram of finite or affine type. By this we mean any of the diagrams in Figure 4.1 4.2. The variety structure of the parameter space 71

0 1 0 n 1 n - 1

2 ... n - 1 0 2 ... n - 2 n

1 3 4 5 6 (1) (1) (1) (a) An (b) Dn (c) E6

2 2

0 1 3 4 5 6 7 1 3 4 5 6 7 8 0

(1) (1) (d) E7 (e) E8

Figure 4.1: The simply laced Dynkin diagrams of affine type. The notation comes from Kac (1990) and the associated finite-type diagrams are obtained by deleting vertex 0. without or with vertex 0, respectively. See Theorems 4.8 and 4.9. In Section 4.4 we study the variety X. After some observations for general Γ, we again specialize to the diagrams of Figure 4.1. For these we prove that X is an affine space, and that for g in an open dense subset of X the Lie algebra L(g) is isomorphic to a fixed Lie algebra. See Theorems 4.11 and 4.13. For the latter of these theorems we need the field to be algebraically closed. This condition was erroroneously ommitted by Draisma and in ’t panhuis (2008). We paraphrase the theorem here.

Theorem 4.2 Let F be an algebraically closed field of characteristic not two, let Γ be any of the simply laced Dynkin diagrams of affine type in Figure 4.1, let Γ0 be the finite- type diagram obtained by removing vertex 0 from Γ, and let Σ be the edge set of Γ. Then the parameter space

X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))} is isomorphic to the afﬁne space of dimension |Σ| + 1 over F, and for g in an open dense subset of X the Lie algebra L(g) is isomorphic to the Chevalley algebra of type Γ0. We conclude with remarks on applications and related work in Section 4.5. There we also discuss an interesting connection with Chapter3.

4.2 The variety structure of the parameter space

In the proof of Theorem 4.1 we use the N-grading ∞ M F = Fd d=1 72 Chapter 4. Constructing simply laced Lie algebras from extremal elements of F with Fd (d ∈ N) the linear span of all monomials of degree d in the elements of Π.

Proof of Theorem 4.1. Cohen et al. (2001) and Zel0manov and Kostrikin (1990) proved that L(0) is finite-dimensional. Moreover, I(0) is homogeneous. Hence, there is a finite-dimensional homogeneous subspace V of F such that F = V ⊕ I(0). Note that the theorem only requires that F = V + I(0). We will argue later why this suffices. Observe that V contains the image of Π in L(0): the abelian Lie algebra spanned by Π is clearly a quotient of L(0), so the component of I(0) in degree 1 is trivial. From the shape of the generators (4.1) it is clear that the homogeneous graded ideal gr(I(g)) as defined in AppendixA contains I(0), so that F = V + I(g) for all g, and F = V ⊕ I(g) if and only if g ∈ X. We will argue that the map

∗ Π Ψ: X → (V ) , g 7→ (gx|V )x∈Π =: g|V is injective, and that its image is a closed subvariety of (V ∗)Π. For each g ∈ X let πg : F → V be the projection onto V along I(g). We prove two slightly technical statements: First, for all u ∈ F there exists a polynomial map ∗ Π Pu :(V ) → V such that

∀g∈X : Pu(g|V ) = πg(u).

∗ Π Moreover, for all x ∈ Π and u ∈ F there exists a polynomial Qx,u :(V ) → F such that

∀g∈X : Qx,u(g|V ) = gx(u),

∀ ∀ ∗ Π : Q (h) = h (u). u∈V h∈(V ) x,u x

We proceed by induction on the degree of u: assume that both statements are true in all degrees less than d, and write u = u1 + u2 + u3 where u1 has degree less than d, u2 ∈ V ∩ Fd, and u3 ∈ I(0) ∩ Fd. Then u3 can be 0 written as a sum of terms of the form xk ··· x1x1u with k ∈ N, {xi | i ∈ [k]} ⊆ Π and u0 of degree d − (k + 1) < d. Modulo I(g) for g ∈ X this term is equal to

0 2g (u )π (x ··· x ) = 2Q 0 (g| )P (g| ), x1 g k 1 x1,u V xk···x1 V

0 where we used the induction hypothesis for u and xk ··· x1. Hence, we obtain a Pu of the form P := P + u + terms of the form 2Q 0 P u u1 2 x1,u xk···x1 has the required property. Similarly, for x ∈ Π and g ∈ X we have

0 0 2g (x ··· x x u )x = xxx ··· x x u = 4Q 0 (g| )Q (g| )x mod I(g), x k 1 1 k 1 1 x1,u V x,xk···x1 V 4.2. The variety structure of the parameter space 73 and since x 6∈ I(g) we conclude that

0 g (x ··· x x u ) = 2Q 0 (g| )Q (g| ). x k 1 1 x1,u V x,xk···x1 V

Hence, we can deﬁne Qx,u such that it satisﬁes

∀ ∗ Π : Q (h) = Q (h)+h (u )+ terms of the form 2Q 0 (h)Q (h). h∈(V ) x,u x,u1 x 2 x1,u x,xk···x1

This shows the existence of Pu and Qx,u. The injectivity of Ψ is now immediate: any g ∈ X is determined by its restriction to V by gx(u) = Qx,u(g|V ). Next, we show that im Ψ is closed. For any tuple h ∈ (V ∗)Π one may try to deﬁne a Lie algebra structure on V by setting

∀u,v∈V :[u, v]h := P[u,v](h).

By construction, if h = g|V for some g ∈ X, then this turns V into a Lie algebra isomorphic to L(g). Moreover, in this case the Lie bracket has the following two properties:

• If v ∈ V is expressed as a linear combination P c x ··· x of x1,...,xd∈Π (xd,...,x1) d 1 monomials in the elements of Π, where the Lie bracket is taken in F , then the expression P c [x , [... [x , x ] ...] ] also equals v. x1,...,xd∈Π (xd,...,x1) d 2 1 h h h

• [x, [x, u]h]h = 2Qx,u(h)x for all x ∈ Π, u ∈ V .

Conversely, suppose that [., .]h indeed defines a Lie algebra on V satisfying (4.2) and (4.2). Then (V, [., .]h) is a Lie algebra of dimension dim(L(0)) that by (4.2) is generated by the image of Π, and by (4.2) this image consists of extremal elements. Hence, there exists an g ∈ X corresponding to this Lie algebra, and its restriction to V is h. Indeed, 2gx(u) is the coefficient of x in [x[x, u]h]h, which is 2Qx,u(h) = 2hx(u) for u ∈ V . Finally, [., .]h satisfying the anti-commutativitiy identities, the Jacobi identities, and both (4.2) and (4.2), are all closed conditions on h. Here, we use the polynomiality of Pu and Qx,u. This proves that im Ψ is closed. Now, if U is any homogeneous subspace containing V , then the restriction map Ψ0 : X → (U ∗)Π is clearly also injective. Moreover, an h0 ∈ (U ∗)Π lies in the image of 0 0 0 this map if and only if h |V lies in im Ψ and hx(u) = Qx,u(h |V ) for all u ∈ U. Thus 0 0 0 0 0 0 im Ψ is closed and the maps im Ψ → im Ψ, h 7→ h |V and im Ψ → im Ψ , h 7→ h 0 0 with hx(u) = Qx,u(h), u ∈ U are inverse morphisms between im Ψ and im Ψ . Similarly, if V 0 is any other homogeneous vector space complement of I(0) contained in U, then the restriction map (U ∗)Π → ((V 0)∗)Π induces an isomorphism between the images of X in these spaces. This shows that the variety structure of X does not depend on the choice of V . Finally, all morphisms indicated here are defined over F. We conclude that we have a F-variety structure on X which is independent of the choice of V . 74 Chapter 4. Constructing simply laced Lie algebras from extremal elements

The type of reasoning in this proof will return in Section 4.4: in the case where Γ is a Dynkin diagram of finite or affine type we show that for g ∈ X the restriction g|V depends polynomially on even fewer values of the gx, thus embedding X into smaller affine spaces. That these embeddings are closed can be proved exactly as we did above.

4.3 The sandwich algebra

For now, Γ is an arbitrary finite graph (not necessarily a Dynkin diagram). Then the Lie algebra L(0) is the so-called sandwich algebra corresponding to Γ. It is a finite- dimensional nilpotent Lie algebra. First we analyze the weight grading of L(0). Then we use this to determine the isomorphism type of L(0) in the case of a simply laced Dynkin diagram of finite or affine type.

4.3.1 Weight grading Π The sandwich algebra L(0) corresponding to a ﬁnite graph Γ carries an N -grading Π deﬁned as follows. The weight of a word (xd, . . . , x1) over Π is the element µ ∈ N with ∀x∈Π : |{i ∈ {1, . . . , d} | xi = x}|.

For such a word the corresponding monomial xd ··· x1 lives in the free Lie algebra on Π, but we use the same notation for its images in F and L(0) when this does not lead to any confusion. We will sometimes say that a monomial xd ··· x1 ∈ L(0) has weight µ. Then we mean that the word (xd, . . . , x1) has weight µ although the monomial xd ··· x1 itself might be 0. Now the free Lie algebra is graded by weight and this grading refines the grading by degree. Like the grading by degree, the grading by weight is inherited by L(0) as all relations defining L(0) are monomials. We write L(0)µ for the space of Π weight µ ∈ N . Then dim(L(0)µ) is the multiplicity of µ. Moreover, for x ∈ Π we write αx for the weight of the word (x), that is, the element with a 1 on position x and Π zeroes elsewhere. We define a symmetric Z-bilinear form h., .i on Z by its values on the standard basis: for x, y ∈ Π we set  2 if x = y,  hαx, αyi := −1 if x ∼ y, and  0 otherwise.

The matrix A := (hαx, αyi)x,y∈Π is the generalized Cartan matrix of Γ. See Appendix Π A. The height of an element of Z is by deﬁnition the sum of the coefﬁcients of the αx, x ∈ Π, in it. The following proposition describes the relation between different components of the weight grading. 4.3. The sandwich algebra 75

Proposition 4.3 Let F be a field with char(F) 6= 2 and let Γ = (Π, ∼) be a finite graph. Moreover, let x ∈ Π and λ ∈ Π satisfy hα , λi = −1. Then L(0) = xL(0) . N x αx+λ λ Before proving this proposition we first recall an elementary property of sandwich elements, to which they owe their name: the sandwich property.

Lemma 4.4 Let x be a sandwich element in the Lie algebra L(0) and let y, z ∈ L(0) be arbitrary. Then xyxz = 0.

Proof. Direct consequence of the Premet identities an the fact that gx = 0.

The next two lemmas give criteria for a monomial to be zero.

Lemma 4.5 Let w = (xd, xd−1, . . . , x1) be a word over Π and let x ∈ Π. Let xi, xj be consecutive occurrences of x in w, that is, i > j, xi = xj = x, and xk 6= x for all k strictly between i and j. Suppose that the letters in w strictly between xi and xj contain at most 1 occurrence of a Γ-neighbour of x, that is,

|{k ∈ {j + 1, . . . , i − 1} | xk ∼ x}| ≤ 1.

Then xdxd−1 ··· x1 = 0 in L(0).

Proof. Set z := xdxd−1 ··· x1. First, using the fact that on F the linear map adx commutes with ady for any y ∈ Π with x 6∼ y, we can move xi in z to the right until it is directly to the left of either xj or the unique xk ∼ x between xi and xj. So, we may assume that this was already the case to begin with. If i = j + 1 then either j = 1 and z is zero by anti-commutativity, or j > 1 and the

xixjxj−1 ··· x1 = xxxj−1 ··· x1 = 0 by the sandwich property of x. Suppose, on the other hand, that xi ∼ xi−1. Then, assuming j = 1 and i − 1 > 2, the monomial z is zero since x2x1 is—indeed, x2 6∼ x1. On the other hand, if i − 1 = 2 or j > 1 then we can move xj in z to the left until it is directly to the right of xi−1. So again, we may assume that it was there right from the beginning. But now

xixi−1xjxj−1 ··· x1 = xxi−1xxj−1 ··· x1.

If j > 1, then this monomial equals zero by Lemma 4.4. If j = 1, then it is zero by the sandwich property of x.

Lemma 4.6 Let (xd, xd−1, . . . , x1) be a word with d ≥ 2 over Π and suppose that the weight µ of (x , x , . . . , x ) satisﬁes hα , µi ≥ 0. Then x x ··· x = 0 in d−1 d−2 1 xd d d−1 1 L(0). 76 Chapter 4. Constructing simply laced Lie algebras from extremal elements

Proof. Set w := (xd, xd−1, . . . , x1) and z := xdxd−1 ··· x1. The condition on the bilinear form can be written as follows:

2|{j ∈ {1, . . . , d − 1} | xj = xd}| ≥ |{j ∈ {1, . . . , d − 1} | xj ∼ xd}|.

First we note that if the right-hand side is 0, then z is trivially zero: then all xi with i < d commute with xd, and there at least d − 1 ≥ 1 such factors. In other words, we may assume that the right-hand side is positive. Consequently, we may assume the same for the left-hand side. Let the set in the left-hand side of this inequality consist of the indices im > im−1 > . . . > i1. By the above, m is positive. In the word w there are m pairs (i, j) satisfying the conditions of Lemma 4.5 with x = xd, namely, (d, im), (im, im−1),..., (i2, i1). Now, if for some such (i, j) there are less than two Γ-neighbours of xd in the interval between xi and xj, then z = 0 by Lemma 4.5. So, we may assume that each of these m intervals contains at least two Γ-neighbours of xd. But then, by the above inequality, these exhaust all Γ-neighbours of xd in w, so in particular there are exactly 2 Γ-neighbours of xd between x and x , and none to the right of x . Now, if i > 1, then z is zero because i2 i1 i1 1 x commutes with everything to the right of it. Hence, assume that i = 1, and note that i1 1 i ≥ 4. If x 6∼ x = x , then again z is trivially 0, so assume that x is a Γ-neighbour 2 2 1 i1 2 of xd = x1. Then we have

x ··· x x x = −x ··· x x x , i2 3 2 1 i2 3 1 2 but in the monomial on the right there is only one Γ-neighbour of x between x and d i2 x1. Hence, it is zero by Lemma 4.5.

Proof of Proposition 4.3. Let w = (xd, . . . , x1) be a word over Π of weight αx +λ. We show that in L(0) the monomial z := xd ··· x1 is a scalar multiple of some monomial of the form xz0, where z0 is a monomial of weight λ. Obviously, x occurs in w. Therefore, let k be maximal with xk = x. If k = 1, then, since d ≥ 2, we may interchange xk = x1 and xk+1 = x2 in z at the cost of a minus sign. So, we may assume that k ≥ 2. Suppose ﬁrst that there occur Γ-neighbours of x = xk to the left of xk in w. We claim that then z = 0. Indeed, let µ, ν be the weights of (xk−1, . . . , x1) and (xd, . . . , xk+1), respectively. Then we have

hαx, µi = hαx, λi − hαx, νi = −1 − hαx, νi ≥ 0, where in the last inequality we use that there are occurrences of neighbours of xk, but none of xk itself, in the word (xd, . . . , xk+1). Now, we ﬁnd xk ··· x1 = 0 by Lemma 4.6 (note that k ≥ 2). Hence, z = 0 as claimed. So, we can assume that there are no Γ-neighbours of xk to the left of xk in w. Then 0 we may move xk in z all the way to the left, hence z is indeed equal to xz for some 0 monomial z of weight λ. 4.3. The sandwich algebra 77

4.3.2 Relation with the root system of the Kac-Moody algebra

Recall the deﬁnition of the Kac-Moody algebra gKM over C corresponding to a ﬁnite- type Dynkin diagram Γ (see also Section 1.1.3): it is the Lie algebra generated by 3 · |Π| generators, denoted ex, fx, hx for x ∈ Π, modulo the relations

 hxhy = 0,  e f = 0,   x y  exfx = hx,  1−Axy ∀ : and ∀ : ade e = 0, x,y∈Π h e = A e , x6=y∈Π x y  x y xy y  1−Axy   ad fy = 0. hxfy = −Axyfy, fx

Π The Lie algebra gKM is endowed with the Z -grading in which ex, fx, hx have weights Π αx, −αx, 0, respectively. Moreover, Φ := {β ∈ Z \{0} | (gKM)β 6= 0} is the root Π system of gKM. It is equal to the disjoint union of its subsets Φ± := Φ ∩ (±N) and contains the simple roots αx, x ∈ Π. Note that a root in Φ is called real if it is in the orbit of some simple root under the Weyl group W of gKM. In that case it has multiplicity 1 in gKM. We now call a root β ∈ Φ+ very real (this is non-standard terminology) if

∃ : β = α + ... + α , x1,...,xd∈Π xd x1 such that ∀ : hα , α + ... + α i = −1. i∈[2,d] xi xi−1 x1 Now, the following proposition compares the multiplicities of weights in the Lie algebra L(0) over F with the multiplicities of weights in the Lie algebra gKM over C.

Proposition 4.7 Let F be a ﬁeld with char(F) 6= 2 and let Γ be a Dynkin diagram of ﬁnite type. Then

∀ Π : L(0)λ = {0}, λ∈N \Φ+

∀ : λ very real ⇒ dim(L(0)λ) ≤ 1. λ∈Φ+

Π Proof. First we focus on the ﬁrst part of the lemma. Therefore, let λ ∈ N \ Φ+ and proceed by induction on the height of λ. Clearly, L(0)λ = {0} for λ of height 1. Suppose now that this is also the case for height d − 1 ≥ 1, and consider a word w = (xd, xd−1, . . . , x1) of weight λ 6∈ Φ+. Set µ := λ − α . If µ 6∈ Φ , then x ··· x = 0 by the induction hypothesis, so xd + d−1 1 we may assume that µ ∈ Φ . This together with µ + α 6∈ Φ implies (by elementary + xd + sl -theory in g ) that hα , µi ≥ 0. Now Lemma 4.6 shows that x ··· x = 0. 2 KM xd d 1 This concludes the ﬁrst part. The second part also follows by induction on the height of λ, using Proposition 4.3 for the induction step. 78 Chapter 4. Constructing simply laced Lie algebras from extremal elements

4.3.3 Simply laced Dynkin diagrams of finite type In this section we assume that Γ is a Dynkin diagram of finite type, that is, one of the diagrams in Figure 4.1 with vertex 0 removed. Then gKM is a finite-dimensional simple Lie algebra over C. The Chevalley basis (see Section 1.1.3) of gKM consists of the images of h , x ∈ Π, and one vector e ∈ (g ) for every root α ∈ Φ, where e and x α KM α αx e may be taken as the images of e and f , respectively. This basis satisfies −αx x x

∀α, β ∈ Φ: α + β ∈ Φ ⇒ eαeβ = ±eα+β.

Moreover, the Chevalley algebra g of type Γ can be obtained by tensoring gKM with F, and has a triangular decomposition

g = n− ⊕ h ⊕ n+, where n := L g and h := g . See also Section 1.1.3. ± β∈Φ± β 0 0 0 0 Finally, note that we write ex, fx , hx for the images in g of ex, fx, hx, respectively.

Theorem 4.8 Let F be a field of characteristic not two, let Γ be a simply laced Dynkin diagram of finite type obtained from a diagram in Figure 4.1 by removing vertex 0, let g be the corresponding Chevalley algebra over the field F of characteristic not two, and 0 0 let n+ be the subalgebra generated by the ex. Then the map sending x ∈ Π to ex induces a isomorphism L(0) → n+. In the proof of this theorem we use the following well-known facts about simply laced Kac-Moody algebras of finite type: first, the bilinear form h., .i coming from the generalized Cartan Matrix only takes the values −1, 0, 1, 2 on Φ+ × Φ+, and second, all roots in Φ+ are very real.

Proof of Theorem 4.8. To prove the existence of a homomorphism π sending x to 0 Ex, we verify that the relations deﬁning L(0) hold in n+. That is, we prove that

0 0 ∀x,y∈Π : x 6∼ y ⇒ exey = 0, 2 ∀x∈Π∀z∈n : ad 0 z = 0. + ex

The first statement is immediate from the relations defining gKM. For the second relation, if z ∈ n+ is a root vector with root λ ∈ Φ+, then hλ, αxi ≥ −1 by the above. In 2 particular, hλ + 2αx, αxi ≥ 3 and λ + 2αx 6∈ Φ+. As a consequence ad 0 z = 0. As root ex vectors span n+, we have proved the existence of π. Now we have to show that π is an isomorphism. It is surjective as n+ is generated 0 by the ex. Hence, it suffices to prove that dim(L(0)) ≤ dim(n+). But by Proposition 4.7 and the fact that all roots are very real we have L(0)λ = {0} for all λ 6∈ Φ+ and dim(L(0)λ) ≤ dim(gλ) for all λ ∈ Φ+. This concludes the proof. 4.3. The sandwich algebra 79

4.3.4 Simply laced Dynkin diagrams of affine type Suppose now that Γ is a simply laced Dynkin diagram of affine type from Figure 4.1. Then the generalized Cartan matrix A has a one-dimensional kernel, spanned by a unique Π primitive vector δ ∈ N . Here, primitive means that the greatest common divisor of the coefficients of δ on the standard basis is 1. Indeed, there always exists a vertex x0 ∈ Π (labelled 0 in Figure 4.1) with coefficient 1 in δ, and all such vertices form an Aut(Γ)- orbit. For later use, we let h be the Coxeter number, which is the height of δ. 0 0 0 Write Π := Π \{x0}, Γ for the induced subgraph on Π (which is a Dynkin diagram of finite type), and Φ0 for the root system of the Chevalley algebra g of type 0 Π0 Γ . This root system lives in the space Z , which we identify with the elements of Π Z that are zero on x0. Retain the notation n± ⊆ g from Section 4.3.3, and consider the semi-direct product u of the n+-module g/n+ and n+ itself. We will prove that it is Π isomorphic to L(0). In our proof we use the following Z -grading of u: the root spaces 0 Π0 0 in n+ have their usual weight in Φ+ ⊆ Z , while for each λ ∈ {0} ∪ Φ− the image of gλ in g/n+ ⊆ u has weight δ + λ. Thus the set of all weights ocurring in u is

0 0 Θ := Φ+ ∪ {δ + λ | λ ∈ Φ−} ∪ {δ}.

Theorem 4.9 Let Γ be a simply laced Dynkin diagram of affine type from Figure 4.1, let Γ0 be the subdiagram of finite type obtained by removing vertex 0, and let g be the Chevalley algebra of type Γ0 over a field of characteristic unequal to 2. For x ∈ Π0 let 0 ex ∈ n+ be the element of the Chevalley basis of g with simple root αx and for the lowest 0 0 root θ ∈ Φ− let eθ ∈ g/n+ be the image of the element in the Chevalley basis of weight 0 0 0 Π θ. Then the map sending x ∈ Π to ex and x0 to eθ induces a Z -graded isomorphism L(0) → n+ n g/n+ of Lie algebras.

Note that over C one can argue directly in the Kac-Moody algebra gKM. Then L(0) is also the quotient of the positive nilpotent subalgebra of gKM by the root spaces with roots of height larger than the Coxeter number h. In the proof one uses root multiplicities (Kac 1990, Proposition 6.3). One might also pursue this approach in positive characteristic using the results of Billig (1990), but we have chosen to avoid deﬁning the Kac-Moody algebra in arbitrary characteristic and use the Chevalley basis instead.

Proof of Theorem 4.9. The proof is close to that of Theorem 4.8. We start by verifying 0 0 0 that the relations deﬁning L(0) hold in u = n+ n g/n+. First, ex and ey with x, y ∈ Π commute when they are not connected in Γ. This follows from the deﬁning equations of 0 0 0 gKM. Second, ex and eθ commute if x ∈ Π is not connected to x0, as θ + αx is then not 0 0 in Φ . Third, each ex is a sandwich element in u: for its action on n+ this follows as in the proof of Theorem 4.8 and for its action on g/n+ it follows from the fact that

2 0 ad 0 g ⊆ ex ⊆ n+. ex F 80 Chapter 4. Constructing simply laced Lie algebras from extremal elements

0 Fourth, eθ is a sandwich element as ad 0 maps u into g/n+, which has trivial multipli- eθ cation. This shows the existence of a homomorphism π : L(0) → u. Moreover π is graded. In particular, the weight of e0 is δ + θ = α . θ x0 0 0 The ex generate n+ and eθ generates the n+-module g/n+. These statements follow from properties of the Chevalley basis and imply that π is surjective. So, we need only show that dim(L(0)) ≤ dim(u). We prove this for each weight in Θ. First, the roots in Φ0 are very real, so their multiplicities in L(0) are at most 1 by 0 Proposition 4.7. Second, we claim that all roots of the form δ + β with β ∈ Φ− are also very real. This follows by induction on the height of β. For β = θ it is clear since δ + θ = α . For β 6= θ it is well-known that there exists an x ∈ Π0 such that x0 hαx, βi = 1. Then we have δ + β = (δ + β − αx) + αx where δ + β − αx ∈ Θ and hα, δ + β − αxi = 0 + 1 − 2 = −1. Here, we use that δ is in the radical of the form h., .i coming from the generalized Cartan Matrix. By induction, δ + β − αx is very real. Hence, so is δ + β. This shows, again by Proposition 4.7, that also the roots of the form 0 δ + β with β ∈ Φ− have multiplicity at most 1 in L(0). Next we show that δ has multiplicity at most |Π0| = dim(h) in L(0). Indeed, we claim that L(0)δ is contained in X [x, L(0) ]. δ−αx x∈Π0 Then, by the above, each of the summands has dimension at most 1, and we are done. The claim is true almost by deﬁnition: any monomial of weight δ must start with some x ∈ Π, so we need only show that monomials starting with x0 are already contained in the sum above. Consider any monomial z := xd ··· x1 of weight δ, where xd = x0. As the coefﬁcient of x0 in δ is 1, none of the xi with i < d is equal to x0. But then an elementary application of the Jacobi identities and induction shows that z is a linear combination of monomials that do not start with x0. Because of Proposition 4.7, we already know

∀ : L(0) = {0}. µ/∈Φ+ µ

Therefore, it remains to show the same statement for µ∈ / Φ+ \ Θ. However, Lemma 4.6 and the fact that

∀x∈Π : hαx, δi = 0 together imply that

∀x∈Π :[x, L(0)δ] = {0}.

So, it sufﬁces to show that if µ ∈ Φ+ is not in Θ, then

“µ can only be reached through δ”. 4.4. The parameter space and generic Lie algebras 81

More precisely: if (xd, . . . , x1) is any word over Π such that X α = µ xj j∈[d] and X ∀ : µ := α ∈ Φ , i∈[d] i xj + j∈[i] then there exists an i such that µi = δ. But this follows immediately from the fact that δ is the only root of height h (Kac 1990, Proposition 6.3). We ﬁnd that every monomial corresponding to such a word is zero, and this concludes the proof of the theorem.

4.4 The parameter space and generic Lie algebras

So far we have only considered the Lie algebras L(0). Now, we will be concerned with the variety X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))}. First, in Sections 4.4.1–4.4.3, we collect some tools for determining X in the case of simply laced Dynkin diagrams. Then in Sections 4.4.4–4.4.5 we ﬁnd an open dense subset of X such that all Lie algebras L(g) with g an element of the open subset are isomorphic.

4.4.1 Scaling First let Γ = (Π, ∼) be arbitrary again, not necessarily a Dynkin diagram. Scaling of ∗ Π the generators xi has an effect on X: given t = (tx)x∈Π in the torus T := (F ) there is a unique automorphism of F that sends x ∈ Π to txx. This gives an action of T on F , and we endow F ∗ with the contragredient action. Finally, we obtain an action of T on X by deﬁning

−1 −1 ∀t∈T ∀g∈X ∀x∈Π∀y∈F :(tg)x(y) := tx gx(t y).

Indeed, note that with this deﬁnition the automorphism of F induced by t sends xxy − 2gx(y)x ∈ F to

2 −1 (tx)(tx)(ty) − 2gx(y)tx = tx(xx(ty) − 2tx gx(y)x) 2 −1 −1 = tx(xx(ty) − 2tx gx(t (ty))x) 2 = tx(xx(ty) − 2(tg)x(ty)x), and the ideal I(g) deﬁning L(g) to I(tg). Therefore, this automorphism of F induces an isomorphism L(g) → L(tg). 82 Chapter 4. Constructing simply laced Lie algebras from extremal elements

This scaling action of T on X will make things very easy in the case of simply laced Dynkin diagrams where X will turn out to be isomorphic to an afﬁne space with linear action of T in which the maximal-dimensional orbits have codimension 0, 1 or 2.

4.4.2 The Premet relations

Our arguments showing that certain monomials m := xd ··· x1 are zero in the sandwich algebra L(0) always depended on the sandwich properties: xxy = 0 and xyxz = 0 whenever x is a sandwich element and y, z are arbitrary elements of the Lie algebra. The Premet identities translate such a statement into the following statement: in L(g) the monomial m can be expressed in terms of monomials of degree less than d − 1 and values of g on monomials of degree less than d − 1. xd

4.4.3 The parameters Recall from Section 4.2 that the restriction map X → (V ∗)Π is injective and has a closed image. A key step in the proof was showing that for g ∈ X the values gx(u), x ∈ Π, u ∈ F depend polynomially on g|V . In what follows, this will be phrased informally as

“g can be expressed in g|V ” or “g|V determines g”.

In this phrase we implicitly make the assumption that g ∈ X, that is, L(g) has the maximal possible dimension. In the case of Dynkin diagrams, we will exhibit a small number of values of g in which g can be expressed. For this purpose, the following proposition, which also holds for other graphs, is useful.

Proposition 4.10 Let Γ = (Π, ∼) be a ﬁnite or afﬁne Dynkin diagram, let g ∈ X, let q = xd ··· x1 be a monomial of degree d ≥ 2 and weight β, and let z ∈ Π be such that hαz, βi ≥ −1. Then gz(q) can be expressed in the parameters gx(m) with monomials m of degree less than d − 1 and x ∈ Π.

Proof. First, if xd is not a Γ-neighbour of z in Γ, then

gz(xd ··· x1) = g(z, xd ··· x1) = −g(xdz, xd−1 ··· x1) = g(0, xd−1 ··· x1) = 0, and we are done. So, assume that xd is a Γ-neighbour of z. Now,

g (x ··· x ) = −g(x z, x ··· x ) = −g(x , zx ··· x ) = g (zx ··· x ). z d 1 d d−1 1 d d−1 1 xd d−1 1

In both cases we have used that the images of z and xd are non-zero in L(g) for g ∈ X. Now, hα , β − α i ≥ 0. So, Lemma 4.6 says that zx ··· x can be expressed in z xd d−1 1 terms of smaller monomials and values gx(m) for x ∈ Π and monomials m of degree less than d − 1. Then by linearity of g the last expression above can also be expressed xd in terms of values gx(m) with x ∈ Π and m of degree less than d − 1. 4.4. The parameter space and generic Lie algebras 83

4.4.4 Simply laced Dynkin diagrams of ﬁnite type Suppose that Γ = (Π, ∼) is a simply laced Dynkin diagram of ﬁnite type. Then we Π ∗ Π identify Z with the character group of T = (F ) in the natural way: we write µ for the character ∗ µ Y µx T → F , t 7→ t = tx . x∈Π

Furthermore, if Σ is the set of edges of Γ, then we write αe instead of αx + αy for e = {x, y} ∈ Σ.

Theorem 4.11 Let Γ = (Π, ∼) be a simply laced Dynkin diagram of ﬁnite type obtained from a diagram in Figure 4.1 by removing vertex 0. Moreover, let Σ be the edge set of Γ, let g be the Chevalley algebra of type Γ over a ﬁeld F of characteristic not two, and set ∗ Π T := (F ) . Then the variety

X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))}

Σ is, as a T -variety, isomorphic to the vector space V := F on which T acts diagonally with character −αe on the component corresponding to e ∈ Σ. For g corresponding to ∗ Σ any element in the dense T -orbit (F ) the Lie algebra L(g) is isomorphic to a fixed Lie algebra. We first need a lemma that will turn out to describe the generic L(g). We retain the 0 0 0 notation ex, fx , hx ∈ g and n+ from Section 4.3.3 and define 0 0 0 ∼ C := {(cx)x∈Π | ∀x∈Π : cx ∈ hex, fx , hxi = sl2 is extremal}.

C is an irreducible variety, that is, it is not the union of two non-empty varieties.

0 Lemma 4.12 For c = (cx)x∈Π in some open dense subset of C the Lie subalgebra g of g generated by the cx has dimension dim(n+). Moreover,

∀ : x ∼ y ⇒ g 6= 0. x,y∈Π cxcy

Proof. By definition g0 is generated by extremal elements, hence it has dimension at most that of L(0), which is isomorphic to n+ by Theorem 4.8. The condition that the cx generate a Lie algebra of dimension less than dim(n+) is closed, and the tuple 0 (ex)x∈Π ∈ C does not fulfill it. Hence, using the irreducibility of C we find that for c 0 0 in some open dense subset of C the Lie algebra g satisfies dim(g ) = dim(n+). This proves the first statement. The second statement follows directly from the same statement for Γ of type A2, i.e., for g = sl3, where it boils down to the statement that the two copies of sl2 in sl3 corresponding to the simple roots are not mutually perpendicular relative to the extremal form in sl3. 84 Chapter 4. Constructing simply laced Lie algebras from extremal elements

Proof of Theorem 4.11. By Theorem 4.8 and Proposition 4.10, any g ∈ X is determined by its values gx(m) with x ∈ Π and monomials m of weights β ∈ Φ+ such that either β has height 1 or hαx, βi ≤ −2. However, since β is a positive root, the latter inequality cannot hold. Hence, β has height 1. In particular, m ∈ Π and the only x ∈ Π for which gx(m) 6= 0 are the Γ-neighbours of m. Moreover, from the symmetry of the extremal form we conclude that gx(m) = gm(x) for x, m ∈ Π neighbours in Γ. Thus, we have found a closed embedding

Σ Ψ: X → F : g 7→ (gx(y)){x,y}∈Σ.

Σ Now, if we let T act on F through the homomorphism

∗ Σ −1 −1 T → (F ) , t 7→ (tx ty ){x,y}∈Σ, then Ψ is T -equivariant by the results of Section 4.4.1. Note that T acts by the character −αe on the component corresponding to e ∈ Σ. The fact that Γ is a tree readily implies that the characters αe, e ∈ Σ, are linearly independent over Z in the character group of ∗ Σ T , so that the homomorphism T → (F ) is surjective. But then T has ﬁnitely many Σ orbits on F , namely of the form

∗ Σ0 Σ\Σ0 0 (F ) × {0} with Σ ⊆ Σ.

Σ Now, as Ψ(X) is a closed T -stable subset of F we are done if we can show that

∗ Σ (F ) ∩ Ψ(X) 6= ∅.

But this is precisely what Lemma 4.12 tells us: there exist maximal dimensional Lie 0 algebras g generated by extremal elements such that all gx(y) with x ∼ y are non-zero. This concludes the proof.

The proof above also implies that all Lie algebras described in Lemma 4.12 are isomorphic. More generally: for any two Lie algebras g0 and g00 with tuples of distinguished, 0 00 extremal generators (cx)x∈Π and (cx)x∈Π such that

0 0 00 00 •∀ x,y∈Π : x 6∼ y ⇒ cxcy = cxcy = 0,

•∀ : x ∼ y ⇒ (g 0 0 = 0 ⇔ g 00 00 = 0), x,y∈Π cxcy cxcy

0 00 • dim(g ) = dim(g ) = dim(n+),

0 00 0 then there exists an isomorphism g → g mapping each cx with x ∈ Π to a scalar 00 multiple of cx. 4.4. The parameter space and generic Lie algebras 85

4.4.5 Simply laced Dynkin diagrams of afﬁne type Suppose now that Γ is a simply laced Dynkin diagram of afﬁne type and let g be the 0 0 Chevalley algebra of type Γ , the graph induced by Γ on Π = Π \{x0}. To state the Π ∗ Π analogue of Theorem 4.11 we again identify Z with the character group of T = (F ) and retain the notation αe = αx + αy for e = {x, y} ∈ Σ, the edge set of Γ.

Theorem 4.13 Let Γ = (Π, ∼) be a simply laced Dynkin diagram of affine type from Figure 4.1 with edge set Σ and let Γ0 be the finite-type diagram obtained by deleting vertex 0 from Γ. Moreover, let g be the Chevalley algebra of type Γ0 over an algebraically ∗ Π closed field F of characteristic not two, and set T := (F ) . Then the variety

X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))}

Σ is, as a T -variety, isomorphic to the vector space V := F × F on which T acts diagonally with character −αe on the component corresponding to e ∈ Σ, and with character −δ on the last component. For all g ∈ X corresponding to points in some open dense Σ subset of F × F the Lie algebra L(g) is isomorphic to g. Unlike for diagrams of ﬁnite type, it is not necessarily true that T has only ﬁnitely many orbits on V . Indeed, the following three situations occur:

(1) (i) The characters αe (e ∈ Σ), δ are linearly independent. This is the case for Deven, (1) (1) E7 , and E8 . Then T has ﬁnitely many orbits on V .

(ii) The characters αe (e ∈ Σ) are linearly independent, but δ is in their Q-linear (1) (1) (1) ∗ Σ ∗ span. This is the case for Aeven , Dodd and E6 . Now the orbits of T in (F ) ×F (1) (1) have codimension 1. For Aeven and E6 the character δ has full support when expressed in the αe. This readily implies that T has finitely many orbits on the ∗ Σ ∗ (1) n−3 complement of (F ) × F . However, for Dn with n odd, 2 edge characters get coefficient 0 when δ is expressed in them. Therefore, T still has infinitely many orbits on said complement.

(1) (iii) The characters αe (e ∈ Σ) are linearly dependent. This is the case only for Aodd, ∗ Σ ∗ and in fact δ is then also in the span of the αe. Now, the T -orbits in (F ) × F have codimension 2, and in the complement there are still inﬁnitely many orbits.

This gives some feeling for the parameter space X. It would be interesting to determine exactly all isomorphism types of Lie algebras L(g) with g ∈ X. However, here we Σ confine ourselves to those with g in some open dense subset of F × F. The proof is very similar to that of Theorem 4.11. Again, we first prove a lemma 0 0 0 that turns out to describe the generic L(g). Therefore, retain the notation ex, fx , hx ∈ g for x ∈ Π0. Moreover, denote the lowest weight in Φ0 by θ, let e0 , f 0 ∈ g be the − x0 x0 86 Chapter 4. Constructing simply laced Lie algebras from extremal elements elements of the Chevalley basis of weights θ and −θ, respectively, set h0 := e0 f 0 , x0 x0 x0 and define 0 0 0 ∼ C := {(cx)x∈Π | ∀x∈Π : cx ∈ hex, fx , hxi = sl2 is extremal}.

This is an irreducible variety.

Lemma 4.14 For c = (cx)x∈Π in some open dense subset of C the cx generate g. More- over, ∀ : x ∼ y ⇒ g 6= 0. x,y∈Π cxcy

0 Proof. The ﬁrst statement is true for c = (ex)x∈Π. This follows from the properties of the Chevalley basis. Hence, by the irreducibility of C it is true for c in some open dense subset of C. The second statement follows, as in Lemma 4.12, from the same statement in sl3.

In the following proof we will show that the choice of (cx)x∈Π as in Lemma 4.14 already (1) gives generic points in X, except for the case where Γ is of type Aodd. For this case we give another construction.

Proof of Theorem 4.13. By Theorem 4.9 and Proposition 4.10, any g ∈ X is determined by its values gx(m) with x ∈ Π and monomials m of weights β ∈ Θ such that either β has height 1 or hαx, βi ≤ −2. In contrast with the case of ﬁnite-type diagrams, there do exist pairs (x, β) ∈ Π × Θ with this latter property, namely precisely those of the form (x, δ −αx). For all x ∈ Π, let mx be a monomial that spans the weight space in L(0) of weight δ − αx. This space is 1-dimensional by Theorem 4.9. We claim that the f (m ) can all be expressed in terms of f (m ) and values f (r) with z ∈ Π and r of x x x0 x0 z degree less than h − 2. Indeed, if x 6= x0, then x0 occurs exactly once in mx. Writing 0 mx = xd ··· x1x0ye ··· y1 with x1, . . . , xd, y1, . . . , ye ∈ Π we ﬁnd

gx(mx) = g(x, xd ··· x1x0ye ··· y1) d+1 = (−1) g(x0x1 ··· xdx, ye ··· y1) d = (−1) g(x0, (x1 ··· xdx)(ye ··· y1)) = (−1)dg ((x ··· x x)(y ··· y )), x0 1 d e 1 and the expression (x ··· x x)(y ··· y ) can be rewritten in terms of m and shorter 1 d e 1 x0 monomials, using values gz(r) with r of degree less than d + e = h − 2. We have now found a closed embedding X → Σ × , g 7→ (g (y)) , g (m ) . F F x {x,y}∈Σ x0 x0 4.4. The parameter space and generic Lie algebras 87

Σ For ease of exposition we will view X as a closed subset of F × F. The theorem Σ follows once we can realise generic parameter values in F × F with extremal elements that generate g. To this end, choose a generic tuple (cx)x∈Π in C as described in Lemma 4.14. The cx (x ∈ Π) generate g and satisfy ∀ : x ∼ y ⇒ g 6= 0. x,y∈Π cxcy Hence, they yield a point in X with

∀ : x ∼ y ⇒ g = g 6= 0. x,y∈Π xy cxcy Furthermore, the parameter g (m ) equals the extremal form evaluated on c and the x0 x0 x0 monomial m evaluated in the c . Express that monomial in the c as x0 x x

ξe0 + ηf 0 + ζh0 x0 x0 x0 plus a term perpendicular to he0 , f 0 , h0 i, x0 x0 x0 and write c as x0 ξ0e0 + η0f 0 + ζ0h0 . x0 x0 x0 0 For the degenerate case where cx = ex for all x we have ξ = ζ = 0 and η 6= 0 (that monomial is a non-zero scalar multiple of the highest root vector f 0 ). So, g (m ) = x0 x0 x0 g(e0 , ηf 0 ) 6= 0. Therefore, this parameter is non-zero generically. Hence, we have x0 x0 ∗ Σ ∗ found a point g ∈ X ∩ ((F ) × F ). In particular,

∗ Σ ∗ X ∩ ((F ) × F ) 6= ∅.

Along the lines of previous remarks we now distinguish three cases:

∗ Σ (i) If the αe(e ∈ Σ) and δ are linearly independent, then T acts transitively on (F ) × ∗ F and we are done.

(ii) If the αe(e ∈ Σ) are linearly independent, but δ lies in their span, then we show that we can alter the point g above in a direction transversal to its T -orbit. Let R be the torus in the adjoint group of g whose Lie algebra is h, and consider the effect on g of conjugation of c with an element r ∈ R, while keeping the other c ﬁxed. This x0 x transforms c in x0 rθξ0e0 + r−θη0f 0 + ζ0h , x0 x0 x0 and therefore it transforms g (m ) into x0 x0

rθξ0η g(e0 , f 0 ) + r−θη0ξ g(f 0 , e0 ) + ζ0ζ g(h0 , h0 ), x0 x0 x0 x0 x0 x0 88 Chapter 4. Constructing simply laced Lie algebras from extremal elements while it keeps the parameters g (y) with x ∼ y unchanged: these only depend on ζ0. x0 0 This shows that we can indeed move g inside X in a direction transversal to its T -orbit. Now, using the fact that F is algebraically closed, we are done.

(1) (iii) Finally, in the case of An−1 with n even we ﬁrst show that tuples in C only give Σ points in a proper closed subset of F × F. Here, g = sln(F) and Γ is an n-cycle whose 0 0 0 points we label points 0, . . . , n−1. Relative to the usual choices of ei , fi , hi the element ci is a matrix with 2 × 2-block 2 αiβi αi 2 −βi −αiβi on the diagonal in rows (and columns) i and i + 1 and zeroes elsewhere. We count the rows and columns modulo n so that row 0 is actually row n. Then we have g(ci, ci+1) = αiβiαi+1βi+1 and

g(c1, c2)g(c3, c4) ··· g(cn−1, c0)

= (α1β1)(α2β2)(α3β3)(α4β4) ··· (αn−1βn−1)(α0β0)

= (α0β0)(α1β1)(α2β2)(α3β3) ··· (αn−2βn−2)(αn−1βn−1)

= g(c0, c1)g(c2, c3) ··· g(cn−2, cn−1). (4.1)

So, the tuple of parameter values of the tuple (ci)i∈[0,n−1] ∈ C lies in a proper closed Σ subset W of F × F. 0 Therefore, we allow the tuple (ci)i∈[0,n−1] to vary in a slightly larger variety C ⊃ C as follows: the conditions on c1, . . . , cn−1 remain the same, but c0 is now allowed to take the shape  2  −α0β0 0 ... 0 −β0  γ α 0 ... 0 γ β   2 0 2 0   . . . .   . . . .    γn−1α0 0 ... 0 γn−1β0 2 α0 0 ... 0 α0β0 (which is extremal since it has rank 1 and trace 0), subject to the equations

∀i∈[2,n−2] : βiγi + αiγi+1 = 0, (4.2) which ensure that c0 commutes with c2, . . . , cn−2. Still, any tuple in an open neigh- 0 bourhood U ⊆ C of our original tuple (ci)i∈[0,n−1] (with generic αi and βi but all γi equal to 0) generates sln. We now argue that the differential d at (ci)i∈[0,n−1] of the map Σ U → X ⊆ F × F sending a tuple to the parameters that it realises has rank |Σ| + 1, as required. Indeed, the T -action already gives a subspace of dimension |Σ| − 1, tangent to 2 2 W . Making γ2 (and hence all γi) non-zero adds −α1γ2α0 to g(c0, c1) and βn−1γn−1b0 4.5. Notes 89

to g(c0, cn−1), and it ﬁxes all other g(ci, cj). This inﬁnitesimal direction is not tangent to W : it adds

2 βn−1γn−1β0(α1β1)(α2β2) ··· (αn−3βn−3)(αn−2βn−2) to the left-hand side of (4.1), and

2 −α1γ2α0(α2β2)(α3β3) ··· (αn−2βn−2)(αn−1βn−1) to the right-hand side. Dividing these expressions by common factors, the ﬁrst becomes βn−1γn−1β0β1 and the second −α1γ2α0αn−1. These expressions are not equal generically, even modulo the equations (4.2) relating the γi to the αi and βi. Indeed, these equations do not involve α0, α1, αn−1, β0, β1, βn−1. Note that varying γ may also effect the parameter g (m ), but in any case the 2 x0 x0 Σ above shows that the composition of the differential d with projection onto F is surjective. On the other hand, conjugation with the torus S as in case (4.4.5) yields a vector in the image of d which is supported only on the factor F corresponding to δ. This concludes the proof that d has full rank.

4.5 Notes

4.5.1 Recognising the simple Lie algebras Going through the proof that

X = {g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))} is an affine variety, one observes that the map g 7→ g|V is not only injective on X, but even on X0 := {g ∈ (F ∗)Π | ∀x ∈ Π: x 6= 0 in L(g)} ⊇ X. The same is true for the map g 7→ (gx(y)){x,y}∈Σ in the case where Γ is a Dynkin diagram of finite type, and for the map g 7→ (f (y)) , g (m ) in the case x {x,y}∈Σ x0 x0 where Γ is a Dynkin diagram of affine type. This shows that, for these Dynkin diagrams, X0 is actually equal to X. Hence, we obtain the following theorem.

Theorem 4.15 Suppose that Γ is a Dynkin diagram of finite or affine type. Let L be any Lie algebra, over a field of characteristic not two, generated by extremal elements cx, x ∈ Π, with ∀x,y∈Π : x 6∼ y ⇒ cxcy = 0. ∗ Π Define g ∈ (F ) by the condition that cxcxu = 2gx(u)cx holds in L. Then g ∈ X and L is a quotient of L(g). 90 Chapter 4. Constructing simply laced Lie algebras from extremal elements

This theorem could well prove useful for recognising the Chevalley algebras g: if g Σ corresponds to a point in the open dense subset of F × F referred to in Theorem 4.13, then one concludes that L is a quotient of g. Hence, if g is a simple Lie algebra, then L is isomorphic to g. It is not clear to us whether, for general Γ, the image of X0 in (V ∗)Π is closed. This is why we chose to work with X instead.

4.5.2 Other graphs Our methods work very well for Dynkin diagrams, but for more general graphs new ideas are needed to determine L(0),X, and L(g) for g ∈ X. The relation with the Kac-Moody algebra of Γ may be much tighter than we proved in Section 4.3.2. General questions of interest are:

• Is X always an afﬁne space?

• Is there always a generic Lie algebra?

We expect the answers to both questions to be negative, but do not have any counterex- amples. The references in ’t panhuis et al. (2009), Postma (2007), and Roozemond (2005) contain other series of graphs which exhibit the same properties as we have proved here: the variety X is an affine space, and generic points in it correspond to Chevalley algebras of types An,Bn,Cn,Dn. In fact, the graph that they find for Cn is just the finite-type Dynkin diagram of type A2n. This also follows from our results: take 2n generic extremal elements (ci)i∈[2n] in sl2n+1(F) as in Lemma 4.12. These generate a subalgebra 2n+1 of sl2n+1(F) of dimension 2 by that same lemma, and their images span a subspace 2n+1 W of dimension 2n in F . It is not hard to write down an explicit, non-degenerate symplectic form f on W with respect to which

∀i∈[2n]∀ 2n+1 : f(ci(p), q) + f(p, ci(q)) = 0. p,q∈F

Hence, the Lie algebra generated by them is sp2n(F).

4.5.3 Geometries with extremal point set

Consider again the Lie algebra sp2n(F) generated by 2n extremal elements (ci)i∈[2n] over the ﬁeld F with

∀i,j∈[2n] : |i − j| > 1 ⇔ cicj = 0. (4.3)

Since the characteristic is not two, this Lie algebra is simple. Moreover, results by Postma (2007) imply that there are no special or strongly commuting pairs of extremal points in the extremal point set E of sp2n(F). Hence, because of Theorem 3.2, the set E 4.5. Notes 91

is the point set of a non-degenerate polar space (E, F) generated by 2n points (Ci)i∈[2n] with ∀i,j∈[2n] : |i − j| > 1 ⇔ Ci is not collinear with Cj. (4.4)

Cohen et al. (2001) proved that sp2n(F) cannot be generated by less than 2n elements. Hence, {Ci | i ∈ [2n]} is a basis of (E, F). Thus, since the symplectic polar space Sp(V, f) with (V, f) a symplectic space of dimension 2n over F also has a basis {Ci | i ∈ [2n]} satisfying (4.4), the non-degenerate polar space (E, F) must be isomorphic to Sp(V, f). Conversely, suppose (E, F) =∼ Sp(V, f) with (V, f) a symplectic space of dimension 2n over F is a polar space coming from a Lie algebra g with extremal point set E as described in Chapter3. Then there must be a set of extremal elements {ci | i ∈ [2n]} satisfying (4.3) and generating g. As we already saw in Section 4.5.2, the Lie algebra g is isomorphic to sp2n(F). Summarizing, we see that a ﬁnite non-degenerate polar space isomorphic to Sp(V, f) for some symplectic space (V, f) of dimension 2n over F can be constructed from a ∼ simple Lie algebra g using the construction of Chapter3 if and only if g = sp2n(F). It remains to be seen whether the same line of reasoning can be followed for the other ﬁnite classical polar spaces. 92 Chapter 4. Constructing simply laced Lie algebras from extremal elements Chapter 5

Classifying the polarized embeddings of a cotriangular space

5.1 Introduction

A delta space is a partial linear space in which each point not on a line is collinear with no, all but one, or all points of that line (Higman 1983). Cuypers (2007) showed how delta spaces can be embedded in projective spaces using so-called polarized embeddings. These embeddings map lines into lines and hyperplanes into hyperplanes. In addition, Cuypers gave a geometric characterization of these polarized embeddings assuming the number of points on a line is at least four. In this chapter, we take a look at delta spaces in which each line contains three points and in which a point not on a line is collinear with no or two points of that line, that is, we consider cotriangular spaces. To be more specific, we consider irreducible cotriangular spaces. These are examples of Fischer spaces which do not contain affine planes. Shult (1974) and Hall (1989) proved that such an irreducible cotriangular space is of one of three possible types: triangular, symplectic, or orthogonal. Moreover, Hall (1983) classified all polarized embeddings into a projective space over the field F2: for each finite irreducible cotriangular space there is a universal embedding over F2 such that each polarized embedding over F2 is a quotient of this embedding. In this chapter we generalize this result to arbitrary fields. We start with the definition of a polarized embedding and some examples. Then we formulate criteria which can be used to determine whether the quotient of a polarized embedding is also polarized. Subsequently, we determine the possible dimensions of a polarized embedding. Finally we consider each of the different types of irreducible cotriangular spaces separately and describe their polarized embeddings. Theorems 5.9 and 5.11 describe the polarized embeddings using the root systems of type An (n > 4) for the spaces of triangular type. For the spaces of symplectic or 94 Chapter 5. Classifying the polarized embeddings of a cotriangular space orthogonal type we have to distinguish between characteristic two and not two. If the characteristic is not two, Theorems 5.22 and 5.32 describe the polarized embeddings using the root systems of type E6, E7, and E8. Otherwise, Theorems 5.22, 5.28 and 5.33 describe the polarized embeddings using the associated symplecic and quadratic forms. As a consequence all polarized embeddings of a finite irreducible cotriangular space are essentially known. As a main result we mention the following theorem, which follows from Theorems 5.9, 5.20, 5.27, 5.32, and 5.33.

Theorem 5.1 Each ﬁnite irreducible cotriangular space admitting a polarized embedding over an arbitrary ﬁeld F has a universal embedding over F.

5.2 Polarized embeddings

Here, in this section, we introduce the notion of a polarized embedding, we deﬁne equivalence of polarized embeddings, and give several examples. However, ﬁrst some notation.

5.2.1 Notation For Π a point-line space we write P(Π) to indicate the point set and L(Π) to indicate the line set. Moreover, for V a vector space we write P(V ) to indicate the projective space corresponding to V . If Π or V are obvious from the context we leave them out. Now, let Π be a cotriangular space, let P be a projective space, let φ be a map from P(Π) to P(P), and let x ∈ X ⊆ P(Π). Then we write ⊥ • ∆x instead of x \{x},

•h ...iφ instead of hφ(...)i,

φ • HX instead of h∪x∈X ∆xiφ, and

φ φ • Hx instead of HX if X = {x}. Moreover, we leave out φ in the last two cases if it is clear which φ is meant. Note, since each point not on a line is perpendicular to at least one point of that line, ⊥ ⊥ both ∆x and x are hyperplanes in Π, that is, each line of Π intersects both ∆x and x non-trivially.

5.2.2 Deﬁnition

Let φ be a map from the point set P(Π) of a cotriangular space Π into the point set P(P) of a projective space P corresponding to a vector space V over a ﬁeld F. Then φ is called 5.2. Polarized embeddings 95 a polarized embedding of Π over F into P if

φ is injective, (5.1)

∀l∈L(Π) : hliφ ∈ L(P) with hliφ ∩ φ(Π) = φ(l), and (5.2) ⊥ ∀x∈P(Π) : Hx ∩ φ(Π) ⊆ φ(x ). (5.3)

Clearly, by deﬁnition, a polarized embedding maps lines into lines. Moreover, since for each point x the subspace Hx cannot be the whole projective space (otherwise the intersection Hx ∩ φ(Π) would equal φ(Π)), each hyperplane ∆x, with x a point, is mapped into a hyperplane. If W is a vector space of dimension m over F and if X is a subspace of P(V ) isomorphic to P(W ), then we say that the rank of X is m and we write rank(X) = m. Moreover, we refer to rank(hφ(Π)i) as the dimension dim(φ) of φ. In particular,

dim(φ) ≤ rank(P(V )).

Now, if we speak of a full polarized embedding φ over F into P(V ), then we mean that dim(φ) = dim(V ), that is, hΠiφ = P(V ). (5.4)

5.2.3 Equivalence

Let F be a ﬁeld admitting an automorphism σ. Moreover, let g be a map between two vector spaces V and W over F, then g is called σ-semi-linear or simply semi-linear if

∀ ∀ : g(αx + βy) = ασg(x) + βσg(y). x,y∈V α,β∈F Let Π be a cotriangular space admitting two full polarized embeddings φ and ψ over F into P(U) and P(V ), respectively. Then φ and ψ are called equivalent if there is an invertible semi-linear transformation g : U → V satisfying ψ = g ◦ φ, that is, the diagram φ Π - P(U)

ψ g

- ? P(V ) commutes. Note, since g is invertible, this is indeed an equivalence relation. Moreover, note that the expression g ◦ φ is well deﬁned since g sends a set of the form Fx, x ∈ U, to Fg(x). In other words, g sends points of P(U) to points of P(V ). The polarized embedding φ is called a universal (polarized) embedding of Π over 96 Chapter 5. Classifying the polarized embeddings of a cotriangular space

F if, for every polarized embedding ϕ of Π over F with P(W ) := hΠiϕ, there is a semi-linear transformation g : U → W which satisﬁes ϕ = g ◦ φ, that is, the diagram

φ Π - P(U)

ϕ g

- ? P(W ) commutes and g is not necessarily bijective. Now, since any two universal embeddings are equivalent, we will speak of “the” instead of “a” universal embedding.

5.2.4 Quotient embeddings Let φ be a full n-dimensional polarized embedding of a cotriangular space Π over a field F into a projective space P(V ) such that φ(x) = Fvx for each point x in Π. Then R for each projective point R/∈ φ(Π) we define φ : P(Π) → P(P(V/R)) as the map sending a point x to the point F(vx +R) in P(V/R). This map will be called the quotient embedding of φ with respect to R. By definition, the dimension of a quotient embedding of φ equals n − 1. Do note that a quotient embedding is not necessarily a polarized embedding. If it is, then we call it polarized. We allow ourselves some abuse of notation: for x a point in Π we write φR(x) = φ(x) + R = Fvx + R instead of F(vx + R).

5.2.5 Natural embedding

Let F be a ﬁeld, let n ∈ N be larger than two, and suppose

± M ∈ {Tn+1, HSp2n(2), NO2n+1(2), NO2n(2)}.

If M admits a polarized embedding over F, then we show how to construct one in a natural way. We call this polarized embedding the natural embedding of M over F. For the ﬁeld F2 this coincides with the natural embedding as introduced by Hall (1983). For Π a cotriangular space isomorphic to M, the composition of the isomorphism at hand and the natural embedding of M over F is a polarized embedding of Π over F. We refer to this map as the natural embedding of Π of type M over F.

Triangular type

Suppose M = Tn+1. Then M, being a cotriangular space of type An (see Lemma 1.38), is isomorphic to R(An). This isomorphism induces an n-dimensional polarized embedding of Tn+1 over F sending a point {i, j} of Tn+1 to the point F(i − i+1) of 5.3. The dimension of a polarized embedding 97

n+1 P(V ) with V the hyperplane of F consisting of those vectors whose coordinates add up to zero. We refer to this embedding as the natural embedding of Tn+1 over F.

Symplectic type

Suppose M is of symplectic type. Then, either M = HSp2n(2) or M = NO2n+1(2). ∼ Note that HSp2n(2) = NO2n+1(2). If M = HSp2n(2), we put m := 2n. Otherwise, we put m := 2n + 1. Suppose char(F) = 2. Then, we deﬁne the natural embedding of M over F as the m map sending a point x of M to the point Fx of P(F ). This map is readily checked to be an m-dimensional polarized embedding of M over F. Next, suppose char(F) 6= 2. Then we will see later on that this can only be the case ∼ if n = 3, that is, M = R(E7) is a cotriangular space of type E7. This isomorphism induces a 7-dimensional polarized embedding of M over F. We refer to this embedding as the natural embedding of M over F.

Orthogonal type

± Suppose M is of orthogonal type. Then, M = NO2n(2). ± Suppose char(F) = 2. Then, we deﬁne the natural embedding of NO2n(2) over F ± 2n as the map sending a point x of NO2n(2) to the point Fx of P(F ). This map is readily ± checked to be a 2n-dimensional polarized embedding of NO2n(2) over F. Next, suppose char(F) 6= 2. Then we will see later on that this can only be the ± + case if n ≤ 4 and M ∈ {N O6 (2), NO8 (2)}. Therefore, deﬁne (X6,X7,X8) := (E6,A7,E8). Then, because of Lemmas 1.34–1.38 and Proposition 1.36 we can assume ∼ that M is a cotriangular space of type Xm if M has generating rank m, that is, M = R(Xm). This isomorphism induces an m-dimensional polarized embedding of M over F. We refer to this embedding as the natural embedding of M over F.

5.3 The dimension of a polarized embedding

Let Π be a ﬁnite irreducible cotriangular space of generating rank n and let F be an arbitrary ﬁeld. Then, because of Theorem 1.33 and Proposition 1.36, we know n ≥ 4 and we can assume that  {T } if n ∈ {4, 5},  n+1  {T , NO−(2)} if n = 6, Π ∈ 7 6  {Tn+1, HSpn−1(2)} if n ≥ 7 is an odd integer, and  ± {Tn+1, NOn (2)} if n ≥ 8 is an even integer.

We will prove that a polarized embedding of Π over F is either (n−1)- or n-dimensional. 98 Chapter 5. Classifying the polarized embeddings of a cotriangular space

5.3.1 Triangular type

Proposition 5.2 Let n ≥ 4 be an integer and let F be a ﬁeld. Then a polarized embedding of Tn+1 over F is (n − 1)- or n-dimensional.

Proof. Let V be a vector space over F. Moreover, let φ be a polarized embedding of Tn+1 into P(V ). By deﬁnition, since Tn+1 is generated by n elements, the dimension of φ over F cannot be larger than n. Therefore, it is sufﬁcient to prove that n − 1 is a lower bound. li := {{1, 2}, {2, i}, {1, i}} is a line in Tn+1 for each i ∈ [3, n]. We will prove by induction that for all i ∈ [3, n]

dimhlj | j ∈ [3, i]iφ ≥ i − 1. (5.5)

For i = 3 this statement is trivial. So, let j ∈ [4, n] and assume (5.5) holds for all i ∈ [3, j − 1]. Next, consider the line lj = {{1, 2}, {2, j}, {1, j}}. Since φ is injective,

⊥ ⊥ lj * {j, j + 1} and φ(lj) * φ({j, j + 1} ). (5.6)

However, j−1 [ lm ⊆ ∆{j,j+1}. m=3 Consequently,

hlk | k ∈ [3, j − 1]iφ ⊆ h∆{j,j+1}iφ = H{j,j+1}. (5.7)

Combining (5.6) and (5.7) with (5.3) results in

φ(lj) * hlk | k ∈ [3, j − 1]iφ. Thus,

dimhlk | k ∈ [3, j]iφ ≥ (j − 2) + 1 = j − 1.

This proves (5.5). In particular,

dim(φ) = dimhΠiφ ≥ dimhlj | j ∈ [3, n]iφ ≥ n − 1.

5.3.2 Symplectic type

Proposition 5.3 Let n ≥ 7 be an odd integer and let F be a ﬁeld. Then a polarized embedding of HSpn−1(2) over a ﬁeld F is (n − 1)- or n-dimensional. 5.4. Polarized quotient embeddings 99

Proof. First we prove that HSpn−1(2) contains a subspace isomorphic to Tn+1. For this reason, deﬁne  1 if i = 1,   i if i is an odd integer in [n − 1], ai :=  i−2 + i if i is an even integer in [3, n], and  n−2 otherwise.

Observe that ∀i,j∈[n+1] : ai ∼ aj ⇔ |j − i| = 1.

The same holds if we replace ai and aj by {i, i + 1} and {j, j + 1}, respectively. Hence, the map sending ai to {i, i + 1} induces an isomorphism of hai | i ∈ [n + 1]i with Tn+1. Now, let φ be a polarized embedding of HSpn−1(2) over F. Since HSpn−1(2) is generated by n elements, the dimension of φ over F is upper bounded by n. Moreover, the dimension is lower bounded by n − 1 because by Proposition 5.2 each polarized embedding of Tn+1 is lower bounded by n − 1.

5.3.3 Orthogonal type

Proposition 5.4 Let F be a ﬁeld, let n ≥ 6 be an even integer, let δ ∈ {+, −}, and δ suppose (n, δ) 6= (6, +). Then, a polarized embedding of NOn(2) over F is (n − 1)- or n-dimensional. Proof. Deﬁne x := n−1 + n. δ ∼ Then x is a point in NOn(2) with ∆x = HSpn−2(2). δ δ Now, let φ be a polarized embedding of NOn(2) over F. Since NOn(2) is generated by n elements, the dimension of φ is upper bounded by n. Moreover, the dimension is lower bounded by n−1 since by Proposition 5.3 the dimension of φ| is lower bounded ∆x by n − 2.

5.4 Polarized quotient embeddings

Let Π be a finite irreducible cotriangular space of generating rank n and let F be an arbitrary field. Then, as we have seen in Section 5.3, a polarized embedding of Π over F is either (n − 1)- or n-dimensional. First, we formulate criteria that can be used to determine whether a quotient embedding is polarized. Then, we assume that the universal embedding u over F exists and we show that each (n − 1)-dimensional polarized embedding of Π over F is equivalent to a quotient embedding of u. The n-dimensional polarized embeddings of Π over F are by definition equivalent with u. 100 Chapter 5. Classifying the polarized embeddings of a cotriangular space

5.4.1 Polarizing criteria Here, we derive the criteria that can be used to determine whether a quotient embedding is polarized. Recall that ∼ is used to indicate that two distinct points are collinear.

Proposition 5.5 Let Π be a ﬁnite irreducible cotriangular space of generating rank n and suppose Π admits a full n-dimensional polarized embedding φ over a ﬁeld F into a R projective space P(V ). Moreover, let R be such that φ is a quotient embedding. Then φR is polarized if and only if

∀x,y,z∈P(Π) : x ∼ y ⇒ R/∈ hx, y, ziφ, and (5.8)

∀x,y∈P(Π) : x ∼ y ⇒ R/∈ h∆x ∪ {y}iφ \ Hx. (5.9)

Proof. φR is a polarized embedding if and only if (5.1) to (5.3) hold (with φ replaced by φR). Thus, it is sufﬁcient to prove

(5.8) ⇒ (5.1), (5.1) ⇒ (5.2a), ((5.1) ∧ (5.2b)) ⇔ (5.8), and ¬(5.3) ⇔ ¬(5.9).

Here,

∀ : hli R ∈ L( (V/R)) l∈L(Π) φ P , and (5.2a) R R ∀ : hli R ∩ φ (Π) ⊆ φ (l) l∈L(Π) φ .(5.2b) are used to indicate the ﬁrst and the second part of (5.2).

(5.8) ⇒ (5.1). Let x, y be two different points of Π. Then using (5.8) we obtain

R/∈ hx, yiφ,R * φ(x) + φ(y), and φ(x) + R 6= φ(y) + R.

Thus φR is injective and (5.1) holds.

(5.1) ⇒ (5.2a). Let x, y, z be three points forming a line l in Π. Since φ is a polarized embedding it follows that φ(z) ⊆ φ(x) + φ(y). This implies

φR(z) = φ(z) + R ⊆ φ(x) + R + φ(y) + R = φR(x) + φR(y).

R R hli R = hx, yi R (V/R) φ (x) 6= φ (y) We conclude that φ φ is a line of P provided that . Thus, (5.1) implies (5.2a).

((5.1) ∧ (5.2b)) ⇔ (5.8). Since (5.1) is implied by (5.8) we can assume that φR is injective. Now, we need to prove (5.2b) ⇔ (5.8). First of all note that (5.2a) holds, that is,

∀ : hli R = hx, yi R . {x,y,u}∈L(Π) φ φ 5.4. Polarized quotient embeddings 101

Therefore,

 R φ (z) ∈/ hli R = hx, yi R  φ φ   ⇔   φ(z) + R * φ(x) + R + φ(y) + R = φ(x) + φ(y) + R ∀l={x,y,u}∈L(Π)∀z∈ /l : ⇔   R * φ(x) + φ(y) + φ(z)   ⇔   R/∈ hx, y, ziφ.

Here, the one but last bi-implication follows from the fact that φ(z) ∈/ hx, yiφ. Thus, assuming that (5.1) holds, we indeed obtain (5.2b) ⇔ (5.8).

¬(5.3) ⇔ ¬(5.9). If x, y are two collinear points in Π, then φ(y) ∈/ Hx and X φ(y) * φ(z). z∈∆x Consequently,

 R φR  φ(y) + R = φ (y) ∈ Hx   ⇔ P ∀x,y∈P(Π) : x ∼ y ⇒ φ(y) ⊆ φ(y) + R ⊆ φ(z) + R z∈∆x  ⇔   R ⊆ (P φ(z) + φ(y)) \ P φ(z). z∈∆x z∈∆x

In other words, (5.3) does not hold if and only if (5.9) does not hold.

5.4.2 Equivalence The following proposition is of great help if we want to determine the equivalence classes of those polarized embeddings which do not have the maximal dimension.

Proposition 5.6 Let Π be a ﬁnite irreducible cotriangular space of generating rank n and suppose Π admits a full n-dimensional universal embedding u over a ﬁeld F into P(V ) and a full (n − 1)-dimensional polarized embedding φ over F into P(W ). Then φ is equivalent to a quotient embedding of u.

A direct consequence of this proposition is the following corollary.

Corollary 5.7 If Π admits a full n-dimensional universal embedding u over a ﬁeld F R into P(V ) and if there is at most one point R in P(V ) such that u is polarized, then all (n − 1)-dimensional polarized embeddings of Π over F are equivalent. 102 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Proof of Proposition 5.6. We choose vx ∈ V , x ∈ P(Π), such that u(x) = Fvx for all points x of Π. Since u is the universal embedding there is a σ-semi-linear transformation g : V → W such that φ = g ◦ u. We deﬁne wx := g(vx) for all points x in Π. As a consequence,

∀x∈P(Π) : φ(x) = Fwx.

Next, we choose a generating set {xi | i ∈ [n]} for Π and we deﬁne

∀ :(v , w ) := (v , w ). i∈[n] i i xi xi

Since dim(V ) = 1 + dim(W ) = n, we obtain that {vi | i ∈ [n]} is a linearly independent set of size n and that {wi | i ∈ [n]} is a linearly dependent set of size n. This n implies that there is a subset {αi | i ∈ [n]} of F such that X 0 = αiwi. i∈[n]

Now, deﬁne R := Fr with X σ−1 r := αi vi. i∈[n] Then

X X X σ−1 σ X σ−1 0 = αiwi = αig(vi) = (αi ) g(vi) = g( αi vi) = g(r). i∈[n] i∈[n] i∈[n] i∈[n]

If we can prove that (5.8) and (5.9) hold with φ replaced by u, then uR is polarized. Suppose (5.8) does not hold. Then there are x, y, z with x ∼ y such that R ∈ hx, y, ziu. If in addition z is on the line through x and y, then

R ∈ hx, y, ziu = hx, yiu which implies there are scalars α, β such that

r = αvx + βvy.

Applying g to both sides of the equation results in

σ σ σ σ 0 = g(r) = g(αvx + βvy) = α g(vx) + β g(vy) = α wx + β wy.

This is a contradiction with the injectivity of φ. So, z is not on the line through x and y and there are scalars α, β, γ such that

r = αvx + βvy + γvz. 5.4. Polarized quotient embeddings 103

Applying g to both sides of the equation gives

σ σ σ 0 = g(r) = g(αvx + βvy + γvz) = α g(vx) + β g(vy) + γ g(vz) σ σ σ = α wx + β wy + γ wz.

Since φ is injective, α, β and γ must all be non-zero. We get

φ(z) = Fwz ∈ hFwx, Fwyi = hx, yiφ.

So, z cannot be on the line through x and y. This is in contradiction with (5.2). We conclude that (5.8) does hold. Suppose (5.9) does not hold. Then there is a subset {βa | a ∈ ∆x ∪ {y}} of F with βy 6= 0 such that X r = βava.

a∈∆x∪{y} Applying g to both sides of the equation gives

X X σ X σ 0 = g(r) = g( βava) = βa g(va) = βa wa.

a∈∆x∪{y} a∈∆x∪{y} a∈∆x∪{y}

Since βy 6= 0, we obtain

φ φ(y) = Fwy ∈ hFwa | a ∈ ∆xi = hφ(a) | a ∈ ∆xi = Hx .

However, φ is a polarized embedding. Hence, we have found a contradiction with (5.3). We conclude that (5.9) does hold. Thus uR is indeed a polarized embedding. It remains to be proven that uR is equivalent to φ. First, recall that

X σ−1 r = αi vi. i∈[n]

This implies that there is an index m ∈ [n] such that αm 6= 0. Without loss of generality we assume m = n. In other words, X X V = Fvi + R and W = Fwi. i∈[n−1] i∈[n−1] Now, deﬁne h : V/R → W as the map sending v + R ∈ V/R to g(v). This map is well deﬁned since g(r) = 0. Moreover, a straightforward check shows that h is both invertible and 1-semi-linear. Thus, since

R h ◦ u (x) = h(F(vx + R)) = Fh(vx + R) = F(g(vx)) = Fwx = φ(x),

R u is a polarized embedding equivalent to φ. 104 Chapter 5. Classifying the polarized embeddings of a cotriangular space

5.5 Equivalence of polarized embeddings: triangular type

Let Π be a cotriangular space of triangular type generated by n points and let F be an ∼ arbitrary ﬁeld. Then n ≥ 4 and Π = Tn+1. We identify Π with Tn+1 and we prove that the natural embedding of Tn+1 over F is the universal embedding. Furthermore, we take a look at the equivalence classes of the (n − 1)-dimensional polarized embeddings of Tn+1 over F.

5.5.1 Characterizing the polarized embeddings Here, we give a characterization of the polarized embeddings of a cotriangular space of triangular type.

Proposition 5.8 Let n ≥ 4 and suppose φ is a polarized embedding of Tn+1 over F into a projective space P(V ). Then there is a subset {ei | i ∈ [n + 1]} of V with e1 = 0 such that each point {i, j} of Tn+1 is sent to F(ei − ej).

Proof. Clearly, there is a subset {fi | i ∈ [2, n + 1]} of V \{0} such that

∀i∈[2,n+1] : φ({1, i}) = Ffi.

Moreover, since φ maps lines into lines,

∀i∈[2,n] : φ({i, i + 1}) ∈ h{1, i}, {1, i + 1}iφ = {F(αfi + βfi+1) | α, β ∈ F}.

∗ Now, deﬁne f1 := 0. Since φ is injective, there is a subset {αi | i ∈ [2, n + 1]} of F such that ∀i∈[2,n] : φ({i, i + 1}) = F(fi − αi+1fi+1). Next, deﬁne i Y ∀i∈[n+1] : ei := βifi with βi := αj. j=2 Then

∀i∈[2,n] : φ({1, i + 1}) = Ffi+1 = Fβifi+1 = Fei+1 = F(e1 − ei+1), and

∀i∈[2,n] : φ({i, i + 1}) = F(fi − αi+1fi+1) = F(βifi − βiαi+1fi+1) = F(ei − ei+1).

Now, we can prove the lemma using induction. Therefore, let l ∈ [2, n] and assume φ({i, j}) = F(ei − ej) for all distinct i, j ∈ [n + 1] with j − i < l. Next, ﬁx distinct i, j ∈ [n + 1] with j − i = l. We need to prove φ({i, j}) = F(ei − ej). 5.5. Equivalence of polarized embeddings: triangular type 105

If ei, ei+1 and ei+l are linearly dependent, then

φ({1, i}) = Fei ∈ hFei+1, Fei+li = h{1, i + 1}, {1, i + l}iφ.

This is a contradiction with (5.2) and the fact that {1, i} is not on the line generated by {1, i + 1} and {1, i + l}. Hence, ei, ei+1 and ei+l must be linearly independent. Because of this linear independence,

φ({i, i + l}) ∈ h{i, i + 1}, {i + 1, i + l}iφ ∩ h{1, i}, {1, i + l}iφ = hF(ei − ei+1), F(ei+1 − ei+l)i ∩ hFei, Fei+li = {F(α(ei − ei+1) + β(ei+1 − ei+l)) | α, β ∈ F} ∩ {F(γei + δei+l) | γ, δ ∈ F} = {F(αei + (β − α)ei+1 − βei+l) | α, β ∈ F} ∩ {F(γei + δei+l) | γ, δ ∈ F}. = {F(αei − αei+l) | α ∈ F} = {F(ei − ei+l)}.

Thus, indeed,

φ({i, j}) = φ({i, i + l}) = F(ei − ei+l) = F(ei − ej).

5.5.2 The universal embedding

Theorem 5.9 Let n ≥ 4 be an integer and F a ﬁeld. Then the natural embedding u of Tn+1 over F is the universal embedding of Tn+1 over F.

n+1 Proof. Let V be the subspace of F consisting of all vectors whose coordinates add up to zero and let φ be a full polarized embedding of Tn+1 over F into P(W ). This implies (see Proposition 5.8) that there is a subset {ei | i ∈ [n + 1]} of W with e1 = 0 such that φ sends each point {i, j} of Tn+1 to F(ei − ej). Now, deﬁne g : V → W as the linear map which is induced by sending each simple root ai = i − i+1 of the root system of type An to ei − ei+1. Then g is surjective and

j−1 X g ◦ u({i, j}) = g(u({i, j})) = g(F(i − j)) = Fg(i − j) = Fg( am) m=i j−1 j−1 X X = F g(am) = F (em − em+1) = F(ei − ej) = φ({i, j}). m=i m=i for all points {i, j} of Tn+1. In other words, u is the universal embedding of Tn+1 over F. 106 Chapter 5. Classifying the polarized embeddings of a cotriangular space

5.5.3 Quotient embeddings Here, we translate the criteria for a quotient embedding to be polarized, as formulated in Proposition 5.5, to the current setting.

Proposition 5.10 Let u be the universal embedding of Tn+1 over a ﬁeld F into P(V ), n+1 where V is the hyperplane of F consisting of all vectors whose coordinates add up to R zero, and suppose there is a projective point R = Fr making u a quotient embedding of u. Then uR is polarized if and only if

(a) the coefﬁcients of r with respect to the standard basis are non-zero, and

(b) there are no two coefﬁcients which add up to zero if n = 4.

Proof. Suppose uR is polarized. Then at least one of the coefficients of r with respect to the standard basis is non-zero. Moreover, suppose that not all of the coefficients are non- zero. Then we assume that the first coefficient is non-zero and that the last coefficient is zero. Otherwise we can proceed in a similar way. Now, X X r ∈ F(i+1 − i) \ F(i+1 − i). i∈[n−1] i∈[2,n−1]

Hence,

R = Fr ∈ hF(i − i+1) | i ∈ [n − 1]i = h{i, i + 1} | i ∈ [n − 1]iu = h{{1, 2}} ∪ {{i, i + 1} | i ∈ [2, n − 1]}iu = h{{1, 2}} ∪ ∆{1,n+1}}iu, but

R = Fr∈ / hF(i − i+1) | i ∈ [2, n − 1]i = h{i, i + 1} | i ∈ [2, n − 1]iu = h∆{1,n+1}iu = H{1,n+1}.

This is in contradiction with Proposition 5.5. Hence, all coefﬁcient of r with respect to the standard basis are non-zero. Next, assume n = 4 and suppose the i-th and j-th coefﬁcient of r add up to zero, where i, j ∈ [n + 1] with i < j. Then there are a, b, c ∈ [n + 1] such that {i, j, a, b, c} = [n + 1] and

Fr ∈ hF(i − j), Fa, Fb, Fci = hF(i − j), F(a − b), F(b − c), Fci.

The coefﬁcients of each element of V add up to zero. Consequently,

R = Fr ∈ hF(i − j), F(a − b), F(b − c)i = h{i, j}, {a, b}, {b, c}iu. 5.5. Equivalence of polarized embeddings: triangular type 107

This is in contradiction with Proposition 5.5. Hence, no two coefﬁcients of r add up to zero. Conversely, suppose R = Fr satisﬁes (a) and (b) and let {i, j} and {a, b} be two points collinear in Tn+1. Moreover, let c ∈ [n + 1] such that {c} = {a, b}\{i, j} and suppose R ∈ h{{i, j}} ∪ ∆{a,b}iu. Then

R = Fr ∈ h{{i, j}} ∪ ∆{a,b}iu = hF(l − m) | l, m ∈ [n] \{c} ∧ l < mi.

In particular, the c-th coefficient of r is zero. This is in contradiction with (a). We conclude R/∈ hu({{i, j}} ∪ ∆{a,b})i. Thus, the second statement of Proposition 5.5 holds. Now, if n > 4, then it is readily checked that the first statement of Proposition 5.5 cannot be violated. Therefore, we assume n = 4 and we consider points x, y, and z = {i, j} in Tn+1 with x ∼ y. If x ∼ z or y ∼ z, then |x ∪ y ∪ z| ≤ 4. Otherwise, z ⊆ [n + 1] \ (x ∪ y). In the former case each point of hx, y, ziu has a coefficient equal to zero. In the latter case the i-th and the j-th coefficient of each point of hx, y, ziu add up to zero. This is in contradiction with (b). Thus, R cannot be an element of hx, y, ziu R and the first statement of Proposition 5.5 holds. In particular, u is polarized.

5.5.4 The equivalence classes Here, we determine the equivalence classes of the polarized embeddings of a cotriangular space of triangular type which are not of maximal dimension by close examination of the polarized quotient embeddings.

Theorem 5.11 Let u be the universal embedding of Tn+1 over a ﬁeld F into P(V ), where n+1 V is the hyperplane of F consisting of all vectors whose coordinates add up to zero. Then a polarized quotient embedding of u exists if and only if

• n = 4 and F ∈/ {F2, F3, F4, F8},

• n > 4, n is even and F 6= F2, or • n > 4 and n is odd. Moreover, two polarized quotient embeddings uR and uS are equivalent if and only if there is a ﬁeld automorphism σ of F and and two sets {αi | i ∈ [n]}, {βi | i ∈ [n]} such that X X (r, s) = ( αi(i − i+1), βi(i − i+1)) ∈ (R \{0}) × (S \{0}) i∈[n] i∈[n] with σ ∀i∈[n] : αi = βi. 108 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Fix u as the universal embedding of Tn+1 over a ﬁeld F into P(V ), where V is the n+1 hyperplane of F consisting of all vectors whose coordinates add up to zero. We prove the proposition step by step. First, we consider the quotient embeddings over F2 for even n.

Lemma 5.12 Suppose F = F2 and n is even. Then none of the quotient embeddings of u are polarized. Proof. Because of Proposition 5.10 we can conclude that the only possibility for r is the sum n+1 X w := i i=1 of all basis elements. Since n + 1 is odd, the coefﬁcients of w do not add up to zero. As a consequence w∈ / V whereas r ∈ V . Thus, none of the quotient embeddings of u are polarized.

Lemma 5.13 Suppose n = 4. Then there exists a polarized quotient embedding of u over F if and only if F ∈/ {F2, F3, F4, F8}. Proof. For each field we check whether there exists an R such that the quotient embed- R ding u is polarized. For F ∈ {F2, F3, F4, F8} we assume R = Fr exists. Because of Proposition 5.10 this implies that all coefficients of r with respect to the standard basis are non-zero. Moreover, no two coefficients add up to zero. Using this we derive a contradiction. For F ∈/ {F2, F3, F4, F8} we find an explicit r ∈ V such that all coefficients of R = Fr with respect to the standard basis are non-zero and no two coefficients add up to zero. Again using Proposition 5.10 we then find that uR is a polarized embedding.

F2. Particular case of Lemma 5.12.

F3. F3 contains only two distinct non-zero elements. So, at least three of the coefﬁ- cients of r are equal. Necessarily, they add up op to zero. But then the remaining two coefﬁcients also add up to zero. This is the required contradiction.

F4. F4 contains only three distinct non-zero elements. So assuming that all coefficients of r are non-zero we find that at least two of the coefficients are equal. We are working in even characteristic. Hence, they must add up to zero. This is the required contradiction.

F8. F8 has even characteristic and therefore no two coefﬁcients of r are equal. As a consequence F8 must have ﬁve distinct non-zero elements adding up to zero. However, the seven distinct non-zero elements of F8 also add up to zero. This implies that F8 has two distinct non-zero elements adding up to zero. F8 does not contain two such elements. This is the required contradiction. 5.5. Equivalence of polarized embeddings: triangular type 109

F ∈/ {F2, F3, F4, F8}. If F has odd characteristic, then 2 6= 0 and −4 6= 0. Otherwise, there is an ω ∈ F such that

|{0, 1, ω, ω2, ω3, ω3 + ω2 + ω + 1}| = 6.

Now, deﬁne R := Fr with 1 + 2 + 3 + 4 − 45 if char(F) 6= 2, and r := 2 3 3 2 1 + ω2 + ω 3 + ω 4 + (ω + ω + ω + 1)5 otherwise.

The coefficients of r add up to zero and therefore r ∈ V . Moreover, all coefficients of r are non-zero and no two coefficients add up to zero. Thus, there exists a quotient embedding of u which is polarized.

Lemma 5.14 Suppose n > 4. Then u has a polarized quotient embedding if and only if n is odd or F 6= F2.

Proof. If n is even and F = F2, then because of Lemma 5.12 we know that u has no polarized quotient embeddings. So, assume F 6= F2 or n is odd. Moreover, if F 6= F2, ∗ then ﬁx an ω ∈ F \{1}. This enables us to deﬁne

 n+1 2  P  (2i−1 − 2i) if n is odd, and i=1 r := n  P2  (2i−1 − 2i) + ω(n − n+1) if n is even. i=1 It is easily checked that the coefﬁcients of r with respect to the standard basis are all non-zero. Moreover, they add up to zero. Thus, we can use Proposition 5.10 to conclude R that u with R = Fr is polarized.

Lemma 5.15 Let uR and uS be two polarized quotient embeddings. Then they are equivalent if and only if there is a ﬁeld automorphism σ of F and two sets

{αi | i ∈ [n]}, {βi | i ∈ [n]} such that X X (r, s) = ( αi(i − i+1), βi(i − i+1)) ∈ (R \{0}) × (S \{0}) i∈[n] i∈[n] with

σ ∀i∈[n] : αi = βi. 110 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Proof. Suppose there is a ﬁeld automorphism σ of F and two sets {αi | i ∈ [n]}, {βi | i ∈ [n]} such that X X (r, s) = ( αi(i − i+1), βi(i − i+1)) ∈ (R \{0}) × (S \{0}) i∈[n] i∈[n]

σ such that αi = βi for all i ∈ [n]. Moreover, deﬁne vi := i − i+1 for all i ∈ [n]. Since dim(V/R) = n − 1, there is an i ∈ [n] such that {vj + R | j ∈ [n] \{i}} is a basis of V/R. Without loss of generality we assume j = n. Otherwise, we can proceed in a similar way. Hence, we can deﬁne g : V/R → V/S as the σ-semi-linear transformation induced by sending vi + R with i ∈ [n − 1] to vi + S. This gives

X −1 X −1 σ σ g(vn + R) = g(− αn αivi + R) = − (αn ) αi vi + S i∈[n−1] i∈[n−1] X −1 = − βn βivi + S = vn + S. i∈[n−1]

Now let {i, j} be a point in Tn+1. Then X u({i, j}) = F(i − j) = F vi. i∈[i,j−1] Consequently, uR and uS are equivalent since

S X X X u ({i, j}) = F( vi + S) = F g(vi + R) = g(F vk + R) i∈[i,j−1] i∈[i,j−1] k∈[i,j−1] R = g(F(i − j) + R) = g(u ({i, j})).

R S Conversely, suppose u and u are equivalent. Then there are subsets {αi | i ∈ [n]} and {βi | i ∈ [n]} of F such that X X (r, s) = ( αi(i − i+1), βi(i − i+1)) ∈ (R \{0}) × (S \{0}) i∈[n] i∈[n] and there is an invertible σ-semi-linear transformation g : V/R → V/S such that

S R ∀i∈[n] : F(1 − i + S) = u ({1, i}) = (g ◦ u )({1, i}) = g(1 − i + R).

This can only be the case if g(S) = R. In particular,

σ ∀i∈[n] : αi = βi.

Proof of Theorem 5.11. Lemmas 5.13, 5.14, and 5.15. 5.6. Equivalence of polarized embeddings: X7 111

5.6 Equivalence of polarized embeddings: X7

Let Π be a cotriangular space of symplectic type generated by 7 points and let F be an ∼ ∼ ∼ arbitrary ﬁeld. Then Π = HSp6(2) = NO7(2) = X7. Therefore, we identify Π with the cotriangular space X7 of Example 1.32. We prove that all polarized embeddings of X7 over F are equivalent with the natural embedding of X7 of type NO7(2) over F if char(F) 6= 2. Moreover, if char(F) = 2, then we prove that there are two equivalence classes: one containing the natural embedding of X7 of type NO7(2) over F, the other containing the natural embedding of X7 of type HSp6(2) over F.

5.6.1 Characterizing the polarized embeddings

Here, we give a characterization of the polarized embeddings of X7: the cotriangular space as described in Example 1.32.

Proposition 5.16 Let φ be a polarized embedding of X7 over a ﬁeld F into a projective space P(V ). Then there is a subset {ei | i ∈ [0, 8]} of V such that X X e1 = 0 = 2e0 + ei − ei i∈[4] i∈[5,8] and X X ∀ : φ(x) = ( e − e ). x∈P(X7) F i i i ∈ x ∩ [0,4] i ∈ x ∩ [5,8]

Fix a ﬁeld F, a projective space P(V ) over F, and a polarized embedding φ of X7 over F into P(V ). Now, we prove the proposition one step at a time.

Lemma 5.17 Suppose there is a subset {fi | i ∈ [0, 8]} of V with f1 = 0, φ({0}) = Ff0, and

∀ : {i, j}= 6 {0} ⇒ φ({i, j}) = (f − f ). {i,j}∈P(X7) F i j

Then, there is a subset {ei | i ∈ [0, 8]} of V with e1 = 0, φ({0}) = Fe0,

∀ : {i, j}= 6 {0} ⇒ φ({i, j}) = (e − e ), {i,j}∈P(X7) F i j and

∀ : φ({0, i, j}) = (e + e − e ). {0,i,j}∈P(X7) F 0 i j

Proof. Let {0, k, l} be a point in X7 with k < l. Then,

φ({0, k, l}) ∈ h{0}, {k, l}iφ = hFf0, F(fk − fl)i. 112 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Consequently, there is a non-zero αk,l ∈ F such that

φ({0, k, l}) = F(f0 + αk,l(fk − fl)).

Let {0, i, j} be another point in X7. We will prove that αi,j = αk,l. Since {{j, l}, {0, i, j}, {0, i, l}}, {{i, k}, {0, i, l}, {0, k, l}} ∈ L(X7), we can assume that {0, i, j} and {0, k, l} are connected: either {i, k} or {j, l} is the third point on the line. We assume {i, k} is the third point. Otherwise, we can proceed analogously. Hence, i 6= k 6= j = l 6= i and

F(fi − fk) = φ({i, k}) ∈ h{0, i, l}, {0, k, l}iφ = hF(f + αi,l(fi − fl)), F(f + αk,l(fk − fl))i.

Consequently, there is a β ∈ F such that

F(fi − fk) = F(f0 + αi,l(fi − fl) + β(f0 + αk,l(fk − fl)) = F((1 + β)f0 + αi,lfi + βαk,lfk − (αi,l + βαk,l)fl).

If β 6= −1, then

φ({0}) = Ff0 ∈ hF(fi − fk), F(fi − fl), F(fk − fl)i ∩ φ(X7) = h{i, k}, {i, l}, {k, l}iφ ∩ φ(X7) = {φ({i, k}), φ({i, l}), φ({k, l})}.

This is in contradiction with the injectivity of φ. Consequently, β = −1 and

F(fi − fk) = F(αi,lfi − αk,lfk − (αi,l − αk,l)fl).

If αi,l 6= αk,l, then

φ({1, l} = Ffl ∈ hF(fi − fk), Ffki ∩ φ(X7) = h{i, k}, {1, k}iφ ∩ φ(X7) = {φ({i, k}), φ({1, k}), φ(1, i)}.

This is in contradiction with the injectivity of φ. Consequently, α := αi,j = αi,l = αk,l. Next, deﬁne e0 := f0 and ei := αfi for all i ∈ [8]. Then,

φ({0}) = Ff0 = Fe0, φ({i, j}) = F(fi − fj) = Fα(fi − fj) = F(αfi − αfj) = F(ei − ej), and φ({0, k, l}) = F(f0 + α(fk − fl)) = F(f0 + αfk − αfl) = F(e0 + ek − el) for all points {i, j} and {0, k, l} in X7 with i < j and k < l. 5.6. Equivalence of polarized embeddings: X7 113

Lemma 5.18 There is a subset {ei | i ∈ [0, 8]} of V such that e1 = 0 and X X ∀ : φ(x) = ( e − e ). x∈P(X7) F i i i ∈ x ∩ [0,4] i ∈ x ∩ [5,8]

Proof. According to Proposition 5.8 there is a subset {fi | i ∈ [8]} such that f1 = 0 and φ({i, j}) = F(fi − fj) for all points {i, j} in X7. Clearly, we also have a non-zero f0 ∈ V such that φ({0}) = Ff0. Thus, Lemma 5.17 gives us a subset {ei | i ∈ [0, 8]} of V with e1 = 0, φ({0}) = Fe0,

∀ : {i, j}= 6 {0} ⇒ φ({i, j}) = (e − e ), {i,j}∈P(X7) F i j and

∀ : {i, j}= 6 {0} ⇒ φ({0, i, j}) = (e + e − e ). {0,i,j}∈P(X7) F 0 i j

Now, consider a point {0, i, j, k, l} in X7 with i < j < k < l. It is sufﬁcient to prove that φ({0, i, j, k, l}) equals F(e0 + ei + ej − ek − el). Since {{i, k}, {0, j, l}, {0, i, j, k, l}} is a line, we obtain

φ({0, i, j, k, l}) ∈ h{i, k}, {0, j, l}iφ = hF(ei − ek), F(e0 + ej − el)i.

∗ Consequently, there is an α1 ∈ F such that

φ({0, i, j, k, l}) = F(α1(ei − ek) + e0 + ej − el) = F(e0 + α1ei + ej − α1ek − el).

In the same way we can use the existence of the lines {{i, l}, {0, j, k}, {0, i, j, k, l}}, {{j, k}, {0, i, l}, {0, i, j, k, l}}, and {{j, l}, {0, i, k}, {0, i, j, k, l}}, to prove the existence of non-zero scalars α2, α3, α4 with

φ({0, i, j, k, l}) = F(e0 + α2ei + ej − ek − α2el), φ({0, i, j, k, l}) = F(e0 + ei + α3ej − α3ek − el), and φ({0, i, j, k, l}) = F(e0 + ei + α4ej − ek − α4el).

If e0, ei, ej, ek, el are linearly independent, then clearly α1 = α2 = α3 = α4 = 1 and φ({0, i, j, k, l}) = F(e + el + ek − ej − ei). Therefore, we assume e0 is linearly dependent of ei, ej, ek, and el. Otherwise we can proceed analogously. This implies there are γ0, γi, γj, γk, γl ∈ F such that e0 = γiei + γjej + γkek + γlel. In particular,

φ({0, i, j, k, l}) = F((α1 + γi)ei + (1 + γj)ej − (α1 − γk)ek − (1 − γl)el), φ({0, i, j, k, l}) = F((α2 + γi)ei + (1 + γj)ej − (1 − γk)ek − (α2 − γl)el), φ({0, i, j, k, l}) = F((1 + γi)ei + (α3 + γj)ej − (α3 − γk)ek − (1 − γl)el), and φ({0, i, j, k, l}) = F((1 + γi)ei + (α4 + γj)ej − (1 − γk)ek − (α4 − γl)el). 114 Chapter 5. Classifying the polarized embeddings of a cotriangular space

First, suppose ei, ej, ek, and el are linearly independent. If 1 + γi 6= 0, then α3 + γj = α4 + γj and α3 − γk = 1 − γk. In other words, α3 = α4 = 1 and φ({0, i, j, k, l}) = F(e0 +ei +ej −ek −el). Therefore, assume 1+γi = 0. This implies α1 +γi = α2 +γi = 0. In other words, α1 = α2 = −γi = 1 and φ({0, i, j, k, l}) = F(e + ei + ej − ek − el). Finally, suppose ei, ej, ek, and el are linearly dependent. It is sufﬁcient to derive a contradiction. The injectivity of φ imply that ej, ek, and el are pairwise linearly independent. Hence, if i = 1, then

φ({1, j}) = Fej ∈ hFek, Feli ∩ φ(X7) = h{1, k}, {1, l}iφ ∩ φ(X7) = {φ({1, k}), φ({1, l}), φ({k, l})}.

This is a contradiction with the injectivity of φ. Consequently, i > 1 and X X 6 ≤ rankhX7iφ = dim( Fei) = dim( Fei) ≤ 6. i∈[0,8] i∈[2,8] P Since 8 ∈/ {j, k, l} we conclude that e8 cannot be an element of i∈[0,7] Fei. However, using the lines {{1, 8}, {0, 2, 7}, {0, 3, 4, 5, 6}} we do obtain

Fe8 = φ({1, 8}) ∈ h{0, 2, 7}, {0, 3, 4, 5, 6}iφ. P In particular, e8 ∈ i∈[0,7] Fei. This is the required contradiction.

Lemma 5.19 Suppose there is a subset {ei | i ∈ [0, 8]} of V such that e1 = 0 and X X ∀ : φ(x) = ( e − e ). x∈P(X7) F i i i ∈ x ∩ [0,4] i ∈ x ∩ [5,8]

Then, X X 2e0 + ei − ei = 0. i∈[4] i∈[5,8] Proof. Clearly, X dim( Fei) = dim(φ) ∈ {6, 7}. i∈[0,8] Moreover, as we have already seen in the proof of Proposition 5.16, we know X e8 ∈ ei. i∈[0,7]

Consequently, X dim(φ) = dim( Fei). i∈[0,7] 5.6. Equivalence of polarized embeddings: X7 115

If φ is 7-dimensional, then the non-zero elements of {ei | i ∈ [0, 7]} are linearly independent. Because of this, the lemma is easier to prove if dim(φ) = 7. Therefore, we only consider the case in which φ is 6-dimensional. This implies there is an i ∈ [2, 7] such that X ei ∈ Fej. j∈[0,7]\{i} We assume i = 7. Otherwise, we can proceed in a similar way. ∗ Fix a subset {γi | i ∈ [0, 6]} of F such that X e7 = γiei. i∈[0,6]

Since ∀i∈[5,7] : {{1, 8}, {0, 2, i}, {0, 3, 4, 5, 6, 7}\{i}} ∈ L(X7), there are non-zero α5, α6, α7 ∈ F such that

Fe8 = φ({1, 8}) = F(e0 + e3 + e4 − e5 − e6 − e7 + ei + αi(e + e2 − ei)) = F((αi + 1)e0 + αie2 + e3 + e4 − e5 − e6 − e7 + (1 − αi)ei)  (α + 1 − γ )e + (α − γ )e + (1 − γ )e  5 0 0 5 2 2 3 3  +(1 − γ )e − (α + γ )e − (1 + γ )e if i = 5,  4 4 5 5 5 6 6    (α + 1 − γ )e + (α − γ )e + (1 − γ )e = 6 0 0 6 2 2 3 3  +(1 − γ4)e4 − (1 + γ5)e5 − (α6 + γ6)e6 if i = 6,     (α7 + 1 − α7γ0)e0 + (α7 − α7γ2)e2 + (1 − α7γ3)e3  +(1 − α7γ4)e4 − (1 + α7γ5)e5 − (1 + α7γ6)e6 if i = 7.

Since φ is 6-dimensional, e0, e2, e3, e4, e5, and e6 are linearly independent. Con- sequently, if 1 − γ3 6= 0, then α5 = α6 = 1. So, assume γ3 = 1. This implies 0 = 1 − α7γ3 = 1 − α7 and α7 = 1. Thus, whether or not 1 − γ5 = 0, it follows that X X Fe8 = F(2e0 + ei − ei). i∈[4] i∈[5,7]

This implies there is a non-zero α ∈ F such that X X e8 = α(2e0 + ei − ei). i∈[4] i∈[5,7]

We show α = 1 using fact that

∀i∈[5,7] : {{0, 1, 8}, {2, i}, {0, 3, 4, 5, 6, 7}\{i}} ∈ L(Π). 116 Chapter 5. Classifying the polarized embeddings of a cotriangular space

In the same way as before we can prove that there are non-zero α5, α6, α7 ∈ F such that

F(e0 − e8) = φ({0, 1, 8}) = F(e0 + e3 + e4 − e5 − e6 − e7 + ei + αi(e2 − ei)) = F(e0 + αie2 + e3 + e4 − e5 − e6 − e7 + (1 − αi)ei)  (1 − γ )e + (α − γ )e + (1 − γ )e  0 0 5 2 2 3 3  +(1 − γ )e − (α + γ )e − (1 + γ )e if i = 5,  4 4 5 5 5 6 6    (1 − γ )e + (α − γ )e + (1 − γ )e = 0 0 6 2 2 3 3  +(1 − γ4)e4 − (1 + γ5)e5 − (α6 + γ6)e6 if i = 6,     (1 − α7γ0)e0 + (α7 − α7γ2)e2 + (1 − α7γ3)e3  +(1 − α7γ4)e4 − (1 + α7γ5)e5 − (1 + α7γ6)e6 if i = 7.

Moreover, in the same way as before, we can prove that α5 = α6 = α7 = 1. This results in X X F(e0 − e8) = F( ei − ei). i∈[0,4] i∈[5,7]

As a consequence X X X X F(e0 − e8) = F( ei − ei) = F(α( ei − ei)) i∈[0,4] i∈[5,7] i∈[0,4] i∈[5,7] X X X = F(α( ei − ei)) − α γiei) i∈[0,4] i∈[5,6] i∈[0,6] X X = F((α − αγ0)e0 + α( (1 − γi)ei − (1 + γi)ei)), and i∈[4] i∈[5,6] X X F(e0 − e8) = F(e0 − α(2e0 + ei − ei)) i∈[4] i∈[5,7] X X X = F((1 − 2α)e0 − α( ei − ei) + α γiei) i∈[4] i∈[5,6] i∈[0,6] X X = F((1 − 2α + αγ0)e0 − α( (1 − γi)ei − (1 + γi)ei)) i∈[4] i∈[5,6] X X = F((2α − 1 − αγ0)e0 + α( (1 − γi)ei − (1 + γi)ei)). i∈[4] i∈[5,6] e0, e2, e3, e4, e5, e6 are linearly independent. Consequently, 2α − 1 − αγ0 = α − αγ0. This can only be the case if α = 1. Thus, indeed, X X 2e0 + ei − ei = 0. i∈[4] i∈[5,8] 5.6. Equivalence of polarized embeddings: X7 117

5.6.2 The universal embedding

Theorem 5.20 Let F be a ﬁeld. Then the natural embedding u of X7 of type NO7(2) over F is the universal embedding of X7 over F.

Proof. Suppose φ is a full polarized embedding of X7 over F into a projective space P(U). Then, because of Proposition 5.16, there are subsets {ei | i ∈ [0, 8]} of U and 7 {fi | i ∈ [0, 8]} of F such that X X φ(x) = F( ei − ei), i ∈ x ∩ [0,4] i ∈ x ∩ [5,8] X X u(x) = F( fi − fi), i ∈ x ∩ [0,4] i ∈ x ∩ [5,8] and X X X X 0 = e1 = f1 = 2e0 + ei − ei = 2f0 + fi − fi. i∈[4] i∈[5,8] i∈[4] i∈[5,8]

Moreover, f0, f2, . . . , f7 are linearly independent since dim(u) = 7. This enables us to 7 deﬁne g as the surjective linear map from F to U induced by sending fi, i ∈ [0, 7], to ei. Obviously, φ = g ◦ u. Thus, u is the universal embedding of X7 over F.

Proof of Proposition 5.16. Lemmas 5.17, 5.18 and 5.19.

5.6.3 Quotient embeddings Here, we derive some criteria which a polarized quotient embedding should satisfy.

Proposition 5.21 Let F be a ﬁeld, let u be the universal embedding of X7 of type R NO7(2) over F, let x ∈ P(X7), and suppose u is polarized quotient embedding of u. Then, R ∈ Hx. Moreover,

u(x) ∈ Hx ⇔ char(F) = 2. Proof. We prove the lemma for x = {0}. The other points can be dealt with in the same way. 7 First of all, because of Proposition 5.16, there is a subset {ei | i ∈ [0, 8]} of F such that X X 0 = e1 = 2e0 + ei − ei i∈[4] i∈[5,8] 118 Chapter 5. Classifying the polarized embeddings of a cotriangular space and X X ∀ : u(x) = ( e − e ). x∈P(X7) F i i i ∈ x ∩ [0,4] i ∈ x ∩ [5,8]

Moreover, since

∆x = ∆{0} = {{i, j} | {i, j} ∈ P(X7) ∧ (i, j < 5 ∨ i, j > 4)}

∪ {{0, i, j, k, l} | {0, i, j, k, l} ∈ P(X7)}, we obtain

Hx = h{F(ei − ej) | {i, j} ∈ P(X7) ∧ (i, j < 5 ∨ i, j > 4)} ∪ {F(e0 + ei + ej − ek − el) | {0, i, j, k, l} ∈ P(X7)}i = h{Fe2, Fe3, Fe4, F(e5 − e6), F(e6 − e7), F(e7 − e8), F(e0 − 2e5)}i = h{Fe2, Fe3, Fe4, F(e5 − e6), F(e6 − e7), F(e0 − 2e5)}i.

Note that F(e7 − e8) can be omitted because X X 0 = 2e0 + ei − ei. i∈[4] i∈[5,8]

In particular, Hx has rank six. Consequently, hHx, u(y)i = h∆x ∪ {y}iu has rank seven for all points y collinear with x. This can only be the case if

∀ : x ∼ y ⇒ h∆ ∪ {y}i = hX i . x,y∈P(X7) x u 7 u

Thus, because of the connectedness of X7 and Proposition 5.5, R ∈ Hx and it remains to prove u(x) ∈ Hx ⇔ char(F) = 2. If char(F) = 2, then

u(x) = Fe0 = F(e0 + e3 + e4 + e5 + e6 + e3 + e4 + e5 + e6) ∈ hF(e0 + e3 + e4 + e5 + e6), F(e3 + e4), F(e5 + e6)i = h{0, 3, 4, 5, 6}, {3, 4}, {5, 6}iu

⊆ H{0} = Hx.

Therefore, assume char(F) 6= 2 and suppose u(x) ∈ Hx. It is sufﬁcient to ﬁnd a contradiction. Since,

u(x) ∈ Hx = h{Fe2, Fe3, Fe4, F(e5 − e6), F(e6 − e7), F(e0 − 2e5)}i, 5.6. Equivalence of polarized embeddings: X7 119 there is a subset {αi | i ∈ [6]} of F such that

e0 = α1e2 + α2e3 + α3e4 + α4(e5 − e6) + α5(e6 − e7) + α6(e0 − 2e5)

= α6e0 + α1e2 + α2e3 + α3e4 + (α4 − 2α6)e5 + (α5 − α4)e6 − α5e7.

In other words,

(1 − α6)e0 = α1e2 + α2e3 + α3e4 + (α4 − 2α6)e5 + (α5 − α4)e6 − α5e7.

Since e0, e2, . . . , e7 are linearly independent,

0 = 1 − α6, and

0 = α4 − 2α6 = α5 − α4 = α5.

In particular,

0 = 1 − α6 = α4 = α5 = α6.

This is a contradiction. Thus, u(x) ∈/ Hx.

5.6.4 The equivalence classes

We prove that there are at most two equivalence classes of polarized embeddings of X7.

Theorem 5.22 Let F be a ﬁeld. If char(F) 6= 2, then all polarized embeddings of X7 over F are equivalent to the natural embedding of X7 of type NO7(2) over F. Otherwise, there are two equivalence classes: the natural embedding of X7 of type NO7(2) over F and the natural embedding of X7 of type HSp6(2) over F.

Proof. We have already seen that the natural embedding u of X7 of type NO7(2) over F is the 7-dimensional universal embedding of X7 over F. Suppose there is an R mak- R ing u polarized. Then, because of Proposition 5.21, R ∈ Hx for all points x in X7. If char(F) 6= 2, then we find a contradiction. Otherwise, since there exists a 6-dimensional polarized embedding, it is sufficient to prove that R is uniquely determined. char(F) = 2. Define \ u H := H{i,i+1}. i∈[6] Then R ∈ H. Consequently, it is sufficient to prove that the rank of H is at most one. This follows from combining dim(u) = 7 with

u \ u ∀i∈[6]∀j∈[5] : u({i + 1, i + 2}) ∈/ H{i,i+1} ∧ u({j + 2, j + 3}) ∈ H{k,k+1}. k∈[j] 120 Chapter 5. Classifying the polarized embeddings of a cotriangular space char(F) 6= 2. Observe that X7 contains a set {xi | i ∈ [7]} of seven distinct points with xi ⊥ xj for all i, j ∈ [7]. For instance,

{{0}, {1, 2}, {3, 4}, {5, 6}, {7, 8}, {0, 1, 2, 5, 6}, {0, 3, 4, 7, 8}}.

We deﬁne \ H := Hu . xi i∈[7] Then R ∈ H. Consequently, it is sufﬁcient to prove that the rank of H is zero. If i, j ∈ [7], then, because of Proposition 5.21, u(x ) ∈/ Hu if and only if i = j. Combining this i xj with dim(u) = 7 results in rank(H) = 0.

5.7 Equivalence of polarized embeddings: symplectic type

Let Π be a cotriangular space of symplectic type generated by n points and let F be ∼ ∼ an arbitrary ﬁeld. Then, n ≥ 7 is odd and Π = HSpn−1(2) = NOn(2). Therefore, we identify Π with NOn(2). Moreover, since we already dealt with the case n = 7 in section 5.6, we assume n ≥ 9. We prove that the existence of a polarized embedding of Π over F implies char(F) = 2. Moreover, we prove that there are two equivalence classes: one containing the natural embedding of NOn(2) over F, the other containing the natural embedding of NOn(2) of type HSpn−1(2) over F.

5.7.1 Field characteristic

We prove that the existence of a polarized embedding implies char(F) = 2.

Proposition 5.23 Let F be a ﬁeld, let n ≥ 9, and suppose NOn(2) admits a polarized embedding φ over F. Then char(F) = 2.

Proof. We assume char(F) 6= 2 and try to ﬁnd a contradiction. − Because of Lemma 1.40 there is a subspace M8 of NOn(2) isomorphic to NO8 (2). − We identify M8 with NO8 (2) and deﬁne

− M7 := h(x1, . . . , x2n+2) ∈ N O8 (2) \{(0,..., 0, 1, 0)} | x2n+2 = 0i, − M6 := h(x1, . . . , x2n+2) ∈ N O8 (2) | (x2n+1, x2n+2) = (0, 0)i,

M2 := {{0,..., 0, 0, 1}, {0,..., 0, 1, 0}, {0,..., 0, 1, 1}}.

Then, M2 ⊆ M8, M6 ⊆ M7 ⊆ M8, and M2 ⊥ M6. 5.7. Equivalence of polarized embeddings: symplectic type 121

Moreover, ∼ + ∼ ∼ ∼ M6 = NO6 (2) = T8 and M7 = NO7(2) = X7. If we deﬁne ∀ : φ := φ| , i∈[6,8] i Mi then, because of Proposition 1.36 and Theorem 5.22,

dim(φ8) ≤ 8 and dim(φ7) = 7.

Now, let x and y be two of the three points making up the line that is M2. Necessarily, they are collinear and x ⊥ M6 ⊥ y. In particular,

hM i ⊆ Hφ8 ∩ Hφ8 . 6 φ8 x y

Next, let z be a point in M8 different from x and y such that x ⊥ z ∼ y. Such a point φ8 φ8 exists since M8 is irreducible. Since z ∈ Hx \ Hy , we obtain that dim(φ6) is upper bounded by 6. Next, because of Proposition 5.16 there is a set {ei | i ∈ [0, 8]} with X X 0 = e1 = 2e0 + ei − ei, i∈[4] i∈[5,8] and

hM i = h e | i ∈ [8]i and hM i = h e | i ∈ [0, 8]i. 6 φ7 F i 7 φ7 F i P Consequently, since e0 ∈ i∈[8] ei,

7 = dim(φ7) = dim(φ6) ≤ 6.

This is the required contradiction.

5.7.2 Dimensionality We already showed that a polarized embedding of an irreducible cotriangular space has two possible values. Here, we show that if the dimension of a polarized embedding of a cotriangular space of symplectic type is known, then also the dimension of the polarized embedding restricted to a subspace of symplectic type is known.

Proposition 5.24 Let F be a ﬁeld of characteristic two, let n ≥ 9, and let φ be a polarized embedding of NOn(2) over F and let Mk be a subspace of NOn(2) isomorphic to NOk(2) for an odd integer k ∈ [7, n]. Then,

dim(φ| ) = dim(φ) − n + k. Mk 122 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Proof. There are n − k points not in Mk which together with Mk generate NOn(2). Consequently, we must have

dim(φ| ) ≥ dim(φ) − (n − k) = dim(φ) − n + k. Mk

Next, observe that Mk is generated by k points. Hence, if dim(φ) = n, then

dim(φ| ) ≤ k = dim(φ) − n + k. Mk Consequently, it is sufﬁcient to prove dim(φ| ) ≤ k − 1 assuming dim(φ) = n − 1. Mk We proceed by induction on k. If k = n, then dim(φ| ) = dim(φ) = n − 1 = k − 1. Mk Hence, we can assume k < n and

dim(φ| ) ≤ k + 1 Mk+2 ∼ for all subspaces Mk+2 = NOk+2(2) of NOn(2). There are collinear points x and y in NOn(2) perpendicular to Mk. Consequently, ∼ Mk+2 := hMk ∪ {z + x | z ∈ P(Mk)} ∪ {z + y | z ∈ P(Mk)}i = NOk+2(2).

This implies dim(φ| ) ≤ k + 1. Mk+2

Furthermore, note that each point z in NOn(2) is contained in a subspace isomorphic to ∼ X7 = NO7(2). Hence, because of Proposition 5.21,

∀ : z ∈ Hφ. z∈P(NOn(2)) z Combining this with the collinearity of x and y we obtain

φ φ φ φ(y) ∈/ Hx , φ(x) ∈ Hx , and φ(x) ∈/ Hy .

φ φ Consequently, since Mk ⊆ Hx ∩ Hy ,

dim(φ| ) ≤ dim(φ| ) − 2 ≤ k − 1. Mk Mk+2

5.7.3 Embedding lines Here, we ﬁnd a way to embed lines.

Proposition 5.25 Let F be a ﬁeld of characteristic two, let n ≥ 9, and suppose φ is a full even-dimensional polarized embedding over F into P(V ). 5.7. Equivalence of polarized embeddings: symplectic type 123

Then there exist a subset {vx | x ∈ P(NOn(2))} of V \{0} such that

∀ : φ(x) = v , x∈P(N (On(2))) F x and

∀ : v + v + v = 0. {x,y,z}∈L(NOn(2)) x y z

Proof. In this proof we refer to the lines in NOn(2) as the hyperbolic lines of NOn(2). In other words, if x and y are two distinct collinear points in NOn(2), then we refer to {x, y, x + y} as the hyperbolic line on x and y. Otherwise, x + y + n is a point in NOn(2) and we refer to {x, y, x + y + n} as the singular line on x and y. Note, the hyperbolic lines make up L(NOn(2)) and the set of hyperbolic lines is disjoint from the set of singular lines. By deﬁnition each hyperbolic line is mapped into a line. First we prove that this also holds for the singular lines. Therefore, let {x1, x3, x5} be a singular line of NOn(2). The collinearity graph of Π has diameter two. This implies that there are two points x2 and x4 of Π such that there is a path x1 ∼ x2 ∼ x3 ∼ x4 ∼ x5. of length ﬁve in the collinearity graph of Π. In HSpn−1(2) it is easily checked that three points corresponding to a singular line of NOn(2) cannot be collinear with the same point. Consequently, the same also holds in NOn(2). As a consequence, if i, j ∈ [5] with i < j, then xi ∼ xj if and only if j = i + 1. This implies ∼ ∼ ∼ M := hx1, x2, x3, x4, x5i = R(A5) = T6 = NO5(2).

Since dim(φ) is even, we can use Proposition 5.24 to conclude dim(φ|M) = 4. More- over, because of Proposition 5.21,

φ({x , x , x }) ∈ H ∩ H . 1 3 5 x1 x3 Consequently, since

φ(x ) ∈/ H , φ(x ) ∈ H , and φ(x ) ∈/ H , 2 x1 4 x1 4 x3 we obtain

rank({x1, x3, x5}) = 2.

So, indeed, each singular line is mapped into a line. Next, deﬁne M as the linear space whose point set is the point set of NOn(2) and whose line set is the set of all singular and hyperbolic lines in NOn(2). This linear 2n space is isomorphic to P(F2 ) and we denote the corresponding isomorphism by ι. In 124 Chapter 5. Classifying the polarized embeddings of a cotriangular space

−1 2n particular, φ ◦ ι is an injective map from P(F2 ) to P(V ) mapping lines into lines. In 2n other words, there exists a semi-linear map g : F2 → V such that

−1 ∀ 2n : φ ◦ ι (F2w) = Fg(w) w∈F2 \{0} (Faure 2002, Theorem 3.1). Since ι is an isomorphism, there is for each point x in 2n NOn(2) a non-zero wx ∈ F2 such that ι(x) = F2wx. Moreover, since the lines 2n in P(F2 ) have cardinality three, wx = wy + wz for each singular or hyperbolic line {x, y, z} in NOn(2). Now, deﬁne vx := g(wx) for all points x in NOn(2). Then

∀ : φ(x) = φ ◦ ι−1 ◦ ι(x) = φ ◦ ι−1( w ) = g(w ) = v x∈P(NOn(2)) F2 x F x F x and vx = g(wx) = g(wy + wz) = g(wy) + g(wz) = vy + vz, for all singular or hyperbolic lines {x, y, z} in NOn(2).

5.7.4 Quotient embeddings Here, we derive properties a polarized quotient embedding should satisfy.

Proposition 5.26 Let F be a ﬁeld of characteristic two, let n ≥ 9, and suppose φ is an n-dimensional polarized embedding of NOn(2) over F. Then there is a unique R such that φR is polarized.

Proof. Since the natural embedding of NOn(2) of type HSpn−1(2) over F is (n − 1)- dimensional, there exists an R such that this natural embedding is equivalent to the polarized quotient embedding φR. R Now, ﬁx an R such that φ is polarized and ﬁx a subspace X of NOn(2) isomorphic ∼ to X7 = NO7(2). Then, because of Proposition 5.24,

R dim(φ |X ) = 6 and dim(φ|X ) = 7.

If R/∈ hX iφ, then

R dim(φ | ) = rankhX i R = rankhX i = 7. X φ φ

Consequently, R ∈ hX i = hX i . Hence, φR| = (φ| )R is a quotient embedding φ φ|X X X of φ|X . Thus, because of Theorem 5.22, R is uniquely determined.

5.7.5 The universal embedding Now, we can prove that a cotriangular space of symplectic type admits a universal embedding if the ﬁeld characteristic is two. 5.7. Equivalence of polarized embeddings: symplectic type 125

Theorem 5.27 Let F be a ﬁeld of characteristic two and let n ≥ 9. Then the natural embedding u of NOn(2) over F is the universal embedding of NOn(2) over F.

Proof. Because of Proposition 5.26 it is sufﬁcient to prove that each n-dimensional polarized embedding of NOn(2) over F is equivalent to u. So, let φ be a full n-dimensional polarized embedding of NOn(2) over F into P(V ). R Moreover, let R = Fr be the unique R making φ polarized. Then, Proposition 5.25 implies that there exists a subset

{wx | x ∈ P(NOn(2))} ⊆ V \{0} such that ∀ : φR(x) = w + R x∈P(NOn(2)) F x and ∀ : w + w + w ∈ R. {x,y,z}∈L(NOn(2)) x y z Consequently, there exist subsets

{vx | x ∈ P(NOn(2))} ⊆ V \{0} and {λx | x ∈ P(NOn(2))} ⊆ F such that ∀ : φ(x) = v ∧ v = w + λ r. x∈P(NOn(2)) F x x x x As a consequence,

∀ : v + v + v = w + w + w + (λ + λ + λ )r ∈ R. {x,y,z}∈L(NOn(2)) x y z x y z x y z Given a line {x, y, z} we know

Fvx + Fvy = Fvx + Fvy + Fvz.

However, since φR is injective,

r∈ / Fvx + Fvy = Fvx + Fvy + Fz.

Consequently, ∀ : v + v + v = 0. {x,y,z}∈L(NOn(2)) x y z In the same way as we did for φ, we can ﬁnd a subset

n {ux | x ∈ P(NOn(2))} ⊆ F \{0} such that ∀ : u(x) = u ∧ u + u + u = 0. {x,y,z}∈L(NOn(2)) F x x y z 126 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Now, let {xi | i ∈ [n]} be a basis for NOn(2) and deﬁne

∀ :(u , v ) := (u , v ). i∈[n] i i xi xi P n P Then, V = i∈[n] Fvi and F = i∈[n] Fui. Therefore, we can deﬁne g as the invertible linear map with ∀i∈[n] : g(vi) = ui.

Moreover, for each point x in NOn(2) we can deﬁne Ix and Jx as the unique sets of indices such that X X (ux, vx) = ( ui, vj).

i∈Ix j∈Jx Clearly, ∀ : I = J . i∈[n] xi xi

Suppose there is a subset X of the point set of NOn(2) such that

∀x∈X : Ix = Jx, and consider a line {x, y, z} with x, y ∈ X . Then (uz, vz) = (ux + uy, vx + vy). In other words, Iz = Jz. Consequently, using induction we can prove

∀ I = J . x∈P(NOn(2)) x x In particular, X X X X ∀ : g(v ) = g( v ) = g( v ) = g(v ) = u = u . x∈P(NOn(2)) x i i i i x i∈Jx i∈Ix i∈Ix i∈Ix

Thus, since NOn(2) is connected,

∀ : u(x) = u = g(v ) = g( v ) = g(φ(x)) = g ◦ φ(x). x∈P(NOn(2)) F x F x F x

In other words, u = g ◦ φ and φ is indeed equivalent to u.

5.7.6 The equivalence classes Combining Propositions 5.23–5.26 with Theorem 5.27 gives the equivalence classes for NOn(2).

Theorem 5.28 Let F be a ﬁeld and let n ≥ 9. Then NOn(2) admits a polarized embedding over F if and only if char(F) = 2. Moreover, if it admits a polarized embedding, then there are two equivalence classes: the natural embedding of NOn(2) over F and the natural embedding of NOn(2) of type HSp2n(2) over F. 5.8. Equivalence of polarized embeddings: orthogonal type 127

5.8 Equivalence of polarized embeddings: orthogonal type

Let Π be a cotriangular space of orthogonal type generated by n points and let F be an arbitrary ﬁeld. Then, n ≥ 6 and there is a δ ∈ {+, −} with (n, δ) 6= (6, +) and ∼ δ δ Π = NOn(2). We identify Π with NOn(2) and prove that all polarized embeddings of δ δ NOn(2) over F are equivalent with the natural embedding of NOn(2) over F.

5.8.1 Field characteristic

The following lemma can be proven in exactly the same manner as Proposition 5.23.

Proposition 5.29 Let F be a ﬁeld, let n ≥ 6, let δ ∈ {+, −} such that (n, δ) 6= (6, +), δ and suppose NOn(2) admits a polarized embedding φ over F. Then char(F) 6= 2 implies (n, δ) ∈ {(6, −), (8, +)}.

5.8.2 Dimensionality

Proposition 5.30 Let F be a ﬁeld, let n ≥ 6, let δ ∈ {+, −} such that (n, δ) 6= (6, +), δ and suppose NOn(2) admits a polarized embedding φ over F. Then, φ is n-dimensional.

Proof. Because of Proposition 1.36, the dimension of φ is upper bounded by n. In other words, it remains to check that n is also a lower bound. We start with small values of n. Suppose (n, δ) = (6, −) and deﬁne

(x1, . . . , x6) := (2 +5, 2 +6, 4 +5, 1 +2 +4 +6, 5, 1 +2 +3 +4 +5 +6).

Then, \ ∀ : x ∈/ H ∧ x ∈ H i∈[5] i i+5 i+1 j +5 j∈[i] In particular, dim(φ) ≥ 6. Suppose (n, δ) = (8, +) and char(F) 6= 2. Moreover, deﬁne x := 7 + 8. Then, ∼ ∼ ∆x = NO7(2) = X7 and φ(∆x) ⊆ Hx.

Consequently, dim(φ) ≥ dim(φ| ) + 1 = 8. Here, the last equality follows from ∆x Theorem 5.22. For the other cases we know that char(F) = 2 and we deﬁne

( + + , + + ) if δ = +, and (x, y) := n−3 n−2 n−1 n−3 n−2 n (n−1, n) if δ = −. 128 Chapter 5. Classifying the polarized embeddings of a cotriangular space

Then, ∼ −δ x ∼ y, M := ∆x ∩ ∆y = NOn−2(2), and hMiφ ⊆ Hx ∩ Hy.

Moreover, because of Proposition 5.21 and the fact that each point is contained in a subspace isomorphic to X7, we obtain

∀ δ : x ∈ Hx. x∈P(NOn(2)) Combining this with

φ(y) ∈/ Hx, φ(x) ∈ Hx, and φ(x) ∈/ Hy, we obtain that the dimension of φ|M is lower bounded by dim(φ) − 2. Thus, the lemma − follows from the fact that the dimension of each polarized embedding of NO6 (2) or + NO8 (2) is lower bounded by 6 or 8, respectively.

5.8.3 Embedding lines in characteristic two Assuming the characteristic is two we prove that each line of a cotriangular space of orthogonal type can be embedded in a projective line.

Proposition 5.31 Let F be a ﬁeld of characteristic two, let n ≥ 6, let δ ∈ {+, −} such δ that (n, δ) 6= (6, +), and suppose NOn(2) admits a polarized embedding φ over F into a projective space P(V ). Then there is a subset

δ {vx | x ∈ P(NOn(2))} ⊆ V such that ∀ δ : φ(x) = Fvx ∧ vx + vy + vz = 0. {x,y,z}∈L(NOn(2)) Proof. Deﬁne ( + + , + + ) if δ = +, and (p, q) = n−3 n−2 n−1 n−3 n−2 n (n−1, n) if δ = −.

Then, ∼ ∼ p ∼ q and ∆p = ∆q = NOn−1(2).

A polarized embedding of NOn−1(2) over F is equivalent with either the natural embedding of NOn−1(2) over F or the natural embedding of NOn−1(2) of type HSpn−2(2) over F. Consequently, there is a subset {vx | x ∈ ∆p} of V such that

∀ : φ(x) = v ∧ v + v + v = 0. {x,y,z}∈L(∆p) F x x y z 5.8. Equivalence of polarized embeddings: orthogonal type 129

For the same reason there is a subset {wx | x ∈ ∆q} of V such that

∀ : φ(x) = w ∧ w + w + w = 0. {x,y,z}∈L(∆q) F x x y z

Because of the fact that φ is injective, vx = wx for all x ∈ ∆p ∩ ∆q. Hence, after having deﬁned vx := wx for all x ∈ ∆q \ ∆p,

∀ : φ(x) = v ∧ v + v + v = 0. {x,y,z}∈L(∆p∪∆q) F x x y z

It remains to check that the lines not contained in ∆p ∪ ∆q embed in projective lines. However, ﬁrst we embed the points outside ∆p ∪ ∆q. First, let x ∈ (p∼ ∩ q∼) \ hp, qi. Then hx, p, qi is isomorphic to a dual afﬁne plane. Consequently, because of Lemma 1.41,

hx, p, qi⊥⊥ = hx, p, qi.

Moreover, since p∈ / ∆q, the intersection of hx, p, qi and ∆q is contained in a line {a, b, c}. Let x1 ∈ ∆q \ (∆x ∪ ∆p ∪ {a, b, c}). Since

⊥ ∆x ∩ ∆p ∩ ∆q = hx, p, qi , there must be a x0 ∈ ∆x ∩ ∆p ∩ ∆q collinear with x1. Otherwise,

⊥⊥ x1 ∈ hx, p, qi ∩ ∆q = {a, b, c} and this is a contradiction. Now, deﬁne Mx := hx, x0, x1i. This is isomorphic to a dual afﬁne plane of order two because of the fact that x ⊥ x0 ∼ x1 ∼ x. Consequently, there are points x2, x−1, x−2 such that

Mx = ({x, x−2, . . . , x2}, {{x0, x1, x2}, {x0, x−1, x−2}, {x1, x, x−1}, {x2, x, x−2}).

Moreover, x2 ∈ ∆q because x0, x1 ∈ ∆q, and x2 ∈/ ∆p because, x0 ∈ ∆p and x1 ∈/ ∆p. Combining this with x∈ / ∆p and the fact that each line other than hp, qi intersects ∆p gives x−1, x−2 ∈ ∆p. Thus,

{xi | i ∈ [−2, 2]} ⊆ ∆p ∪ ∆q and P(M) \ (∆p ∪ ∆q) = {x}.

Therefore, we define v := v + v . x x−2 x2 Finally, let r be such that hp, qi = {p, q, r}, fix two lines {p, a, b} and {q, b, c} distinct from hp, qi, and define

(vp, vq, vr) := (va + vb, vb + vc, va + vc). 130 Chapter 5. Classifying the polarized embeddings of a cotriangular space

It remains to check

∀ δ : φ(x) = Fvx ∧ vx + vy + vz = 0. {x,y,z}∈L(P(NOn(2))\(∆p∪∆q)) We distinguish four cases. x∈ / ∆p ∪ ∆q ∪ hp, qi and y, z ∈ ∆p ∪ ∆q. Recall the deﬁnition of Mx. Then, since φ(x) ∈ hx±1i ∩ hx±2i, there are non-zero scalars α, β such that

(v + αv ) = φ(x) = (v + βv ) = (v + (1 + β)v + βv ). F x−2 x2 F x−1 x1 F x−2 x0 x2 Combining this with

dimhv , v , v i = rankhM i = 3 x−2 x0 x2 x φ results in α = β = 1. In particular,

φ(x) = (v + αv ) = (v + v ) = v . F x−2 x2 F x−2 x2 F x

Note that we can assume {x, y, z} ∈/ Mx. Otherwise, we are done. Moreover, since x∈ / ∆p ∪ ∆q, we can assume y ∈ ∆p \ ∆q and z ∈ ∆q \ ∆p. Thus, there is a point w ∈ ∆p ∩ ∆q such that

M := ({w, x−2, x, x2, y, z}, {{w, x−2, y}, {w, x2, z}, {x−2, x, x2}, {y, x, z}}) is isomorphic to a dual afﬁne plane of order two. In particular, φ(x) ∈ hy, zi ∩ hx±2i. This implies there are non-zero scalars α, β such that

(v + αv ) = φ(x) = (v + βv ) = (v + (1 + β)v + βv ). F x−2 x2 F y z F x−2 w x2 Combining this with

dimhv , v , v i = rankhMi = 3 x−2 w x2 φ results in α = β = 1. In particular,

Fvx = φ(x) = F(vy + αvz) = F(vy + vz).

This proves the existence of a non-zero scalar γ such that

v = γ(v + v ) = γ(v + v + v + v ) = γv . x y z w x−2 w x2 x

Clearly, γ = 1 and vx + vy + vz = 0. x∈ / ∆p ∪ ∆q ∪ hp, qi, y ∈ ∆p ∪ ∆q and z∈ / ∆p ∪ ∆q. Since y ∼ x, we can assume 5.8. Equivalence of polarized embeddings: orthogonal type 131

x−2 ∼ y ⊥ x2. Consequently, there is a point w such that

M := ({w, x−2, x, x2, y, z}, {{w, x−2, y}, {w, x2, z}, {x−2, x, x2}, {y, x, z}}) is isomorphic to a dual afﬁne plane of order two. Since the lines other than {x, y, z} all have at most one point outside ∆p ∪ ∆1 we can follow the same reasoning as above to conclude

φ(x) = Fvx and vx + vy + vz = 0. x∈ / ∆p ∪ ∆q ∪ hp, qi and y, z∈ / ∆p ∪ ∆q. This is in contradiction with the fact that each line intersects ∆p ∪ ∆q. x ∈ hp, qi. Because of symmetry we can assume x ∈ hp, qi \ {q}, y∈ / ∆p ∪ ∆q, and z ∈ ∆q \ ∆p. Now, ﬁx an arbitrary line {a, b, z} in ∆q. Then there is a point c such that M := ({a, b, c, x, y, z}, {{z, y, x}, {z, b, a}, {b, c, x}, {a, c, y}) is isomorphic to a dual afﬁne plane of order two. Since the lines other than {x, y, z} do not contain p or q we can follow the same reasoning as above to conclude

φ(x) = Fvx and vx + vy + vz = 0.

5.8.4 The equivalence classes and the universal embedding: characteristic not two Assuming the characteristic is not two we prove that all polarized embeddings of a cotriangular space of orthogonal type are equivalent. In particular this proves that the natural embedding is the universal embedding.

Theorem 5.32 Let F be a ﬁeld of characteristic not two, let n ≥ 6, let δ ∈ {+, −} such δ that (n, δ) 6= (6, +), and suppose NOn(2) admits a full polarized embedding φ over F into a projective space (P(V ). Then (n, δ) ∈ {(6, −), (8, +)} and φ is equivalent to the δ δ natural embedding u of NOn(2) over F. This is the universal embedding of NOn(2) over F. − Proof. Since NO6 (2) is isomorphic to a subspace of HSp6(2), the case (n, δ) = (6, −) can be proven in exactly the same manner as Theorem 5.20. Therefore, because of Proposition 5.29 we can assume (n, δ) = (8, +). Let {ai | i ∈ [8]} be the simple system of the root system of type E8, and let + {xi | i ∈ [n]} be a basis of NO8 (2) with

∀i ∈ [8] : u(xi) = Fai. 132 Chapter 5. Classifying the polarized embeddings of a cotriangular space

+ We prove there are a set {Ix | x ∈ P(NO8 (2))} of subsets of [8] and a basis {vi | i ∈ [8]} of V such that X X ∀ + : φ(x) = vi ∧ u(x) = ai. x∈P(NO8 (2)) F F i∈Ix i∈Ix

8 Then the invertible linear map g : F → V induced by sending ai to vi ensures that φ and u are indeed equivalent. + P Observe that there is a subset {bx | x ∈ P(NO8 (2))} of A := i∈[8] Fai such that

∀ + : u(x) = bx. x∈P(NO8 (2)) F

+ Since {ai | i ∈ [8]} is a simple system, there must be a set {Ix | x ∈ P(NO8 (2))} of subsets of [8] such that X ∀ + : bx = ± ai. x∈P(NO8 (2)) i∈Ix In particular, X X ∀ + : u(x) = bx = (± ai) = ai. x∈P(NO8 (2)) F F F i∈Ix i∈Ix

Moreover, if we combine this with the fact that char(F) 6= 2, then,

∀ + ∃r,s,t∈{x,y,z} : {x, y, z} = {r, s, t} ∧ Ir = Is ] It. {x,y,z}∈L(NO8 (2)) + ∼ The subspace M1 := hxi | i ∈ [7]i of NO8 (2) is isomorphic to R(E7) = X7. Con- sequently, there is a 7-dimensional vector space V over such that φ| is a full po- 1 F M1 larized embedding of M into (V ). Next, deﬁne A := P a . Then u| is 1 P 1 1 i∈[7] F i M1 also a full polarized embedding of M1 into P(A1) and, since char(F) 6= 2, this embedding is equivalent with φ| . Hence, there is an invertible semi-linear transformation M1 g : A → V such that φ| = g ◦ u| . Now, we let {v | i ∈ [7]} be such that 1 1 1 M1 M1 i

∀i∈[7] : vi = g1(ai) := vi.

Then,

∀ : φ(x ) = φ| (x ) = g (u| (x )) = g ( a ) = g (a ) = v . i∈[7] i M1 i 1 M1 i 1 F i F 1 i F i In addition, X X ∀ : φ(x) = φ| (x) = g (u| (x)) = g ( a ) = v . x∈P(M1) M1 1 M1 1 F i F i i∈Ix i∈Ix 5.8. Equivalence of polarized embeddings: orthogonal type 133

Moreover, deﬁne

(c1, . . . , c7) := (a1, a2, a3, a4 + a5, a6, a7, a8) and let {yi | i ∈ [7]} be such that

∀i∈[7] : u(yi) = Fci.

In other words, yi = xi if i ∈ [3], yi = xi + xi+1 if i = 4, and yi = xi+1 if i ∈ [5, 7].

Note that x4 + x5 is the third point on the line through x4 and x5 in M1. Hence, u(x4 + x5) = F(a4 + a5). ∼ M2 := hyi | i ∈ [7]i is also isomorphic to R(E7) = X7. In other words, in the P same way as before we ﬁnd a subspace V2 of V , a subspace A2 = i∈[7] ci of A, and an invertible semi-linear transformation g : A → V such that φ| = g ◦ u| . 2 2 2 M2 2 M2 Now, let {wi | i ∈ [7]} be such that

∀i ∈ [7] : g2(ci) = wi. Then,

∀ : φ(y ) = φ| (y ) = g ◦ u| (y ) = g ( c ) = g (c ) = w . i∈[7] i M2 i 2 M2 i 2 F i F 2 i F i Moreover,  P P  φ(x) = φ|M (x) = g2 ◦ u|M (x) = g2(F i∈I ai) = F i∈I g2(ai)  2 2 x x  ∀ : x∈P(M2) ( !  P w4 if 4, 5 ∈ Ix  = F wi +  0 4, 5 ∈/ I  i∈Ix∩[3,8]\[4,5] if x

The intersection of M1 and M2 is hx[3] ∪ x[6,7] ∪ {x4 + x5}i. So, there is a subset {αi | i ∈ [6]} of F \{0} with

wi = αivi, if i ∈ [3], w4 = α4(v4 + v5), if i = 4, and wi = αi+1vi+1. if i ∈ [5, 6].

Without loss of generality we can assume α4 = 1. Since x3 + x4 + x5 is the third point on the line through x3 and x4 + x5 in Π1 ∩ Π2, we must have

I = I ] I = {3, 4, 5}. x3+x4+x5 x3 x4+x5 134 Chapter 5. Classifying the polarized embeddings of a cotriangular space

In particular,

F(v3 + v4 + v5) = φ(x3 + x4 + x5) = F(w3 + w4) = F(α3v3 + v4 + v5).

Hence, α3 = 1. Continuing this line of reasoning we ﬁnd

∀i∈[6] : αi = 1.

Now, deﬁne v8 := w7. Then

∀i∈[8] : φ(xi) = Fvi and X ∀ : φ(x) = v . x∈P(M1∪M2) F i i∈Ix Next, deﬁne

M3 := h{yi | i ∈ [6]} ∪ {x2 + x5 + x7 + x8}i,

M4 := h{xi | i ∈ [6]} ∪ {x3 + x5 + x6 + x8}i,

M5 := h{xi | i ∈ [6]} ∪ {x3 + x5 + x6 + x7}i,

M6 := h{yi | i ∈ [7] \ [2, 2]}} ∪ {x2 + x6 + x7 + x8}i, and

M7 := h{yi | i ∈ [7] \ [1, 2]} ∪ {x2 + x6 + x7 + x8, x7 + x8}i.

Let (i, j) ∈ {(2, 3), (1, 4), (1, 5), (2, 6), (6, 7)}. Then, in the same way as before, we can prove that X ∀ : φ(x) = v . x∈P(Mi∪Mj ) F i i∈Ix

+ Since NO8 (2) = ∪i∈[7]Mi, the lemma follows.

5.8.5 The equivalence classes and the universal embedding: characteristic two Assuming the characteristic is two we prove that all polarized embeddings of a cotriangular space of orthogonal type are equivalent. In particular this proves that the natural embedding is the universal embedding.

Theorem 5.33 Let F be a ﬁeld of characteristic two, let n ≥ 6, let δ ∈ {+, −} such that δ (n, δ) 6= (6, +), and suppose NOn(2) admits a full polarized embedding φ over F into δ a projective space P(V ). Then φ is equivalent with the natural embedding u of NOn(2) δ over F. This is the universal embedding of NOn(2) over F. 5.8. Equivalence of polarized embeddings: orthogonal type 135

δ n Proof. Because of Proposition 5.31 there are a subset {ux | x ∈ P(NOn(2))} of F δ and a subset {vx | x ∈ P(NOn(2))} of V such that  u(x) = u ,  F x ∀ δ : {x,y,z}∈L(NO (2)) φ(x) = Fvx, and n  ux + uy + uz = vx + vy + vx = 0.

δ Now, let {xi | i ∈ [n]} be a basis for NOn(2) and let {ui | i ∈ [n]} and {vi | i ∈ [n]} such that ∀ :(u , v ) = (u , v ). i∈[n] i i xi xi n This makes {ui | i ∈ [n]} a basis of F and {vi | i ∈ [n]} a basis of V . We prove there δ is a set {Ix | x ∈ P(NOn(2))} of subsets of [n] such that  u(x) = P u , and  F i i∈Ix ∀ δ : x∈P(NOn(2)) φ(x) = P v .  F i i∈Ix

n Then the invertible linear map g : F → V induced by sending ui to vi ensures that u and φ are indeed equivalent. For each point x in NOn(2) we can deﬁne Ix and Jx as the unique subsets of [n] such that X X (ux, vx) = ( ui, vj).

i∈Ix j∈Jx Clearly, ∀ : I = J . i∈[n] xi xi

Suppose that Ix = Jx for all x in a subset X of the point set of NOn(2) and consider a line {x, y, z} with x, y ∈ X . Then

(uz, vz) = (ux + uy, vx + vy).

In other words, I = J because I = I and I = I . So, indeed there is a set z z ux vx uy vy δ {Ix | x ∈ P(NOn(2))} of subsets of [n] such that  u(x) = P u , and  F i i∈Ix ∀ δ : x∈P(NOn(2)) φ(x) = P v .  F i i∈Ix

n Thus, the invertible linear map g : F → V induced by sending ui to vi ensures that u and φ are equivalent. 136 Chapter 5. Classifying the polarized embeddings of a cotriangular space Appendix A

Basic terminology

A.1 Afﬁne varieties and polynomial maps

n Let n ∈ N and let F be a field. Then, affine space A (F) is the space {(α1, . . . , αn) | n n α1, . . . , αn ∈ F}. It is also denoted by F or, if it is clear which field F is meant, A . n Moreover, A is equipped with the Zariski topology. The closed sets are the sets

n V (I) := {a ∈ A | p(a) = 0 for all p ∈ I}, where I is an ideal in the polynomial ring F[X1,...,Xn]. n An afﬁne variety X over a ﬁeld F is just a closed set V (I) of A . For any extension F of F, the set of F-rational points (also called F-points) on X is

n X(F) := {a ∈ A (F) | p(a) = 0 for all p ∈ I}.

n For X(F) the Zariski topology on A (F) induces a topology on X(F). The open sets are n those subsets of X(F) which equal X(F) ∩ Y for some open set Y of A (F). A subset Y of X(F) is called dense if it intersects every non-empty open subset of X(F). m n A polynomial map over F is a map p : F → F (m, n ∈ N) such that for all points n a ∈ F p(a) = (p1(a), . . . , pm(a)) for suitable polynomials p1, . . . , pm ∈ F[X1,...,Xn].

A.2 Generalized Cartan matrices and Dynkin diagrams

For a more in-depth discussion of generalized Cartan matrices and Dynkin diagrams we refer to Kac (1990), Humphreys (1978), and Humphreys (1990). A square matrix A = (Ai,j)i,j∈[n] with integer entries is called a generalized Cartan matrix if there is a diagonal matrix D and a symmetric matrix S such that A = DS and, 138 Appendix A. Basic terminology for all i, j ∈ [n] with i 6= j,

Ai,i = 2,Ai,j ≤ 0, and Ai,j = 0 ⇒ Aj,i = 0.

Each generalized Cartan matrix (Ai,j)i,j∈[n] can be replaced by a graph. The vertex set of the graph is [n] and two distinct vertices i and j are connected by Ai,jAj,i edges. A These edges are directed from i to j if and only if i,j < 1. Each graph which can be Aj,i obtained in this way from a generalized Cartan matrix is called a Dynkin diagram.A Dynkin diagram without any directed edges is called simply laced. For A an invertible matrix, the Dynkin diagram is said to be of finite type. For A a matrix whose null space is 1-dimensional, we say that the Dynkin diagram is of affine type. The remaining Dynkin diagrams are said to be of indefinite type. One can prove that the generalized Cartan matrix can be recovered from the corresponding Dynkin diagram if it the Dynkin diagram in question is not of indefinite type. There are four infinite families of connected Dynkin diagrams of finite type: (An)n>1, (Bn)n>2, (Cn)n>3, and (Dn)n>4. They are the Dynkin diagrams of classical type. Moreover, there are five exceptional cases: E6, E7, E8, F4, and G2. They are depicted in Figures A.1-A.9 using the vertex labeling introduced by Bourbaki (1968).

1 2 ... n - 1 n 1 2 ... n - 1 n

Figure A.1: An Figure A.2: Bn

n - 1

1 2 ... n - 1 n

1 2 ... n - 2 n

Figure A.3: Cn Figure A.4: Dn

2 2

1 3 4 5 6 1 3 4 5 6 7

Figure A.5: E6 Figure A.6: E7

1 3 4 5 6 7 8 1 2 3 4 1 2

Figure A.7: E8 Figure A.8: F4 Figure A.9: G2 A.3. Root systems 139

Each of the connected Dynkin diagrams of finite type can be extended by adding a vertex 0 in such a way that we obtain the diagrams of Figures A.10-A.19. If X is a Dynkin diagram of finite type, then we denote the extended diagram by X(1). In fact, this extended diagram is in all cases a Dynkin diagram of affine type.

1 0 n

0 1

2 ... n - 1

(1) (1) Figure A.10: A1 Figure A.11: An

0 1 2 ... n - 1 n

1 2 ... n - 1 n

(1) (1) Figure A.12: Bn Figure A.13: Cn

0 1 n - 1

0 2 ... n - 2 n

1 3 4 5 6

1 (1) Figure A.14: Dn Figure A.15: E6

2 2

0 1 3 4 5 6 7 1 3 4 5 6 7 8 0

(1) (1) Figure A.16: E7 Figure A.17: E8

0 1 2 3 4 1 2 0

(1) (1) Figure A.18: F4 Figure A.19: G2

A.3 Root systems

We refer to Humphreys (1978) for the notions introduced here. Let V be a vector space over R endowed with a positive deﬁnite symmetric bilinear 140 Appendix A. Basic terminology form (·, ·), that is,

∀ ∀ :(x, x) > 0 ∧ (x, y) = (y, x) ∧ (αx + βy, z) = α(x, z) + β(y, z). x,y,z∈V α,β∈R A reﬂection is a linear operator r on V which sends some nonzero vector x to its negative while ﬁxing pointwise the hyperplane orthogonal to x. We may write r = rx. Then 2(y, x) ∀ ∀ ∀ ∗ : r = r ∧ r (y) = y − x. x∈V \{0} y∈V α∈R x αx x (x, x)

Now, Φ is called a root system if it is a ﬁnite spanning set of nonzero vectors in V called roots satisfying the following conditions:

(i) ∀x∈Φ :Φ ∩ Rx = {x, −x}.

(ii) ∀x∈Φ : rxΦ = Φ.

2(x,y) (iii) ∀x,y∈Φ : hx, yi := (y,y) ∈ Z.

A root system is called irreducible if it cannot be written as a union Φ1 ∪ Φ2 where (x, y) = 0 for all x ∈ Φ1 and all y ∈ Φ2. A subset ∆ of Φ is called a simple system if ∆ is a vector space basis for the R-span of Φ in V and if moreover each x ∈ Φ is a linear combination of elements of ∆ with coefficients all of the same sign. Suppose {ai | i ∈ [n]} is a simple system of Φ, then X X Φ+ := Φ ∩ R≥0am and Φ− := Φ ∩ R≤0am m∈[n] m∈[n] are called the positive root system and the negative root system of Φ, respectively. Rela- P P tive to a simple system ∆ we define the height of a root x = y∈∆ αyy as | y∈∆ αy|. If {ai | i ∈ [n]} is a simple system of a root system Φ, then (hai, aji)i,j is a generalized Cartan matrix. Moreover, the corresponding Dynkin diagram is of finite type and no two root systems give rise to the same Dynkin diagram.

n n Example A.1 For each n ∈ N, let (·, ·): R × R → R be the standard inner product, that is, the restriction of the inner product to the standard basis equals the Kronecker delta. Using this we can give constructions of the irreducible root systems corresponding to the simply laced Dynkin diagrams of types An, Dn, and En. For each construction one can check that Φ is a root system with simple system ∆ = {ai | i ∈ [n]}. Note that whenever we write here combinations such as ±i ± j, it is understood that the signs may be chosen arbitrarily.

n+1 An: Let n > 1, let V be the hyperplane in R consisting of those vectors whose coordinates sum up to zero and deﬁne Φ as the set

{i − j | i, j ∈ [n + 1] ∧ i 6= j}. A.3. Root systems 141

For ∆ take {ai | i ∈ [n]} with, for all i ∈ [n],

ai = i − i+1.

n Dn: Let n > 4, let V = R , and deﬁne Φ as the set

{±i ± j | i, j ∈ [n] ∧ i 6= j}.

For ∆ take {ai | i ∈ [n]} with i − i+1 if i < n, ai = n−1 + n otherwise.

8 E8: Let V = R and deﬁne Φ as the set 1 X {± ± | i, j ∈ [8] ∧ i < j} ∪ { ± | even number of + signs}. i j 2 i i∈[8]

For ∆ take {ai | i ∈ [8]} with

 1 ( − − − − − − + ) if i = 1,  2 1 2 3 4 5 6 7 8 ai = 1 + 2 if i = 2,  i−1 − i−2 otherwise.

E7: Starting with the root system of type E8 just constructed, let V be the span of 8 ∆ := {ai | i ∈ [7]} in R , deﬁne Φ as the set

{±i ± j | i, j ∈ [6] ∧ i < j}

∪ {±(7 − 8)} 1 X ∪ {± ( − + ± ) | odd number of − signs}. 2 7 8 i i∈[6]

E6: Starting again with the root system of type E8, let V be the span of ∆ := {ai | i ∈ 8 [6]} in R and deﬁne Φ as the set

{±i ± j | i, j ∈ [5] ∧ i < j} 5 1 X ∪ {± ( − − + ± ) | odd number of − signs}. 2 8 7 6 i i=1

In this example, given a simple system ∆ = {ai | i ∈ [n]}, the vertex set of the corresponding Dynkin diagram is [n] and for all distinct i, j ∈ [n] there exists an edge {i, j} if and only if (ai, aj) 6= 0. 142 Appendix A. Basic terminology

A.4 Algebras and modules

An algebra is a vector space A over a ﬁeld F equipped with a bilinear multiplication. If the multiplication is associative, then the algebra is called associative. Moreover, if

AA = {ab | a, b ∈ A} = {0}, then A is said to be abelian. A subset I of an algebra A that is closed under multiplication is called a subalgebra of A. If in addition AI ⊆ I, then I is called an ideal of A. An ideal I of A with {0}= 6 I 6= A is said to be proper. An algebra is called simple if it has no proper ideals and if it is not abelian. If we are given a vector space V over a ﬁeld F and an algebra A over the same ﬁeld F together with a map A × V → V (denoted (a, v) → a · v), then V is called an A-module if

∀ ∀ ∀ :(αx + βy) · v = α(x · v) + β(y · v), α,β∈F a,b∈A v∈V ∀ ∀ ∀ : a · (αv + βw) = α(a · v) + β(a · w), α,β∈F a∈A v,w∈V ∀a,b∈A∀v∈V :(ab) · v = a · b · v − b · a · v.

If A is an algebra containing an ideal B and a subalgebra C with

A = B + C = {b + c | (b, c) ∈ B × C} and B ∩ C = {0}, then A is said to be the semi-direct product of B and C. We write A = B o C or A = C n B. On the other hand, if we are given an algebra C and a C-module B. Then we can deﬁne A := B ⊕ C. Clearly, A is a vector space and we can equip A with a bilinear multiplication:

∀a,b∈B∀c,d∈C :(a + c, b + d) 7→ c · b + d · a + cd.

In particular, on C the multiplication corresponds to the ordinary multiplication on C and on B the multiplication is trivial. This makes B an abelian ideal inside A and A = B o C the semi-direct product of B and C. We refer to A as the semi-direct product corresponding to B and C. Let S be a subset of an algebra A. Then we denote the intersection of all subalgebras of A containing S by hSi. It is the smallest subalgebra of A containing S. The elements of S are called the generators of hSi and hSi is said to be generated by S. Let A be an algebra and M and A-module, then M is said to be generated as an A-module by a set S if

X ∀ ∃ : m = a · s. m∈M {as:s∈S}⊆A s s∈S A.5. Gradings 143

A map φ : A → B between two algebras A and B over a ﬁeld F is called a homomorphism of algebras if it is a homomorphism of vector spaces, that is,

∀ ∀ : φ(αx + βy) = αφ(x) + βφ(y), x,y∈A α,β∈F and if

∀x,y∈A : φ(xy) = φ(x)φ(y).

A homomorphism φ : A → B is called an endomorphism if A = B, an isomorphism if φ is a bijection, and an automorphism if φ is an bijection from A to A. Let S, T be subsets of an algebra A. The normalizer NS(T ) of T in S is the subset

{s ∈ S | {s}T ⊆ T } and the centralizer CS(T ) of T in S is the subset

{s ∈ S | {s}T = {0}}.

If S = T = A, then we write C(A) instead of CA(A). This is the center of A.

A.5 Gradings

Let S be a set. Then, an S-graded vector space is a vector space V which can be written as a direct sum of subspaces, that is, M V = Vi, s∈S where Vs is a vector space for each s ∈ S. For all s ∈ S, the elements of Vs are called the homogeneous elements of weight s. For each weight s ∈ S, the dimension of Vs is called the multiplicity of s. If W is a subspace of an S-graded vector space V , then W is called homogeneous if M W = (Vs ∩ W ) . s∈S In this case W is also an S-graded vector space. Suppose S has a partial order and let V be an S-graded vector space. Then, for each element x ∈ V there exist a unique list of elements (xs)s∈S (with only a ﬁnite number of them nonzero) called the homogeneous components of x such that X x = xs and ∀s∈S : xs ∈ Vs. s∈S 144 Appendix A. Basic terminology

We use this to deﬁne the map

gr : V → V, x 7→ xd, where

d = max{s | s ∈ S ∧ xs 6= 0} is the degree of x. If I is an ideal of V , then we deﬁne the graded ideal gr(I) associated to I as the ideal generated by {gr (x) | x ∈ I}. Suppose S = (S, +) is a commutative monoid and let A be an algebra such that the underlying vector space is S-graded, that is, M A = As. s∈S

Then, A is called a S-graded algebra if and only if

∀s,t∈S : AsAt ⊆ As+t.

Finally, a homomorphism between two S-graded algebras is called graded if it respects the grading.

A.6 Symplectic, orthogonal and Hermitian spaces

Let V be a vector space over a ﬁeld F. Then a bilinear map f : V × V → F is called a symplectic form if it is alternating, that is,

∀x∈V : f(x, x) = 0.

A symplectic form f is non-degenerate if

Rad(f) := {x ∈ V | f(x, V ) = {0}} = {0}.

If f is a non-degenerate symplectic form on V , then (V, f) is called a symplectic space. A map Q : V → F is called a quadratic form if

f : V × V → F, (x, y) 7→ Q(x + y) − Q(x) − Q(y) is a bilinear form, and ∀ ∀ : Q(αx) = α2Q(x). x∈V α∈F Here, f is called the associated bilinear form. It is a symplectic form if char(F) = 2.A quadratic form Q is non-degenerate if

Rad(Q) := {x ∈ Rad(f) | Q(x) = 0} = {0}. A.6. Symplectic, orthogonal and Hermitian spaces 145

If Q is a non-degenerate quadratic form on V , then (V,Q) is called an orthogonal space. A map f : V × V → F is called a Hermitian form relative to an involution σ of F if   f(w + x, y + z) = f(w, y) + f(w, z) + f(x, y) + f(x, z), ∀ ∀ : f(αx, βy) = αβσf(x, y), α,β∈F w,x,y,z∈V  f(x, y) = f(y, x)σ.

A Hermitian form f is non-degenerate if

Rad(f) := {x ∈ V | f(x, V ) = {0}} = {0}.

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Symbols E ...... 6 C(A) ...... 143 Ei ...... 7 CS(T ) ...... 143 E≤i ...... 7 E ...... 6 HU n(2) ...... 15 Ei ...... 7 HU(V, f) ...... 15 E≤i ...... 7 HSp(V, f) ...... 14 E ...... 3 HSp2n(2) ...... 14 i,j + φ N O(V,Q) ...... 15 Hx ...... 94 − Hφ ...... 94 N O(V,Q) ...... 15 X NO(V,Q) L(g) ...... 70 ...... 14 NO (2) ...... 14 N (T ) ...... 143 2n+1 S NO+ (2) ...... 15 [k, m] ...... 2 2n NO− (2) ...... 15 [n] ...... 2 2n R(X ) ...... 18 ∆ ...... 13, 94 m x T (Ω) ...... 14 h·i ...... 11, 142 T ...... 14 h·i ...... 94 n φ X ...... 17 ÷ ...... 17 7 g ...... 62 ÷c ...... 17 P,A gKM ...... 5 n ...... 142 gP ...... 62 I(g) ...... 70 h ...... 78 adx ...... 1 iP,A ...... 62 Rad(Q) ...... 144 n± ...... 78 Rad(f) ...... 144, 145 nn(F) ...... 2 -points ...... 137 F tn(F) ...... 2 A ...... 62 gln(F) ...... 2 A ...... 62 h(F) ...... 2 ⊥ ...... 12, 13, 16 hn(F) ...... 2 o ...... 142 o2n+1(F) ...... 3 ∼ ...... 13, 16 o2n(F) ...... 4 gxyz ...... 9 sln+1(F) ...... 3 gxy ...... 9 sp2n(F) ...... 3 gx ...... 6 sun(F, f) ...... 4 ⊥ x ...... 12, 13 un(F, f) ...... 4 152 Index

A commuting pair ...... 8 abelian ...... 142 connected ...... 11 admissible vanishing set ...... 63 cotriangular space ...... 16 affine plane ...... 12 irreducible ...... 16 affine space ...... 137 orthogonal type ...... 18 affine vanishing set ...... 63 symplectic type ...... 18 affine variety ...... 137 triangular type ...... 18 algebra ...... 142 type Xm ...... 18 abelian ...... 142 Coxeter number ...... 79 associative ...... 142 graded ...... 144 D simple ...... 142 degree ...... 144 alternating form ...... 144 delta space ...... 93 anti-commutativity identity ...... 1 dense ...... 137 associated bilinear form ...... 144 dimension ...... 95 automorphism ...... 143 dual affine plane ...... 12 Dynkin diagram ...... 138 B affine type ...... 138 basis ...... 11 classical type ...... 138 bilinear form ...... 140 exceptional type ...... 138 associative ...... 9 extended ...... 139 positive definite ...... 140 finite type ...... 138 symmetric ...... 140 indefinite type ...... 138 building ...... 23 simply laced ...... 138

C E Cartan subalgebra ...... 2 endomorphism ...... 143 center ...... 143 equivalent ...... 95 centralizer ...... 143 exceptional type ...... 138 Chevalley algebra ...... 5 exponential map ...... 10 classical type ...... 5 extended Dynkin diagram ...... 139 type An ...... 5 extremal element ...... 6 type Bn ...... 5 extremal form ...... 9 type Cn ...... 5 extremal functional ...... 9 type Dn ...... 5 extremal identity ...... 6 Chevalley basis ...... 5 extremal point ...... 6 classical polar spaces ...... 13 classical type ...... 2,5, 138 F closed sets ...... 137 ﬁltration ...... 21 co-collinearity graph ...... 11 Fischer space ...... 13 co-connected ...... 11 irreducible ...... 13 collinear ...... 11 full polarized embedding ...... 95 Index 153

G L general linear Lie algebra ...... 2 Lie algebra ...... 1 generalized Cartan matrix ...... 74, 137 associated ...... 1 generalized hexagon ...... 22 classical type ...... 2 generating rank ...... 11 nilpotent ...... 2 generating set ...... 11, 142 root system ...... 4 generators ...... 11, 142 semi-simple ...... 2 geometry of F-transvection groups . . . . 61 simply laced ...... 5 graded algebra ...... 144 solvable ...... 2 graded homomorphism ...... 144 type An ...... 3 graded ideal ...... 144 type Bn ...... 3 graded vector space ...... 143 type Cn ...... 3 type Dn ...... 4 H line ...... 11 height ...... 140 linear Lie algebra ...... 2 Heisenberg Lie algebra ...... 2 linear space ...... 11 Hermitian form ...... 4, 145 non-degenerate ...... 145 M Hermitian space ...... 145 module ...... 142 homogeneous ...... 143 generators ...... 142 homomorphism ...... 143 multiplicity ...... 74, 143 graded ...... 144 N hyperbolic line ...... 8, 123 natural embedding ...... 96 hyperbolic pair ...... 8 negative root system ...... 140 hyperbolic path ...... 8 nilpotent ...... 2 hyperplane ...... 11 non-degenerate ...... 12, 21, 144, 145 non-degenerate polar space ...... 12 I non-degenerate root filtration space . . . 21 ideal ...... 142 normalizer ...... 143 infinitesimal Siegel transvection ...... 7 infinitesimal transvection ...... 7 O irreducible ...... 13, 16 open sets ...... 137 irreducible root system ...... 140 order ...... 11, 12 irreducible variety ...... 83 orthogonal Lie algebra ...... 3,4 isomorphism ...... 11, 143 orthogonal polar space ...... 13 isotropic line ...... 50 orthogonal space ...... 145 orthogonal type ...... 18 J Jacobi identity ...... 1 P parameter space ...... 70 K partial linear space ...... 11 Kac-Moody algebra ...... 5 plane ...... 11 154 Index point ...... 11 semi-direct product ...... 142 point-line space ...... 11 semi-linear ...... 95 polar graph ...... 12 semi-simple ...... 2 polar pair ...... 8 simple ...... 142 polar space ...... 12 simple system ...... 140 affine ...... 59 simply laced Dynkin diagram ...... 138 classical ...... 13 simply laced Lie algebra ...... 5 polarized embedding ...... 95 singular line ...... 123 polarized quotient embedding ...... 96 singular subspace ...... 12, 21 polynomial map ...... 137 solvable ...... 2 positive definite bilinear form ...... 140 special linear Lie algebra ...... 3 positive root system ...... 140 special pair ...... 8 Premet identity ...... 6 special unitary Lie algebra ...... 4 primitive vector ...... 79 sporadic Fischer spaces ...... 15 projective plane ...... 11 strongly commuting pair ...... 8 proper ...... 11, 142 subalgebra ...... 142 subspace ...... 11 Q symmetric bilinear form ...... 140 quadratic form ...... 144 symmetric difference ...... 16 non-degenerate ...... 144 symplectic form ...... 144 quotient embedding ...... 96 non-degenerate ...... 144 polarized ...... 96 symplectic Lie algebra ...... 3 R symplectic polar space ...... 13 rank ...... 12, 21, 95 symplectic space ...... 144 rational points ...... 137 symplectic triple ...... 8 real roots ...... 77 symplectic type ...... 18 reflection ...... 140 T root ...... 4, 140 transvection (sub)groups ...... 61 real ...... 77 transversal coclique ...... 12 very real ...... 77 triangular type ...... 18 root filtration space ...... 21 non-degenerate ...... 21 U root space decomposition ...... 5 unitary Lie algebra ...... 4 root system ...... 140 unitary polar space ...... 13 irreducible ...... 140 unitary triple ...... 8 of a Lie algebra ...... 4 universal embedding ...... 96

S V sandwich algebra ...... 74 vanishing set ...... 62 sandwich element ...... 8 admissible ...... 63 sandwich point ...... 8 afﬁne ...... 63 sandwich property ...... 75 variety Index 155

afﬁne ...... 137 very real roots ...... 77

W weight ...... 74, 143

Y Young’s geometry ...... 12

Z Zariski topology ...... 137 156 Index Acknowledgements

This thesis is the result of a ten-year stay at Eindhoven University of Technology, the last four years of which I spent as a Ph.D. student in the Discrete Algebra and Geometry group at Eindhoven University of Technology. It would not have been possible without the help of many people. First and foremost I would like to thank my supervisors Hans Cuypers and Arjeh Cohen not only for giving me the opportunity to carry out this Ph.D. project, but also for investing so much time in me and for providing me with sound advice, good ideas, and many interesting problems. In addition I want to express my gratitude to Jan Draisma who, although not officially my supervisor, was always willing to lend a helping hand. I benefited greatly from his ideas and insights. Chapter4 of this thesis would not have been possible if it was not for him. What made these four years especially enjoyable was the great working atmosphere within the Discrete Mathematics group and the many contacts with people from CASA, Combinatorial Optimization, and Security. My special thanks go out not only to Dan and Erik for a collaboration which laid the ground work for many results in this thesis, but also to Çiçek and Shona for providing me with food, Erwin for being Erwin, Max for some last-minute programming, and Maxim for improving my Dutch. Many thanks also to all the other current and former colleagues of which there are too many to mention. I very much enjoyed the many lunches, the coffee breaks, the study groups, the squash and football games, the occasional ball fights, and the rice waffles that I have shared with many of them. Of course our secretaries Anita and Rianne deserve a special word of thanks for assisting me in many different ways. Rianne in particular I would like to thank for the many interesting conversations we have had. Doing sports and especially playing football has been my favourite pastime and formed a welcome distraction to my teaching and research activities. For that I am thankful to the members of Pusphaira and Old Soccers. We did not have much success, but I am confident that will change in the future. My defense committee is formed by Andries Brouwer, Arjeh Cohen, Hans Cuypers, Jan Draisma, Bettina Eick, Tom De Medts, and Bernhard Muhlherr.¨ I would like to 158 Acknowledgements thank them for the time invested, their willingness to judge my work, and the valuable suggestions which improved my thesis considerably. Lastly, but definitely not least, I am deeply grateful to my friends and my family. My parents I cannot thank enough as they made me what I am today. Jan and Ria I am grateful for the warmth with which they welcomed me. Also I would like to thank my sisters Dorris and Hellen, my twin brother Peter, and the newcomers in our family: Tonnie, Joram, and Jessey. Finally, I thank Marjanne not only for ten years of love, support and patience, but also for giving me Sep, my pride and joy, who puts a smile upon my face every time I see him.

Jos in ’t panhuis Heeze, August 2009 Curriculum Vitae

Jos in ’t panhuis was born in Roermond, the Netherlands, on July 10, 1981. In 1999 he finished his pre-university education in Sittard at RKSG Serviam (1993-1998; name change into Serviam-College in 1998-1999) at vwo-gymnasium level. In that same year he enrolled at Eindhoven University of Technology to study mathematics and computer science. After finishing the first year successfully in both subjects, he continued in mathematics. In 2005 he obtained his Master’s degree (ir.) in Industrial and Aplied Mathematics after writing a Master’s thesis entitled Planar Diagrams and Combinatorial Tensor Cat- egories under the supervision of dr. H.J.M. Sterk and prof. dr. A.M. Cohen. During his studies he also carried out an internship at ASML in Veldhoven under the supervision of dr. H.J.M. Sterk. From 2005 until 2009 he was a Ph.D. student at Eindhoven University of Technology under supervision of dr. F.G.M.T. Cuypers and prof. dr. A.M. Cohen. The present thesis is the result of his work in this period. Besides his work as a Ph.D. student, he participated in several study groups at uni- versities in the Netherlands (Eindhoven, Utrecht, and Twente) and Denmark (Lyngby), in which industrial scientists worked alongside mathematicians on problems of direct industrial relevance. Moreover, he was a member of the departmental council and one of the organizers of the EIDMA Seminar Combinatorial Theory. Starting from november 2009 he will be working in the field of risk management at ABN AMRO. 160