The Book of Involutions

Max-Albert Knus

Alexander Merkurjev

Markus Rost

Jean-Pierre Tignol

Author address:

Dept. Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland

E-mail address: knus@math.ethz.ch
URL: http://www.math.ethz.ch/~knus/

Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA

E-mail address: merkurev@math.ucla.edu
URL: http://www.math.ucla.edu/~merkurev/

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516/

Département de mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@agel.ucl.ac.be
URL: http://www.math.ucl.ac.be/tignol/

Contents

Préface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . xi Chapter I. Involutions and Hermitian Forms . . . . . . . . . . . . . 1 Chapter II. Invariants of Involutions . . . . . . . . . . . . . . . . . 3

§1. The Index . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter III. Similitudes . . . . . . . . . . . . . . . . . . . . . . 5

§2. General Properties . . . . . . . . . . . . . . . . . . . . . . 5 Chapter IV. Algebras of Degree Four . . . . . . . . . . . . . . . . 7

§3. Exceptional Isomorphisms . . . . . . . . . . . . . . . . . . 7 Chapter V. Algebras of Degree Three . . . . . . . . . . . . . . . . 9

§4. Étale and Galois Algebras . . . . . . . . . . . . . . . . . . 9 Chapter VI. Algebraic Groups . . . . . . . . . . . . . . . . . . . 11

§5. Hopf Algebras and Group Schemes . . . . . . . . . . . . . . 12

5.A. Classification of irreducible root systems . . . . . . . . . 13 Chapter VII. Galois Cohomology . . . . . . . . . . . . . . . . . . 15 Chapter VIII. Composition and Triality . . . . . . . . . . . . . . . 17 Chapter IX. Cubic Jordan Algebras . . . . . . . . . . . . . . . . . 19

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter X. Trialitarian Central Simple Algebras . . . . . . . . . . . 23 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

iv CONTENTS

Préface

Quatre des meilleurs algébristes d’aujourd’hui (j’aimerais dire, comme jadis, ≪géomètres^{≫}, au sens noble mais hélas désuet du terme) nous donnent ce beau Livre des Involutions, qu’ils me demandent de préfacer.

Quel est le propos de l’ouvrage et à quels lecteurs s’adresse-t-il? Bien sûr il y est souvent question d’involutions, mais celles-ci sont loin d’être omniprésentes et le titre est plus l’expression d’un état d’âme que l’affirmation d’un thème central. En fait, les questions envisagées sont multiples, relevant toutes de domaines importants des mathématiques contemporaines ; sans vouloir être exhaustif (ceci n’est pas une introduction), on peut citer :

- les formes quadratiques et les algèbres de Clifford,

- les algèbres associatives centrales simples (ici les involutions, et notamment celles de seconde espèce, se taillent une place de choix !) mais aussi les algèbres alternatives et les algèbres de Jordan,

- les algèbres de Hopf,

- les groupes algébriques, principalement semi-simples,

- la cohomologie galoisienne. Pour ce qui est du public concerné, la lecture ou la consultation du livre sera profitable à un large éventail de mathématiciens. Le non-initié y trouvera une introduction claire aux concepts fondamentaux des domaines en question ; exposés le plus souvent en fonction d’applications concrètes, ces notions de base sont présentées de façon vivante et dépouillée, sans généralités gratuites (les auteurs ne sont pas adeptes de grandes théories abstraites). Le lecteur déjà informé, ou croyant l’être, pourra réapprendre (ou découvrir) quelques beaux théorèmes jadis ^{≪}bien connus^{≫} mais un peu oubliés dans la littérature récente, ou au contraire, voir des résultats qui lui sont en principe familiers exposés sous un jour nouveau et éclairant (je pense par exemple à l’introduction des algèbres trialitaires au dernier chapitre). Enfin, les spécialistes et les chercheurs auront à leur disposition une référence précieuse, parfois unique, pour des développements récents, souvents dûs aux auteurs eux-mêmes, et dont certains sont exposés ici pour la première fois (c’est par exemple le cas pour plusieurs résultats sur les invariants cohomologiques, donnés à la fin du chapitre 7).

Malgré la grande variété des thèmes considérés et les individualités très marquées des quatre auteurs, ce Livre des Involutions a une unité remarquable. Le ciment un peu fragile des involutions n’est certes pas seul à l’expliquer. Il y a aussi, bien sûr, les interconnections multiples entre les sujets traités ; mais plus déterminante encore est l’importance primordiale accordée à des structures fortes, se prêtant par exemple à des théorèmes de classification substantiels. Ce n’est pas un hasard si les algèbres centrales simples de petites dimensions (trois et quatre), les groupes exceptionnels de type G_{2} et F_{4} (on regrette un peu que Sa Majesté E_{8} fasse ici figure de parent pauvre), les algèbres de composition, …, reçoivent autant d’attention.

On l’a compris, ce Livre est tout à la fois un livre de lecture passionnant et un ouvrage de référence d’une extrême richesse. Je suis reconnaissant aux auteurs de l’honneur qu’ils m’ont fait en me demandant de le préfacer, et plus encore de m’avoir permis de le découvrir et d’apprendre à m’en servir.

Jacques Tits

Introduction

For us an involution is an anti-automorphism of order two of an algebra. The most elementary example is the transpose for matrix algebras. A more complicated example of an algebra over ℚ admitting an involution is the multiplication algebra of a Riemann surface (see the notes at the end of Chapter I for more details). The central problem here, to give necessary and sufficient conditions on a division algebra over ℚ to be a multiplication algebra, was completely solved by Albert (1934/35). To achieve this, Albert developed a theory of central simple algebras with involution, based on the theory of simple algebras initiated a few years earlier by Brauer, Noether, and also Albert and Hasse, and gave a complete classification over ℚ. This is the historical origin of our subject, however our motivation has a different source. The basic objects are still central simple algebras, i.e., “forms” of matrix algebras. As observed by Weil (1960), central simple algebras with involution occur in relation to classical algebraic simple adjoint groups: connected components of automorphism groups of central simple algebras with involution are such groups (with the exception of a quaternion algebra with an orthogonal involution, where the connected component of the automorphism group is a torus), and, in their turn, such groups are connected components of automorphism groups of central simple algebras with involution.

Even if this is mainly a book on algebras, the correspondence between algebras and groups is a constant leitmotiv. Properties of the algebras are reflected in properties of the groups and of related structures, such as Dynkin diagrams, and vice versa. For example we associate certain algebras to algebras with involution in a functorial way, such as the Clifford algebra (for orthogonal involutions) or the λ-powers and the discriminant algebra (for unitary involutions). These algebras are exactly the “Tits algebras,” defined by Tits (1971) in terms of irreducible representations of the groups. Another example is algebraic triality, which is historically related with groups of type D_{4} (É. Cartan) and whose “algebra” counterpart is, so far as we know, systematically approached here for the first time.

In the first chapter we recall basic properties of central simple algebras and involutions. As a rule for the whole book, without however going to the utmost limit, we try to allow base fields of characteristic 2 as well as those of other characteristic. Involutions are divided up into orthogonal, symplectic and unitary types. A central idea of this chapter is to interpret involutions in terms of hermitian forms over skew fields. Quadratic pairs, introduced at the end of the chapter, give a corresponding interpretation for quadratic forms in characteristic 2.

In Chapter II we define several invariants of involutions; the index is defined for every type of involution. For quadratic pairs additional invariants are the discriminant, the (even) Clifford algebra and the Clifford module; for unitary involutions we introduce the discriminant algebra. The definition of the discriminant algebra is prepared for by the construction of the λ-powers of a central simple algebra. The last part of this chapter is devoted to trace forms on algebras, which represent an important tool for recent results discussed in later parts of the book. Our method of definition is based on scalar extension: after specifying the definitions “rationally” (i.e., over an arbitrary base field), the main properties are proven by working over a splitting field. This is in contrast to Galois descent, where constructions over a separable closure are shown to be invariant under the Galois group and therefore are defined over the base field. A main source of inspiration for Chapters I and II is the paper [26] of Tits on “Formes quadratiques, groupes orthogonaux et algèbres de Clifford.”

In Chapter III we investigate the automorphism groups of central simple algebras with involutions. Inner automorphisms are induced by elements which we call similitudes. These automorphism groups are twisted forms of the classical projective orthogonal, symplectic and unitary groups. After proving results which hold for all types of involutions, we focus on orthogonal and unitary involutions, where additional information can be derived from the invariants defined in Chapter II. The next two chapters are devoted to algebras of low degree. There exist certain isomorphisms among classical groups, known as exceptional isomorphisms. From the algebra point of view, this is explained in the first part of Chapter IV by properties of the Clifford algebra of orthogonal involutions on algebras of degree 3, 4, 5 and 6. In the second part we focus on tensor products of two quaternion algebras, which we call biquaternion algebras. These algebras have many interesting properties, which could be the subject of a monograph of its own. This idea was at the origin of our project.

Algebras with unitary involutions are also of interest for odd degrees, the lowest case being degree 3. From the group point of view algebras with unitary involutions of degree 3 are of type A_{2}. Chapter V gives a new presentation of results of Albert and a complete classification of these algebras. In preparation for this, we recall general results on étale and Galois algebras.

The aim of Chapter VI is to give the classification of semisimple algebraic groups over arbitrary fields. We use the functorial approach to algebraic groups, although we quote without proof some basic results on algebraic groups over algebraically closed fields. In the central section we describe in detail Weil’s correspondence [28] between central simple algebras with involution and classical groups. Exceptional isomorphisms are reviewed again in terms of this correspondence. In the last section we define Tits algebras of semisimple groups and give explicit constructions of them in classical cases.

The theme of Chapter VII is Galois cohomology. We introduce the formalism and describe many examples. Previous results are reinterpreted in this setting and cohomological invariants are discussed. Most of the techniques developed here are also needed for the following chapters.

The last three chapters are dedicated to the exceptional groups of type G_{2}, F_{4} and to D_{4}, which, in view of triality, is also exceptional. In the Weil correspondence, octonion algebras play the algebra role for G_{2} and exceptional simple Jordan algebras the algebra role for F_{4}.

Octonion algebras are an important class of composition algebras and Chapter VIII gives an extensive discussion of composition algebras. Of special interest from the group point of view are “symmetric” compositions. In dimension 8 these are of two types, corresponding to algebraic groups of type A_{2} or type G_{2}. Triality is defined through the Clifford algebra of symmetric 8-dimensional compositions. As a step towards exceptional simple Jordan algebras, we introduce twisted compositions, which are defined over cubic étale algebras. This generalizes a construction of Springer. The corresponding group of automorphisms in the split case is the semidirect product Spin_{8} ⋊S_{3}.

In Chapter IX we describe different constructions of exceptional simple Jordan algebras, due to Freudenthal, Springer and Tits (the algebra side) and give interpretations from the algebraic group side. The Springer construction arises from twisted compositions, defined in Chapter VIII, and basic ingredients of Tits constructions are algebras of degree 3 with unitary involutions, studied in Chapter III. We conclude this chapter by defining cohomological invariants for exceptional simple Jordan algebras.

The last chapter deals with trialitarian actions on simple adjoint groups of type D_{4}. To complete Weil’s program for outer forms of D_{4} (a case not treated by Weil), we introduce a new notion, which we call a trialitarian algebra. The underlying structure is a central simple algebra with an orthogonal involution, of degree 8 over a cubic étale algebra. The trialitarian condition relates the algebra to its Clifford algebra. Trialitarian algebras also occur in the construction of Lie algebras of type D_{4}. Some indications in this direction are given in the last section.

Exercises and notes can be found at the end of each chapter. Omitted proofs sometimes occur as exercises. Moreover we included as exercises some results we like, but which we did not wish to develop fully. In the notes we wanted to give complements and to look at some results from a historical perspective. We have tried our best to be useful; we cannot, however, give strong guarantees of completeness or even fairness.

This book is the achievement of a joint (and very exciting) effort of four very different people. We are aware that the result is still quite heterogeneous; however, we flatter ourselves that the differences in style may be viewed as a positive feature.

Our work started out as an attempt to understand Tits’ definition of the Clifford algebra of a generalized quadratic form, and ended up including many other topics to which Tits made fundamental contributions, such as linear algebraic groups, exceptional algebras, triality, … Not only was Jacques Tits a constant source of inspiration through his work, but he also had a direct personal influence, notably through his threat — early in the inception of our project — to speak evil of our work if it did not include the characteristic 2 case. Finally he also agreed to bestow his blessings on our book sous forme de préface. For all that we thank him wholeheartedly.

This book could not have been written without the help and the encouragement of many friends. They are too numerous to be listed here individually, but we hope they will recognize themselves and find here our warmest thanks. Richard Elman deserves a special mention for his comment that the most useful book is not the one to which nothing can be added, but the one which is published. This no-nonsense statement helped us set limits to our endeavor. We were fortunate to get useful advice on various points of the exposition from Ottmar Loos, Antonio Paques, Parimala, Michel Racine, David Saltman, Jean-Pierre Serre and Sridharan. We thank all of them for lending helping hands at the right time. A number of people were nice enough to read and comment on drafts of parts of this book: Eva Bayer-Fluckiger, Vladimir Chernousov, Ingrid Dejaiffe, Alberto Elduque, Darrell Haile, Luc Haine, Pat Morandi, Holger Petersson, Ahmed Serhir, Tony Springer, Paul Swets and Oliver Villa. We know all of them had better things to do, and we are grateful. Skip Garibaldi and Adrian Wadsworth actually summoned enough grim self-discipline to read a draft of the whole book, detecting many shortcomings, making shrewd comments on the organization of the book and polishing our broken English. Each deserves a medal. However, our capacity for making mistakes certainly exceeds our friends’ sagacity. We shall gratefully welcome any comment or correction.

Tignol had the privilege to give a series of lectures on “Central simple algebras, involutions and quadratic forms” in April 1993 at the National Taiwan University. He wants to thank Ming-chang Kang and the National Research Council of China for this opportunity to test high doses of involutions on a very patient audience, and Eng-Tjioe Tan for making his stay in Taiwan a most pleasant experience. The lecture notes from this crash course served as a blueprint for the first chapters of this book.

Our project immensely benefited by reciprocal visits among the authors. We should like to mention with particular gratitude Merkurjev’s stay in Louvain-la-Neuve in 1993, with support from the Fonds de Développement Scientifique and the Institut de Mathématique Pure et Appliquée of the Université catholique de Louvain, Rost’s stays in Zürich at the Mathematics Research Institute (FIM) in winter 1995–96 and at the Mathematics Department of the ETH Zürich for the winter semester of 1996–97, and Tignol’s stay at the Mathematics Department of the ETH Zürich for the winter semester of 1995–96, both with support from the Eidgenössische Technische Hochschule. Moreover, Merkurjev gratefully acknowledges support from the Alexander von Humboldt foundation and the hospitality of the Bielefeld university for the year 1995–96, and Tignol is grateful to the National Fund for Scientific Research of Belgium and to the European Commission for partial support under the “crédit aux chercheurs” and the TMR programmes respectively (contract ERB-FMRX-CT97-0107).

The four authors enthusiastically thank Herbert Rost (Markus’ father) for the design of the cover page, in particular for his wonderful and colorful rendition of the Dynkin diagram D_{4}. They also give special praise to Sergei Gelfand, Director of Acquisitions of the American Mathematical Society, for his helpfulness and patience in taking care of all their wishes for the publication.

Conventions and Notations

Maps. The image of an element x under a map f is generally denoted f(x); the notation x^{f} is also used however, notably for homomorphisms of left modules. In that case, we also use the right-hand rule for mapping composition; for the image of x ∈ X under the composite map X ⟶^{f} Y ⟶^{g} Z we set either g ∘ f(x) or x^{fg} and the composite is thus either g ∘ f or fg.

xii CONVENTIONS AND NOTATIONS

CHAPTER I Involutions and Hermitian Forms

Our perspective in this work is that involutions on central simple algebras are twisted forms of symmetric or alternating bilinear forms up to a scalar factor. To motivate this point of view, we consider the basic, classical situation of linear algebra.

Let V be a finite dimensional vector space over a field F of arbitrary characteristic. A bilinear form b: V × V → F is called nonsingular if the induced map

b̂: V → V^{∗} = Hom_{F} (V, F)

defined by

b̂(x)(y) = b(x, y) for x, y ∈ V

is an isomorphism of vector spaces. For any f ∈ End_{F} (V) we may then define σ_{b}(f) ∈ End_{F} (V) by

σ_{b}(f) = b̂^{−1} ∘ f^{t} ∘ b̂

where f^{t} ∈ End_{F} (V^{∗}) is the transpose of f, defined by mapping φ ∈ V^{∗} to φ ∘ f. Alternately, σ_{b}(f) may be defined by the following property:

b(x, f(y)) = b(σ_{b}(f)(x), y) for x, y ∈ V. (∗)

The map σ_{b}: End_{F} (V) → End_{F} (V) is then an anti-automorphism of End_{F} (V) which is known as the adjoint anti-automorphism with respect to the nonsingular bilinear form b. The map σ_{b} clearly is F -linear.

2 I. INVOLUTIONS AND HERMITIAN FORMS

CHAPTER II Invariants of Involutions

In this chapter, we define various kinds of invariants of central simple algebras with involution (or with quadratic pair) and we investigate their basic properties. The invariants considered here are analogues of the classical invariants of quadratic forms: the Witt index, the discriminant, the Clifford algebra and the signature. How far the analogy can be pushed depends of course on the nature of the involution: an index is defined for arbitrary central simple algebras with involution or quadratic pair, but the discriminant is defined only for orthogonal involutions and quadratic pairs, and the Clifford algebra just for quadratic pairs. The Clifford algebra construction actually splits into two parts: while it is impossible to define a full Clifford algebra for quadratic pairs, the even and the odd parts of the Clifford algebra can be recovered in the form of an algebra and a bimodule. For unitary involutions, the notion of discriminant turns out to lead to a rich structure: we associate in §?? a discriminant algebra (with involution) to every unitary involution on a central simple algebra of even degree. Finally, signatures can be defined for arbitrary involutions through the associated trace forms. These trace forms also have relations with the discriminant or discriminant algebra. They yield higher invariants for algebras with unitary involution of degree 3 in Chapter V and for Jordan algebras in Chapter IX.

The invariants considered in this chapter are produced by various techniques. The index is derived from a representation of the algebra with involution as the endomorphism algebra of some hermitian or skew-hermitian space over a division algebra, while the definitions of discriminant and Clifford algebra are based on the fact that scalar extension reduces the algebra with quadratic pair to the endomorphism algebra of a quadratic space. We show that the discriminant and even Clifford algebra of the corresponding quadratic form can be defined in terms of the adjoint quadratic pair, and that the definitions generalize to yield invariants of arbitrary quadratic pairs. A similar procedure is used to define the discriminant algebra of a central simple algebra of even degree with unitary involution. Throughout most of this chapter, our method of investigation is thus based on scalar extension: after specifying the definitions “rationally” (i.e., over an arbitrary base field), the main properties are proven by extending scalars to a splitting field. This method contrasts with Galois descent, where constructions over a separable closure are shown to be invariant under the action of the absolute Galois group and are therefore defined over the base field.

§1. The Index

According to (??), every central simple F -algebra with involution (A, σ) can be represented as (End_{D}(V), σ_{h}) for some division algebra D, some D-vector space V and some nonsingular hermitian form h on V. Since this representation is essentially unique, it is not difficult to check that the Witt index w(V, h) of the hermitian space (V, h), defined as the maximum of the dimensions of totally isotropic subspaces of V, is an invariant of (A, σ). In this section, we give an alternate definition of this invariant which does not depend on a representation of (A, σ) as (End_{D}(V), σ_{h}), and we characterize the involutions which can be represented as adjoint involutions with respect to a hyperbolic form. We define a slightly more general notion of index which takes into account the Schur index of the algebra. A (weak) analogue of Springer’s theorem on odd degree extensions is discussed in the final subsection.

CHAPTER III Similitudes

In this chapter, we investigate the automorphism groups of central simple algebras with involution. The inner automorphisms which preserve the involution are induced by elements which we call similitudes, and the automorphism group of a central simple algebra with involution is the quotient of the group of similitudes by the multiplicative group of the center. The various groups thus defined are naturally endowed with a structure of linear algebraic group; they may then be seen as twisted forms of orthogonal, symplectic or unitary groups, depending on the type of the involution. This point of view will be developed in Chapter VI. Here, however, we content ourselves with a more elementary viewpoint, considering the groups of rational points of the corresponding algebraic groups.

After a first section which contains general definitions and results valid for all types, we then focus on quadratic pairs and unitary involutions, where additional information can be derived from the algebra invariants defined in Chapter II. In the orthogonal case, we also use the Clifford algebra and the Clifford bimodule to define Clifford groups and spin groups.

§2. General Properties

To motivate our definition of similitude for an algebra with involution, we first consider the split case, where the algebra consists of endomorphisms of bilinear or hermitian spaces.

6 III. SIMILITUDES

CHAPTER IV Algebras of Degree Four

Among groups of automorphisms of central simple algebras with involution, there are certain isomorphisms, known as exceptional isomorphisms, relating algebras of low degree. (The reason why these isomorphisms are indeed exceptional comes from the fact that in some special low rank cases Dynkin diagrams coincide, see §??.) Algebras of degree 4 play a central rôle from this viewpoint: their three types of involutions (orthogonal, symplectic, unitary) are involved with three of the exceptional isomorphisms, which relate them to quaternion algebras, 5-dimensional quadratic spaces and orthogonal involutions on algebras of degree 6 respectively. A correspondence, first considered by Albert [1], between tensor products of two quaternion algebras and quadratic forms of dimension 6 arises as a special case of the last isomorphism.

The exceptional isomorphisms provide the motivation for, and can be obtained as a consequence of, equivalences between certain categories of algebras with involution which are investigated in the first section. In the second section, we focus on tensor products of two quaternion algebras, called biquaternion algebras, and their Albert quadratic forms. The third section yields a quadratic form description of the reduced Whitehead group of a biquaternion algebra, making use of symplectic involutions. Analogues of the reduced Whitehead group for algebras with involution are also discussed.

§3. Exceptional Isomorphisms

The exceptional isomorphisms between groups of similitudes of central simple algebras with involution in characteristic different from 2 are easily derived from Wonenburger’s theorem (??), as the following proposition shows:

8 IV. ALGEBRAS OF DEGREE FOUR

CHAPTER V Algebras of Degree Three

The main topic of this chapter is central simple algebras of degree 3 with involutions of the second kind and their étale (commutative) subalgebras. To every involution of the second kind on a central simple algebra of degree 3, we attach a 3-fold Pfister form, and we show that this quadratic form classifies involutions up to conjugation. Involutions whose associated Pfister form is hyperbolic form a distinguished conjugacy class; we show that such an involution is present on every central simple algebra of degree 3 with involution of the second kind, and we characterize distinguished involutions in terms of étale subalgebras of symmetric elements.

We start with Galois descent, followed by a general discussion of étale and Galois algebras. These are tools which will also be used in subsequent chapters.

§4. Étale and Galois Algebras

Throughout this section, let F be an arbitrary base field, let F_{sep} be a separable closure of F and let Γ be the absolute Galois group of F:

Γ = Gal(F_{sep}/F).

The central theme of this section is a correspondence between étale F -algebras and Γ-sets, which is set up in the first subsection. This correspondence is then restricted to Galois algebras and torsors. The final subsection demonstrates the special features of étale algebras of dimension 3.

10 V. ALGEBRAS OF DEGREE THREE

CHAPTER VI Algebraic Groups

It turns out that most of the groups which have occurred thus far in the book are groups of points of certain algebraic group schemes. Moreover, many constructions described previously are related to algebraic groups. For instance, the Clifford algebra and the discriminant algebra are nothing but Tits algebras for certain semisimple algebraic groups; the equivalences of categories considered in Chapter IV, for example of central simple algebras of degree 6 with a quadratic pair and central simple algebras of degree 4 with a unitary involution over an étale quadratic extension (see §??), reflect the fact that certain semisimple groups have the same Dynkin diagram (D_{3} ≃ A_{3} in this example).

The aim of this chapter is to give the classification of semisimple algebraic groups of classical type without any field characteristic assumption, and also to study the Tits algebras of semisimple groups.

In the study of linear algebraic groups (more generally, affine group schemes) we use a functorial approach equivalent to the study of Hopf algebras. The advantage of such an approach is that nilpotents in algebras of functions are allowed (and they really do appear when considering centers of simply connected groups over fields of positive characteristic); moreover many constructions like kernels, intersections of subgroups, are very natural. A basic reference for this approach is Waterhouse [27]. The classical view of an algebraic group as a variety with a regular group structure is equivalent to what we call a smooth algebraic group scheme.

The classical theory (mostly over an algebraically closed field) can be found in Borel [5], Humphreys [12], or Springer [23]. We also refer to Springer’s survey article [24]. (The new (1998) edition of [23] will contain the theory of algebraic groups over non algebraically closed fields.) We use some results in commutative algebra which can be found in Bourbaki [7], [8], [9], and in the book of Matsumura [14].

The first three sections of the chapter are devoted to the general theory of group schemes. In §?? we define the families of algebraic groups related to an algebra with involution, a quadratic form, and an algebra with a quadratic pair. After a short interlude (root systems, in §??) we come to the classification of split semisimple groups over an arbitrary field. In fact, this classification does not depend on the ground field F, and is essentially equivalent to the classification over the algebraic closure F_{alg} (see Tits [25], Borel-Tits [6]).

The central section of this chapter, §??, gives the classification of adjoint semisimple groups over arbitrary fields. It is based on the observation of Weil [28] that (in characteristic different from 2) a classical adjoint semisimple group is the connected component of the automorphism group of some algebra with involution. In arbitrary characteristic the notion of orthogonal involution has to be replaced by the notion of a quadratic pair which has its origin in the fundamental paper [26] of Tits. Groups of type G_{2} and F_{4} which are related to Cayley algebras (Chapter VIII) and exceptional Jordan algebras (Chapter IX), are also briefly discussed.

In the last section we define and study Tits algebras of semisimple groups. It turns out that for the classical groups the nontrivial Tits algebras are the λ-powers of a central simple algebra, the discriminant algebra of a simple algebra with a unitary involution, and the Clifford algebra of a central simple algebra with an orthogonal pair—exactly those algebras which have been studied in the book (and nothing more!).

§5. Hopf Algebras and Group Schemes

This section is mainly expository. We refer to Waterhouse [27] for proofs and more details.

Hopf algebras. Let F be a field and let A be a commutative (unital, associative) F -algebra with multiplication m: A ⊗_{F} A → A. Assume we have F -algebra homomorphisms

c: A → A ⊗_{F} A (comultiplication)
i: A → A  (co-inverse)
u: A → F  (co-unit)

which satisfy the following:

(a) The diagram

A ⟶^{c} A ⊗_{F} A
c↓  ↓^{c⊗Id}
A ⊗_{F} A ⟶^{Id⊗c} A ⊗_{F} A ⊗_{F} A

commutes.

(b) The map

A ⟶^{c} A ⊗_{F} A ⟶^{u⊗Id} F ⊗_{F} A = A

equals the identity map Id: A → A.

(c) The two maps

A ⟶^{c} A ⊗_{F} A ⟶^{i⊗Id} A ⊗_{F} A ⟶^{m} A

A ⟶^{u} F ⟶^{·1} A

coincide. An F -algebra A together with maps c, i, and u as above is called a (commutative) Hopf algebra over F. A Hopf algebra homomorphism f: A → B is an F -algebra homomorphism preserving c, i, and u, i.e., (f ⊗f)∘c_{A} = c_{B} ∘f, f ∘i_{A} = i_{B} ∘f, and u_{A} = u_{B} ∘ f. Hopf algebras and homomorphisms of Hopf algebras form a category.

§5. HOPF ALGEBRAS AND GROUP SCHEMES 13

is the cone of dominant weights in Λ (relative to Π). We introduce a partial ordering on Λ: χ > χ′ if χ − χ′ is sum of simple roots. For any λ ∈ Λ/Λ_{r} there exists a unique minimal dominant weight χ(λ) ∈ Λ_{+} in the coset λ. Clearly, χ(0) = 0.

5.A. Classification of irreducible root systems. There are four infinite families A_{n}, B_{n}, C_{n}, D_{n} and five exceptional irreducible root systems E_{6}, E_{7}, E_{8}, F_{4}, G_{2}. We refer to Bourbaki [10] for the following datas about root systems.

Type A_{n}, n ≥ 1. Let V = ℝ^{n+1}/(e_{1} + e_{2} + ⋯ + e_{n+1})ℝ where {e_{1}, …, e_{n+1}} is the canonical basis of ℝ^{n+1}. We denote by \overline{e}_{i} the class of e_{i} in V.

Root system  : Φ = \{ \overline{e}_{i} − \overline{e}_{j} | i ≠ j \}, n(n + 1) roots.
Root lattice  : Λ_{r} = \{ ∑ a_{i}\overline{e}_{i} | ∑ a_{i} = 0 \}.
Weight lattice  : Λ = ∑ \overline{e}_{i}ℤ, Λ/Λ_{r} ≃ ℤ/(n + 1)ℤ.
Simple roots  : Π = \{\overline{e}_{1} − \overline{e}_{2}, \overline{e}_{2} − \overline{e}_{3}, …, \overline{e}_{n} − \overline{e}_{n+1}\}.
Dynkin diagram : [figure: 1; 2; n]
Aut(Dyn(Φ))  : {1} if n = 1, {1, τ} if n ≥ 2.
Dominant weights : Λ_{+} = \{ ∑ a_{i} · \overline{e}_{i} ∈ Λ | a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n+1} \}.
Minimal weights : \overline{e}_{1} + \overline{e}_{2} + ⋯ + \overline{e}_{i}, i = 1, 2, …, n + 1.
Type B_{n}, n ≥ 1. Let V = ℝ^{n} with canonical basis {e_{i}}.
Root system  : Φ = { ±e_{i}, ±e_{i} ± e_{j} | i > j }, 2n^{2} roots.
Root lattice  : Λ_{r} = ℤ^{n}.
Weight lattice  : Λ = Λ_{r} + \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n})ℤ, Λ/Λ_{r} ≃ ℤ/2ℤ.
Simple roots  : Π = {e_{1} − e_{2}, e_{2} − e_{3}, …, e_{n−1} − e_{n}, e_{n}}.
Dynkin diagram :  > [figure: 1; 2; n−1; n]
Aut(Dyn(Φ))  : {1}.
Dominant weights : Λ_{+} = { ∑ a_{i}e_{i} ∈ Λ | a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} ≥ 0 }.
Minimal weights : 0, \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n}).
Type C_{n}, n ≥ 1. Let V = ℝ^{n} with canonical basis {e_{i}}.
Root system  : Φ = { ±2e_{i}, ±e_{i} ± e_{j} | i > j }, 2n^{2} roots.
Root lattice  : Λ_{r} = { ∑ a_{i}e_{i} | a_{i} ∈ ℤ, ∑ a_{i} ∈ 2ℤ }.
Weight lattice  : Λ = ℤ^{n}, Λ/Λ_{r} ≃ ℤ/2ℤ.
Simple roots  : Π = {e_{1} − e_{2}, e_{2} − e_{3}, …, e_{n−1} − e_{n}, 2e_{n}}.
Dynkin diagram :  < [figure: 1; 2; n−1; n]
Aut(Dyn(Φ))  : {1}.
Dominant weights : Λ_{+} = { ∑ a_{i}e_{i} ∈ Λ | a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} ≥ 0 }.
Minimal weights : 0, e_{1}.

14 VI. ALGEBRAIC GROUPS

Type D_{n}, n ≥ 3. (For n = 2 the definition works but yields A_{1} + A_{1}.) Let V = ℝ^{n} with canonical basis {e_{i}}.

Root system  : Φ = { ±e_{i} ± e_{j} | i > j }, 2n(n − 1) roots.
Root lattice  : Λ_{r} = { ∑ a_{i}e_{i} | a_{i} ∈ ℤ, ∑ a_{i} ∈ 2ℤ }.
Weight lattice  : Λ = ℤ^{n} + \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n})ℤ,

Λ/Λ_{r} ≃ \begin{cases}ℤ/2ℤ ⊕ ℤ/2ℤ & if n is even, \\ ℤ/4ℤ & if n is odd. \end{cases}

Simple roots  : Π = {e_{1} − e_{2}, …, e_{n−1} − e_{n}, e_{n−1} + e_{n}}.
Dynkin diagram :  \begin{matrix}n−1 \\ n \end{matrix} [figure: 1; 2; n−2]
Aut(Dyn(Φ))  : S_{3} if n = 4, {1, τ} if n = 3 or n > 4.
Dominant weights : Λ_{+} = { ∑ a_{i}e_{i} ∈ Λ | a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n}, a_{n−1} + a_{n} ≥ 0 }.
Minimal weights : 0, e_{1}, \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n−1} ± e_{n}).

Exceptional types.

E_{6}: Aut(Dyn(Φ)) = {1, τ}, Λ/Λ_{r} ≃ ℤ/3ℤ.
[figure]
E_{7}: Aut(Dyn(Φ)) = {1}, Λ/Λ_{r} ≃ ℤ/2ℤ.
[figure]
E_{8}: Aut(Dyn(Φ)) = {1}, Λ/Λ_{r} = 0.
[figure]
F_{4} : Aut(Dyn(Φ)) = {1}, Λ/Λ_{r} = 0.  > [figure]
G_{2}: Aut(Dyn(Φ)) = {1}, Λ/Λ_{r} = 0.  < [figure]

CHAPTER VII Galois Cohomology

In the preceding chapters, we have met groupoids M = M(F) of “algebraic objects” over a base field F, for example finite dimensional F -algebras or algebraic groups of a certain type. If over a separable closure F_{sep} of F the groupoid M(F_{sep}) is connected, i.e., all objects over F_{sep} are isomorphic, then in many cases the objects of M are classified up to isomorphism by a cohomology set H^{1}(Gal(F_{sep}/F), A), where A is the automorphism group of a fixed object of M(F_{sep}). The aim of this chapter is to develop the general theory of such cohomology sets, to reinterpret some earlier results in this setting and to give techniques (like twisting) which will be used in later parts of this book.

There are four sections. The basic techniques are explained in §??, and §?? gives an explicit description of the cohomology sets of various algebraic groups in terms of algebras with involution. In §?? we focus on the cohomology groups of μ_{n}, which are used in §?? to reinterpret various invariants of algebras with involution or of algebraic groups, and to define higher cohomological invariants.

16 VII. GALOIS COHOMOLOGY

CHAPTER VIII Composition and Triality

The main topic of this chapter is composition algebras. Of special interest from the algebraic group point of view are symmetric compositions. In dimension 8 there are two such types: Okubo algebras, related to algebras of degree 3 with unitary involutions (type A_{2}), and para-Cayley algebras related to Cayley algebras (type G_{2}). The existence of these two types is due to the existence of inequivalent outer actions of the group ℤ/3ℤ on split simply connected simple groups of type D_{4} (“triality” for Spin_{8}), for which the fixed elements define groups of type A_{2}, resp. G_{2}. Triality is defined here through an explicit computation of the Clifford algebra of the norm of an 8-dimensional symmetric composition. As a step towards exceptional simple Jordan algebras, we introduce in the last section twisted compositions, generalizing a construction of Springer. The corresponding group of automorphisms is the semidirect product Spin_{8} ⋊S_{3}.

18 VIII. COMPOSITION AND TRIALITY

CHAPTER IX Cubic Jordan Algebras

The set of symmetric elements in an associative algebra with involution admits the structure of a Jordan algebra. One aim of this chapter is to give some insight into the relationship between involutions on central simple algebras and Jordan algebras. After a short survey on central simple Jordan algebras in §??, we specialize to Jordan algebras of degree 3 in §??; in particular, we discuss extensively “Freudenthal algebras,” a class of Jordan algebras connected with Hurwitz algebras and we describe the Springer construction, which ties twisted compositions with cubic Jordan algebras. On the other hand, cubic Jordan algebras are also related to cubic associative algebras through the Tits constructions (§??). Of special interest, and the main object of study of this chapter, are the exceptional simple Jordan algebras of dimension 27, whose automorphism groups are of type F_{4}. The different constructions mentioned above are related to interesting subgroups of F_{4}. For example, the automorphism group of a split twisted composition is a subgroup of F_{4} and outer actions on Spin_{8} (triality) become inner over F_{4}. Tits constructions are related to the action of the cyclic group ℤ/3ℤ on Spin_{8} which yields invariant subgroups of classical type A_{2}, and Freudenthal algebras are related to the action of the group S_{3} on Spin_{8} which yields invariant subgroups of exceptional type G_{2}.

Cohomological invariants of exceptional simple Jordan algebras are discussed in the last section.

Notes

§??. Tits constructions for fields of characteristic not 2 first appeared in print in Jacobson’s book [13], as did the fact, also due to Tits, that any Albert algebra is a first or second Tits construction. These results were announced by Tits in a talk at the Oberwolfach meeting “Jordan-Algebren und nicht-assoziativen Algebren, 17–26.8.1967.” With the kind permission of J. Tits and the Research Institute in Oberwolfach, we reproduce Tits’ Résumé:

Exceptional simple Jordan Algebras (I) Denote by k a field of characteristic not 2, by A a central simple algebra of degree 3 over k, by n: A → A, tr: A → A the reduced norm and reduced trace, and by ×: A×A → A the symmetric bilinear product defined by (x×x)x = n(x). For x ∈ A, set \overline{x} = \frac{1}{2} (tr(x) − x). Let c ∈ k^{∗}. In the sum A_{0} + A_{1} + A_{2} of three copies of A, introduce the following product:

x_{0}  y_{1}  z_{2}
x′_{0}  \frac{1}{2} (xx′ + x′x)_{0} (\overline{x′}y)_{1}  (z\overline{x′})_{2}
y′_{1}  (\overline{x}y′)_{1}  c(y × y′)_{2}  (\overline{y′}z)_{0}
z′_{2}  (z′\overline{x})_{2}  (\overline{yz′})_{0}  \frac{1}{c} (z × z′)_{1}

(II) Denote by ℓ a quadratic extension of k, by B a central simple algebra of degree 3 over ℓ, and by σ: B → B an involution of the second kind kind such that k = { x ∈ ℓ | x^{σ} = x }. Set B^{Sym} = { x ∈ B | x^{σ} = x }. Let b ∈ B^{Sym} and c ∈ l^{∗} be such that n(b) = c^{σ}c. In the sum B^{Sym} + B_{∗} of B^{Sym} and a copy B_{∗} of B, define a product by

x  y
x′  \frac{1}{2} (xx′ + x′x)  (\overline{x′}y)_{∗}
y′  (\overline{x}y′)_{∗}  (\overline{yby′^{σ} + y′by^{σ}})+(c^{σ}(y^{σ} × y′^{σ})b^{−1})_{∗}

Theorem 1. The 27-dimensional algebras described under (I) and (II) are exceptional simple Jordan algebras over k. Every such algebra is thus obtained.

Theorem 2. The algebra (I) is split if c ∈ n(A) and division otherwise. The algebra (II) is reduced if c ∈ n(B) and division otherwise.

Theorem 3. There exists an algebra of type (II) which does not split over any cyclic extension of degree 2 or 3 of k. (Notice that such an algebra is necessarily division and is not of type (I)).

(For more details, cf. N. Jacobson. Jordan algebras, a forthcoming book). J. Tits

Observe that the ×-product used by Tits is our ×-product divided by 2. The extension of Tits constructions to cubic structures was carried out by McCrimmon [15]. Tits constructions were systematically used by Petersson and Racine, see for example [17] and [16]. Petersson and Racine showed in particular that (with a few exceptions) simple cubic Jordan structures can be constructed by iteration of the Tits process ([16], Theorem 3.1). The result can be viewed as a cubic analog to the theorem of Hurwitz, proved by iterating the Cayley-Dickson process.

Tits constructions can be used to give simple examples of exceptional division Jordan algebras of dimension 27. The first examples of such division algebras were constructed by Albert [2]. They were significantly more complicated than those through Tits constructions. Assertions (??) and (??) of Theorem (??) and (??) are due to Tits. The nice cohomological proof given here is due to Ferrar and Petersson [11] (for first Tits constructions).

§??. The existence of the invariants f_{3} and f_{5} was first noticed by Serre (see [22] and [21, Fith Edition, Annexe, §3]). The direct computation of the trace form given here, as well as Propositions (??) and (??) are due to Rost. Serre suggested the existence of the invariant g_{3}. Its definition is due to Rost [20]. An elementary approach to that invariant can be found in Petersson-Racine [18] and a description in characteristic 3 can be found in Petersson-Racine [19].

22 IX. CUBIC JORDAN ALGEBRAS

CHAPTER X Trialitarian Central Simple Algebras

We assume in this chapter that F is a field of characteristic not 2. Triality for PGO_{8}^{+}, i.e., the outer action of S_{3} on PGO_{8}^{+} and its consequences, is the subject of this chapter. In the first section we describe the induced action on H^{1}(F, PGO_{8}^{+}). This cohomology set classifies ordered triples (A, B, C) of central simple algebras of degree 8 with involutions of orthogonal type such that (C(A, σ_{A}), \underline{σ}) ⟶^{∼} (B, σ_{B}) × (C, σ_{C}). Triality implies that this property is symmetric in A, B and C, and the induced action of S_{3} on H^{1}(F, PGO_{8}^{+}) permutes A, B, and C. As an application we give a criterion for an orthogonal involution on an algebra of degree 8 to decompose as a tensor product of three involutions.

We may view a triple (A, B, C) as above as an algebra over the split étale algebra F × F × F with orthogonal involution (σ_{A}, σ_{B}, σ_{C}) and some additional structure (the fact that (C(A, σ_{A}), \underline{σ}) ⟶^{∼} (B, σ_{B}) × (C, σ_{C})). Forms of such “algebras,” called trialitarian algebras, are classified by H^{1}(F, PGO_{8}^{+} ⋊S_{3}). Trialitarian algebras are central simple algebras with orthogonal involution of degree 8 over cubic étale F -algebras with a condition relating the central simple algebra and its Clifford algebra. Connected components of automorphism groups of such trialitarian algebras give the outer forms of simple adjoint groups of type D_{4}.

Trialitarian algebras also occur in the construction of Lie algebras of type D_{4}. Some indications in this direction are given in the last section.

24 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

Bibliography

Page references at the end of entries indicate the places in the text where the entry is quoted.

1. A. A. Albert, On the Wedderburn norm condition for cyclic algebras, Bull. Amer. Math. Soc. 37 (1931), 301–312, (also: [3, pp. 185–196]). 7.

2. , A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958), 1–28, (also: [4, pp. 701–728]). 21.

3. , Collected mathematical papers. Part 1, American Mathematical Society, Providence, RI, 1993, Associative algebras and Riemann matrices, Edited by R. E. Block, N. Jacobson, J. Marshall Osborn, D. J. Saltman and D. Zelinsky.

4. , Collected mathematical papers. Part 2, American Mathematical Society, Providence, RI, 1993, Nonassociative algebras and miscellany, Edited by R. E. Block, N. Jacobson, J. Marshall Osborn, D. J. Saltman and D. Zelinsky.

5. A. Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. 11.

6. A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965), no. 27, 55–150. 11.

7. N. Bourbaki, Éléments de mathématique, Masson, Paris, 1985, Algèbre commutative. Chapitres 1 à 4. 11.

8. , Éléments de mathématique, Masson, Paris, 1985, Algèbre commutative. Chapitres 5 à 7. 11.

9. , Éléments de mathématique, Masson, Paris, 1983, Algèbre commutative. Chapitres 8 et 9. 11.

10. , Éléments de mathématique, Hermann, Paris, 1975, Groupes et algèbres de Lie. Chapitre 7 et 8. 13.

11. J. C. Ferrar and H. P. Petersson, Exceptional simple Jordan algebras and Galois cohomology, Arch. Math. (Basel) 61 (1993), no. 6, 517–520. 21.

12. J. E. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. 21. 11.

13. N. Jacobson, Structure and representations of Jordan algebras, American Mathematical Society, Providence, R.I., 1968, American Mathematical Society Colloquium Publications, Vol. XXXIX. 20.

14. H. Matsumura, Commutative algebra, second ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. 11.

15. K. McCrimmon, The Freudenthal-Springer-Tits constructions revisited, Trans. Amer. Math. Soc. 148 (1970), 293–314. 20.

16. H. P. Petersson and M. L. Racine, Classification of algebras arising from the Tits process, J. Algebra 98 (1986), no. 1, 244–279. 20.

17. , Jordan algebras of degree 3 and the Tits process, J. Algebra 98 (1986), no. 1, 211–243. 20.

18. , An elementary approach to the Serre-Rost invariant of Albert algebras, Indag. Mathem., N.S. 7 (1996), no. 3, 343–365. 21.

19. , The Serre-Rost invariant of Albert algebras in characteristic 3, Indag. Mathem., N.S. 8 (1997), no. 4, 543–548. 21.

20. M. Rost, A (mod 3) invariant for exceptional Jordan algebras, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 12, 823–827. 21.

21. J.-P. Serre, Cohomologie galoisienne, cinquième édition, révisée et complétée, fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994, english transl.: Galois cohomology, Springer-Verlag, Berlin, 1997. 21.

22. , Cohomologie galoisienne: progrès et problèmes, Astérisque (1995), no. 227, Exp. No. 783, 4, 229–257, Séminaire Bourbaki, Vol. 1993/94. 21.

23. T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, Birkhäuser Boston, Mass., 1981, new edition to appear. 11.

24. , Linear algebraic groups, Algebraic Geometry IV, Linear Algebraic Groups, Invariant Theory (Berlin) (A. N. Parshin and I. R. Shafarevich, eds.), Encyclopaedia of Mathematical Sciences, vol. 55, Springer-Verlag, 1994, pp. 1–121. 11.

25. J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) (A. Borel and G. D. Mostow, eds.), vol. 9, Amer. Math. Soc., Providence, R.I., 1966, 1966, pp. 33–62. 11.

26. , Formes quadratiques, groupes orthogonaux et algèbres de Clifford, Invent. Math. 5 (1968), 19–41. viii, 11.

27. W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York, 1979. 11, 12.

28. A. Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960), 589–623 (1961), (also: [29, pp. 413–447]). viii, 11.

29. , Scientific works. Collected papers. Vol. II (1951–1964), Springer-Verlag, New York, 1979.

Index

absolute Galois group, 9 adjoint anti-automorphism, 1 cone of dominant weights, 13 Hopf algebra, 12 Hopf algebra homomorphism, 12 minimal dominant weight, 13 nonsingular bilinear form, 1 transpose of a linear map, 1 triality, 17, 23

28 INDEX

Notation

A_{n} (root system), 13 B_{n} (root system), 13 C_{n} (root system), 13 D_{n} (root system), 14 E_{6} (root system), 14 E_{7} (root system), 14 E_{8} (root system), 14 F_{4} (root system), 14 F_{alg} (algebraic closure), 11 f^{t} (transpose of f), 1 G_{2} (root system), 14 Γ (absolute Galois group), 9 σ_{b} (involution adjoint to b), 1 x^{f} (left module homomorphism), xi χ(λ), 13