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
Document status and date: Published: 01/01/2009 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
Printed by Printservice Technische Universiteit Eindhoven. Cover: dual affine plane of order two.
Design by Oranje Vormgevers, Eindhoven.
A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-1912-5
and Geometries
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
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.
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 ele-ments are those nonzero eleele-ments x for which [x, [x, g]] ⊆ Fx. Extremal eleele-ments for which [x, [x, g]] = {0} are called sandwich elements. The definitions of extremal ele-ments and sandwich eleele-ments 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 ele-ments 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
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 con-nected. 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 cor-respond to extremal elements inside a Lie algebra: the class of finite irreducible cotri-angular spaces. Each such cotricotri-angular 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, symplec-tic, 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
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.
Preface v
Contents ix
1 Preliminaries 1
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
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
5.5.2 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: charac-teristic not two . . . 131
5.8.5 The equivalence classes and the universal embedding: charac-teristic 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
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 transforma-tions. 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 adxto 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 ALieis 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)].
If there is an n ∈ N with gn = {0}, then g is called nilpotent. Moreover, if there is an n ∈ N with g(n)= {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.
Example 1.3 For n ∈ N and F a field define hn(F) as the vector space F2n+1together
with the multiplication induced by
ij =
(j − i)2n+1 if |j − i| = 1 and i, j ∈ [2n],
0 otherwise.
Here, (and in the remainder of this thesis) [m] and [k, m] are defined such that
∀k<m∈Z : {k, . . . , m} =
[1, m] = [m] if k = 1, and [k, m] otherwise.
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 algebraand theorems by Ado and Iwasawa (Jacobson 1962, Chapter 6) prove that every (finite-dimensional) Lie algebra is isomorphic to some linear Lie algebra.
If V = Fn, for a certain n ∈ N, then dim gl(V ) = n2and 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 Examples1.4–1.7.
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
n2+ 2n over F.
Example 1.5 Let F be a field, let n ∈ N, let In ∈ gln(F) be the identity matrix, and
define f to be the bilinear form on F2n+1defined 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-isfying f (Ax, y) = −f (x, Ay) for all x, y ∈ Fn. The following matrices form a basis:
• Ei+1,j+1− En+j+1,n+i+1for i, j ∈ [n],
• Ei+1,n+j+1− Ej+1,n+i+1for i, j ∈ [n] with i < j, and
• En+i+1,j+1− En+j+1,i+1for i, j ∈ [n] with i < j.
The Lie algebra o2n+1(F) is said to be of type Bnand is referred to as the (odd)
orthog-onal Lie algebraof dimension 2n2+ n over F.
Example 1.6 Let F be a field, let n ∈ N, and define f to be the bilinear form on F2n defined by 0 In −In 0 .
We define sp2n(F) as the subalgebra of gl2n(F) consisting of those matrices A which
satisfy f (Ax, y) = −f (x, Ay) for all x, y ∈ Fn. The following matrices form a basis:
• Ei,n+ifor i ∈ [n],
• En+i,ifor i ∈ [n],
• Ei,j− En+j,n+ifor i, j ∈ [n],
• Ei,n+j+ Ej,n+ifor i, j ∈ [n] with i < j, and
• En+i,j+ En+j,ifor i, j ∈ [n] with i < j.
The Lie algebra sp2n(F) is said to be of type Cnand is referred to as the symplectic Lie
Example 1.7 Let F be a field, let n ∈ N, and define f to be the bilinear form on F2n defined by 0 In In 0 .
We define o2n(F) as the subalgebra of gl2n(F) consisting of those matrices A which
satisfy f (Ax, y) = −f (x, Ay) for all x, y ∈ Fn. The following matrices form a basis:
• Ei,j− En+j,n+ifor i, j ∈ [n],
• Ei,n+j− Ej,n+ifor i, j ∈ [n] with i < j, and
• En+i,j− En+j,ifor i, j ∈ [n] with i < j.
The Lie algebra o2n(F) is said to be of type Dnand is referred to as the (even) orthogonal
Lie algebraof dimension 2n2− n over F.
The linear Lie algebra in the next example will return in Chapter2.
Example 1.8 Let n ∈ N and let F be a field which is the fixpoint set of an involution σ of a field F. This makes V = Fnan 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 of dimension n2 over F. Intersecting un(F, f ) with sln(F) gives another Lie algebra
sun(F, f ) called the special unitary Lie algebra of dimension n2− 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 gα:= {x ∈ g | ∀h∈g0 : hx = α(h)x} 6= {0}. This makes
g= g0⊕M
α∈Φ
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−αfor 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
∀F a field: g⊗ZF is a Lie algebra over F.
If Γ is the Dynkin diagram corresponding to Φ, then this Lie algebra is called the Cheval-ley algebraover 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 F a field the Chevalley algebra g ⊗ZF
is often simple, but not always (Seligman 1967, Strade 2004).
Examples of Chevalley algebras are the Lie algebras introduced in Examples1.4–
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
gen-eralized 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 ∀x,y∈Π : hxhy = 0, exfx= hx, hxey = Axyey, hxfy = −Axyfy, and ∀x6=y∈Π : exfy = 0, ad1−Axy ex ey = 0, ad1−Af xy x fy = 0.
For x ∈ Π, assign to ex, fx, hxthe weights αx, −αx, 0 ∈ ZΠ, 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
grading of gKMby weight:
gKM= M
β∈ZΠ
(gKM)β.
Here, for each β ∈ ZΠ, the summand (gKM)β is the weight space consisting of all monomials of weight β. In fact, the root system Φ of gKMof type Γ satisfies
Φ = {β ∈ ZΠ\ {0} | (gKM)β 6= {0}}. It contains the simple roots αxfor x ∈ Π.
A Chevalley basis of gKM consists of the images of hx, x ∈ Π, and one vector eα ∈ (gKM)α for every root α ∈ Φ, where eαx and e−αx may be taken as the images of exand 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 char-acteristic 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 fol-low 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 x :=0 1 0 0 and y :=0 0 1 0 .
Then, g = sl2(F) and either all nonzero matrices in g are extremal or only the matrices
of rank 1. To be more specific,
E ∪ {0} = Fx ∪ Fy ∪ S δ∈F∗
F(δx + δ−1y + xy) if char(F) 6= 2, and
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 infinitesi-mal transvections on g are extreinfinitesi-mal and generate g. Otherwise, all infinitesiinfinitesi-mal Siegel transvections on g are extremal and generate g. See for instance Postma (2007).
For x, y ∈ E we write (x, y) ∈ E−2 ⇐⇒ Fx = Fy, E−1 ⇐⇒ xy = 0, (x, y) /∈ E−2, and Fx + Fy ⊆ E ∪ {0}, 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.
charac-teristic two the definition of E−1above 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, polarif 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 onX 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 tripleif (Y, Z) ∈ E2. Here, a hyperbolic path is simply a path in (E, E2). Example 1.13 The Lie algebra of Example1.11satisfies
E × E =
E−2⊕ E−1⊕ E1 if char(F) = 2, and 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 ele-ments and sandwich points, respectively. Again, if it clear which Lie algebra g is meant, we omit g.
Example 1.14 The Lie algebra of Examples1.11and1.13satisfies S ∪ {0} =
Fxy if char(F) = 2, and {0} otherwise.
Lemma 1.15 (Cohen and Ivanyos 2006) If g is a Lie algebra generated by extremal elements, then the Lie subalgebrahSi generated by the sandwich elements is an ideal of g.
If gxcan 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 gxis 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.
Lemma 1.16 (Cohen and Ivanyos 2006) Let x ∈ E. With the restriction that gx is chosen to be identically zero ifx is a sandwich element, the map gxis a uniquely defined functional on g.
For x ∈ E we call gxthe 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 byE. 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 formg 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 elementsx 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) ∈ E1if 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
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 E0of extremal elements commuting with x such that
z = v + X
w∈E0
w.
Hence, since x ∈ E(L) commutes with E0, xyxz = xyxv + X
w∈E0
xyxw = xyxv = g(x, yv)x − g(x, v)xy − g(x, y)xv = g(x, yv)x − g(x, v)xy − g(x, y)xv
+ X
w∈E0
(−g(y, xw)x − g(x, w)xy − g(x, y)xw) = g(x, yv)x − g(x, v)xy − g(x, y)xv
+ X
w∈E0
(g(x, yw)x − g(x, w)xy − g(x, y)xw) = g(x, y(v + X w∈E0 w)x − g(x, v + X w∈E0 w)xy − g(x, y)x(v + X w∈E0 w) = 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
exp(x, α)y = y + αxy + α2gxyx. If x ∈ E, then we often write exp(x) instead of exp(x, 1). Note that
∀x∈E : char(F) 6= 2 ⇒ exp(x) =
∞ X n=0 1 n!ad n x.
Lemma 1.20 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F containing an extremal elementx. Then
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 orderr.
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 Figures1.1 and1.2. There the lines are coloured in such a way that two lines intersect if and only if they have different colours. Note that in Figure1.1each 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 Xiare singular subspaces. Here, a subspace is called singular if
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 1.21–1.23are 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-transpo-sitions. 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.
Theorem 1.24 (Buekenhout 1974) Each connected Fischer space is isomorphic to a Fischer space coming from a3-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 irre-ducible 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 Tninstead 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 ).
If V = F2n2 (n ∈ N), then we can take f as the symplectic form with
((x1, . . . , x2n), (y1, . . . , y2n)) 7→
n
X
i=1
(x2i−1y2i+ y2i−1x2i),
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 N O(V, Q).
can assume Q is the quadratic form with (x1, . . . , x2n+1) 7→ n X i=1 x2i−1x2i+ x22n+1.
If V = F2n2 (n ∈ N), then there are two possibilities. Either we can take Q as the
quadratic form with
(x1, . . . , x2n) 7→
n
X
i=1
x2i−1x2i,
and we write N O2n+(2) instead of N O(V, Q), or we can take Q as the quadratic form with (x1, . . . , x2n) 7→ n X i=1 x2i−1x2i+ x22n−1+ x22n, and we write N O−2n(2) instead of N O(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 NO(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 ).
If V = Fn4, then we write HUn(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
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, thenΦ is isomophic to
• T (Ω) for a set Ω,
• HSp(V, f ) for a symplectic space (V, f ) over the field F2,
• N O(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, orΩ(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 Example1.32will be considered. This cotriangular space will turn out to be a convenient description of N O7(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 ÷cy = {0} ÷ ([0, 8] \ (x ÷ y)).
This enables us to define
L = [
x,y∈P
{{x, y, x ÷ y}, {x, y, x ÷cy}} ∩ 2P.
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 xyinduces the isomorphism
X7 ∼= N O7(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,
• HSp(V, f ) for a symplectic space (V, f ) of dimension at least 6 over the field F2,
• N O(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
N O+6(2) ∼= T8,
and
∀n∈N: HSp2n(2) ∼= N O2n+1(2).
Proof. The first isomorphism is readily checked. The last isomorphism follows from
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) ∼= N O2n+1(2),
• orthogonal type if there is an n ≥ 3 and an ∈ {±} with (, n) 6= (+, 3) such that (P, L) ∼= N O2n (2).
Thus, if we restrict ourselves to the finite case, as in Chapter 5, 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 cotri-angular space.
Proposition 1.36 (Hall 1983) Let n ≥ 4 be an integer. Then Tn+1has generating rank n, HSp2n−2(2) has generating rank 2n − 1, N O2n±(2) has generating rank 2n, and N O−6(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 Appendix Afor 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 Xnwith 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) := (N O−6(2), N O7(2), N O8+(2)).
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 collinear-ity graph ofΠ 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 ∼= N O2n+1(2),
M2n ∼= N O2n∓(2), M2n+1 ∼= N O2n+1(2), M2n+2 ∼= N O ±
2n+2(2) of N O2n+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, ∆xis a cotriangular space of symplectic type generated by m − 1 points.
There-fore, it is sufficient to prove that there are subspaces M2n ∼= N O ∓
2n(2) and M2n+1 ∼=
N O2n+1(2) of N O∓2n+2(2) with
M2n⊆ M2n+1 ⊆ N O±2n+2(2). Now, for N O+2n+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 O−2n+2(2) \ {(0, . . . , 0, 1, 0)} | x2n+2 = 0i, M2n := h(x1, . . . , x2n+2) ∈ N O−2n+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,
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 ∼= N O−4(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 N O4−(2) the span of a subspace isomorphic to a dual affine plane of order two and a point outside this subspace is N O−4(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
Then (P, L) is called a root filtration space with filtration (Pi)i∈[−2,2]if • P−2is equality on P,
• P−1is 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≤1is 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
• P26= ∅ 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.
Examples1.42–1.47give 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−1as 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 P2as the set consisting of all pairs
of non-collinear points, and define P0as the complement in P × P of P−2] P2. Then
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 point-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
∀(x,H),(y,K)∈P : ((x, H), (y, K)) ∈ P−2 ⇐⇒ x = y ∧ H = K, P≤−1 ⇐⇒ x = y ∨ H = 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
∀l,m∈P : (l, m) ∈ P−2 ⇐⇒ l = m, P−1 ⇐⇒ hl, mi is a singular plane,
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 byE. Moreover, define F as the set of projective lines all of whose points belong toE.
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).
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).
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 Lemma1.18, 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 gen-erated by three (possibly coinciding) extremal elementsx, y, z then Table2.1determines the multiplication onL.
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.
x y z xy xz yz xy z y xz x y z xy xz y z xy z y xz 0 xy xz 2 gxy x 2 gxz x xy z 2 gxy z x gxy z x − gxz xy − gxy xz 0 y z − 2 gxy y y xz 2 gy z y − gxy z y + gyz xy − 2 gxy z y − gxy y z 0 − xy z + y xz − 2 gxz z − 2 gyz z − gxy z z − gyz xz gxy z z − gy z xz − gxz y z − gxz y z 0 gxy z x + gxz xy gxy z y + gy z xy 2 gxy gyz x + gxy z xy − 2 gxy gxz y − gxy z xy − gxy xz + gxy y z − gxy xy z + gxy y xz 0 gxy z z − gy z xz − 2 gxz gy z x + gxy z xz − 2 gxz gyz x − 2 gxy gxz z + gxz y z − gxz xy z +2 gxy z xz − 2 gxz xy z + gxz y xz 0 − 2 gxz gyz y − 2 gxy gy z z − 2 gxz gyz y − gxy z y z − 2 gxy z y z + gyz xy z − gyz y xz +2 gyz y xz 0 − gxy z gyz x − gxy z gxz y − gxy z gxy z − 2 gxz gyz xy +2 gxy gy z xz − 2 gxy gxz y z 0 T able 2.1: The multiplication table of L
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 Lemma1.18, the following result is obvious.
Proposition 2.2 If g is a Lie algebra over a field F containing a Lie subalgebra L gen-erated by a hyperbolic pair(Fx, Fy), then
• L = Fx + Fy + Fxy ∼= sl2(F), and
• C(L) =
Fxy if char(F) = 2, and {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 Example1.11we 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
(E ∩ L) ∪ {0} = [
λ,µ∈F
F(λ2x + gxyµ 2
y + λµxy).
The extremal elements described in this proposition can be identified with the traceless 2 × 2-matrices of rank 1.
Proof of Proposition2.3. Because of Proposition2.2, L = Fx + Fy + Fxy ∼= sl2(F) and ∀λ,µ∈F∗ : λ2x + xy + g xyµ 2 y = λ2exp(y, −λ−1µ)x ∈ E.
Hence, [ λ,µ∈F F(λ2x + gxyµ 2 y + λµxy) ⊆ (E ∩ L) ∪ {0}.
If char(F) 6= 2, then we are done. See also Example1.11. 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,
∀u,v∈E :
(xy)(xy)u = 0, and
(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,
αz + gxyy = α(αx + xy) + gxyy = α2x + gxyy + αxy ∈ E.
This is in contradiction with the fact that the sum of two non-commuting extremal ele-ments 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
αz + (gxy − αβ)y = α(αx + βy + xy) + (gxy− αβ)y = α2x + gxyy + αxy ∈ E. If gxy 6= αβ, then we obtain a contradiction with the fact that the sum of two non-commuting extremal elements is not extremal. Hence, gxy = αβ and αz = α
2 y + gxyy + αxy. We conclude [ λ,µ∈F F(λ2x + gxyµ2y + λµxy) ⊇ (E ∩ L) ∪ {0}.
2.4
Lie subalgebras generated by symplectic triples
First we determine the possible isomorphism types of Lie subalgebras generated by sym-plectic 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.
2.4.1 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 M = α β η −η γ −α θ −θ θ −η ζ −ζ θ −η ζ −ζ α, β, γ, η, θ, ζ ∈ F
is the Lie subalgebra of the symplectic Lie algebra over F defined by the symplectic form
f = 0 1 0 0 −1 0 0 0 0 0 0 1 0 0 −1 0 ,
then the map induced by
(x, y, z) 7→ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , 0 1 −1 1 0 0 0 0 0 1 −1 1 0 1 −1 1 induces an isomorphism betweenL and M .
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 Proposition2.4is the following lemma. Lemma 2.5
L = R + T and T ∼= sl2(F).
Proof. Clearly,
R + T ⊆ L.
Moreover, substituting xz = 0 in Table2.1gives the following multiplication table for R + T . x y xy a b c x y xy a b c 0 xy −2x −b 0 0 −xy 0 2y 0 −a 0 2x −2y 0 −a b 0 b 0 a 0 2c 0 0 a −b −2c 0 0 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.
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.
Suppose a = b = 0 and suppose char(F) 6= 2. Then c = 12ab = 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
∀ω∈F∗
\{1} : ω
2
x + z = exp(y, (ω + 1)−1)exp(x, ω)exp(y, 1)z ∈ E. Hence, (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, a 6= 0 6= b.
Next, suppose |F| > 2 and c = 0. Then, ∀ω∈F∗
\{1} : (1 + ω)x + ω(ω + 1)z = exp(y, (ω + 1) −1
)exp(z, ω)exp(y, 1)x ∈ E. 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,
Fx + Fz ⊆ E(g ⊗FF).
In particular x + z is extremal in g. We conclude (x, z) ∈ E−1= ∅. This is a
contradic-tion. 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 Lemma2.6, 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 .
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, 0 = −1
2(−2βx)y + 1
4xx(4γy) = βxy − 2γx = −δb, and 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 Lemma2.6, 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.
