1 INTRODUCTION 1 Introduction As the title suggests, this thesis is aboutMoufang setsand some of their prop-erties. In fact, we mainly use two structures which are very closely related: Moufang sets andJordan algebras. Moufang sets are pairs consisting of a set and some subgroups of the per-mutation group of the set which follow some basic operation rules: LetM:= (X,(Ux)x∈X) be a pair whereXis a set andUxis a subgroup of the permu-tation group (denoted by Sym(X)) for allx∈X call. WeMaMoufang setif Uxstabilizesxand operates sharply transitively onX\{x}for allx∈X, and in addition,Uxis mapped ontoUyunder conjugation by an elementα∈Uzif α(x) =y. The groupsUxare calledroot groups. On the other hand, Jordan algebrasJare modules over a ringRwith unital element 1∈Jwith a quadratic multiplication, denoted byU, which has to fulﬁll some equations: letUabdenote the quadratic multiplication ofawithb, then U1= idJ, UxVy,x=Vx,yUx=UUxy,x, UUxy=UxUyUx for allx, y∈JwhereVy,xdenotes the linearization ofU,Vx,y(z) :={xyz}:= Ux,z(y(see 2.6.1 for the exact deﬁnition of the) U-multiplication and quadratic Jordan algebras). Well-known examples are the special Jordan algebrasA+whereAis an associa-tive algebra, and the+denotes the given quadratic multiplicationUxy:=xyx, or the Jordan Cliﬀord algebras over a given Cliﬀord algebra. We will see that special Jordan algebras are in a one-to-one correspondence with some kinds of Moufang sets. For example, Moufang sets of the polar line can be seen as so-calledample subspacesof special Jordan algebras. The main result of this thesis is the following: Main Result.LetMandM0be two Moufang sets of one of the following types: a Moufang set of the projective lineP1(K0), a Moufang set of the polar lineP L(K, K0, σ), an orthogonal Moufang setO(k, L0, q), or a mixed Moufang setM(k0, L0). IfM∼=M0, then all isomorphisms are known and induce – up to some excep-tions – isomorphisms of the underlying skew ﬁelds. The motivation for this theorem was the result of my diploma thesis [8], which proved the isomorphism problem for two Moufang sets of the polar line. In my diploma thesis, I could state the result for the polar lines only when the underlying ﬁelds are neither quaternion nor biquaternion algebras. By now, we could solve these cases as well. Moreover, we have a complete solution for all Moufang sets which arise from ﬁelds and have abelian root groups. The main result as you can see above is stated in a very general way. The isomorphisms are described more detailed in the sectionResultsof the third chapter.
1 INTRODUCTION
2
Besides this result we give a proof about the relation of simple Jordan alge-bras to Moufang sets, and we analyze the root groups of Moufang sets of the polar line so that we can prove their uniqueness. In chapter 2, we start with the basic deﬁnitions and notations for this the-sis. We explain the sesquilinear and quadratic forms as well as the polarities which we need for the main result. Furthermore, we give the deﬁnitions of some algebras like thebiquaternion algebrasand theCliﬀord algebras describe. We thequaternion algebrasin detail and illustrate the main concepts of Moufang sets and Jordan algebras. Here, we prove for both of these basic facts which hold for arbitrary Moufang sets and Jordan algebras. We introduce the notion of aZelmanov polynomialand explain at last thespecial universal envelopeof a Jordan algebra. The third chapter describes the Moufang sets which we study in the main result. We ﬁrst deﬁne the underlying structures:Skew ﬁeldswhich lead toinvo-lutory sets;quadratic spaceswhich arise from quadratic forms, andmixed pairs arising from (commutative) ﬁelds. We explain whatsimilarity, a weaker form of an isomorphic relation, means. With these information we create Moufang sets with abelian root groups: the Moufang sets of theprojective linegiven over skew ﬁelds, Moufang sets of thepolar linegiven over involutory sets,orthogonal Moufang sets deﬁned over quadratic spaces andmixedMoufang sets given over mixed pairs. After that, we state the theorems which lead to the main result. In chapter 4, we prove the main result step by step for the Moufang sets described above. Westart with two Moufang sets of the projective line and fol-low the order of the theorems from the section before. The problem of Moufang sets of the projective line is already solved by Hua’s well-known theorem, see [4]. We go on with the isomorphism between two mixed Moufang sets. In this case, we ﬁnd a direct construction of the ﬁelds out of the data of the Moufang set. Next we investigate the isomorphism problem for Moufang sets of diﬀerent types: Moufang sets of the projective line, mixed Moufang sets and orthogonal Moufang sets. These cases merely result from some lemmas. The next and more diﬃcult problem is the isomorphism of two orthogonal Moufang sets, where the isomorphism problem can be solved in two ways; ﬁrstly via a lemma of J. Tits from [15] where we only have to show that the requirements of the lemma are indeed fulﬁlled, and secondly by a direct construction of the ﬁelds, similar to the one of the mixed Moufang sets. At last we look at the Moufang sets of the polar line, being the biggest problem at all. We ﬁrst investigate the cases of the isomorphism of a Moufang set of the polar line and a Moufang set of another type. Then we go on to the problem for two Moufang sets of the polar line, which – up to some exceptions – is already proved in my diploma thesis [8]. It appeared that these exceptions, when the underlying skew ﬁeld is either a quaternion or a biquaternion algebra, were not easy to handle. By now, they are solved as well. In the ﬁfth chapter we show some further results on Moufang sets. We ﬁrst prove that the special simple Jordan algebras (following the classiﬁcation by K. McCrimmon and E. Zelmanov, [13]) are linked to Moufang sets, namely to all Moufang sets with abelian root groups we investigated before. This part is
1 INTRODUCTION
3
more a collection of facts than a concrete proof since all work has been done in the sections before. Afterwards we proof the uniqueness of theU-group for a Moufang set of the polar line. This proof, which is the detailed version of the proof in the paper [1], was developed during a stay in Ghent in cooperation with the other authors of the paper.
At last I want to thank all people who helped me ﬁnishing this thesis, ﬁrst of all my promotor Prof. Dr. Strothand my colleagues from Halle University, in particular Barbara Baumeister. This thesis would not have been completed like this without my several stays in Ghent, for which I want to thank Hendrik Van Maldeghem and Tom De Medts, the last one especially for his comments via email, and all the other colleagues from Ghent University who supported me during my times in Belgium.
Halle (Saale), October 6th, 2005 Rafael Knop
4
2 BASIC DEFINITIONS 2 Basic deﬁnitions We start with the basic deﬁnitions of structures that are important for this thesis, beginning with some basic facts aboutquadratic and sesquilinear forms and with deﬁnitions ofpolaritiesandalgebras we look at the special. Afterwards structures that we need later: the deﬁnitions of aMoufang set, aJordan algebra and thespecial universal envelope Beforeof a Jordan algebra. that, we start with explaining some notations and make conventions: 2.1 Notations and conventions In this thesis we often look at (skew) ﬁelds, vector spaces and projective geom-etry. If not explained in detail, that means: •Askew ﬁeldor adivision ring(usually denoted byKorF) is a ring with unit element and inverses for all non-zero elements. •Aﬁeld(usually denoted bykorf) is always commutative. •Avector spaceover a skew ﬁeld is meant as a left vector space such that the product of a matrixMand a vectorvis written asM∙vas usual. Scalarsαare multiplied from the left to a vectorv,α∙v. •In the projective geometry, classes of vectors are denoted with square brackets. [v] denotes the (projective) class of vectors for whichvis a representative. 2.2 Involutions, quadratic and sesquilinear forms Deﬁnition 2.2.1.LetK An anti-automorphismbe a skew ﬁeld.σ:K→K is called aninvolutionofKifσ2= id. An involutionσofKis calledinvolution of the ﬁrst kindifσﬁxesZ(K); it is calledof the second kindotherwise. LetKbe a skew ﬁeld andMn(K) denote the ring ofn×n-matrices overK. Then the involutions ofMn(K) are well known, see [5, p.189f]: The transpose of matrices is an involution, usually denoted bytand called the transpose involution map. TheL7→S(tL)S−1withS= diag{Q, . . . , Q}and Q:=−1010is an involution as well, called thesymplectic involution. IfAis a non-simple algebra, we haveA=B⊕Bopfor a simple algebraBand the mapε: (b1, b2)7→(b2, b1) is an involution, theexchange involution(see [5, p.187]). Deﬁnition 2.2.2.Letkbe a (commutative) ﬁeld,L0a vector space overk. Aquadratic formqonL0is a function fromL0toksuch that 1.q(λa) =λ2q(a) for allλ∈kanda∈L0 2. the functionf:L0×L0→kgiven byf(a, b) :=q(a+b)−q(a)−q(b) for alla, b∈L0is bilinear
2 BASIC DEFINITIONS
5
Thedefectofqis the set def(q) :={v∈L0|f(v, L0) = 0}. Aquadratic spaceis a triple (k, L0, q) withk, L0, qas above. The formqis anisotropicifq(a) = 0 if and only ifa= 0.qisnon-defectiveif def(q) ={0} anddefective def(otherwise. Ifq) =L0, we call the formqtotally defective. Note that if the formqis anisotropic and charK6= 2,qis always non-defective. A quadratic space isanisotropicifqis anisotropic, it isnon-defective(resp. defectiveortotally defective) ifqis. Two quadratic spaces (k, L0, q) and (f, L1, q∗) areisomorphicif there exists an isomorphismϕ:L0→L1of vector spaces and an isomorphismψ:k→fof ﬁelds such thatq∗◦ϕ=ψ◦q. We will mostly denote two isomorphic quadratic spaces just as (k, q)∼= (f, q∗) sinceL0resp.L1are determined bykandq(resp. fandq∗). TheWitt indexmof a quadratic form is the maximum of the dimensions of all isotropic subspaces ofL0: m:= max{dimU|U≤L0subspace, q(U) = 0} The bilinear formfis uniquely determined and called the bilinear form as-sociated toq. We will sometimes denote it byβq(especially in the cases of polarities, see 2.3). A generalization of the bilinear forms is given by the sesquilinear forms. Deﬁnition 2.2.3.LetKbe a skew ﬁeld,Vbe a right vector space overK andσ:K→Kan involution ofK function. Af:V×V→Kis called a sesquilinear form relative toσ, if for allu, v, x, y∈Vanda, b∈K 1.f(u+v, x+y) =f(u, x) +f(u, y) +f(v, y) +f(v, y) 2.f(ax, by) =aσ∙f(x, y)∙b A sesquilinear formfrelative toσis calledreﬂexiveiff(x, y) = 0⇔f(y, x) = 0. The formfis calledhermitian(resp.skew-hermitian), iff(x, y)σ=f(y, x) (resp.f(x, y)σ=−f(y, x)). A sesquilinear formfrelative toσis calledtrace-valuedif for allx∈V f(x, x)∈ {t+tσ|t∈K} 2.3 Polarities The theory of polarities is needed to prove the main result of the isomorphism problem for orthogonal Moufang sets, see 4.6. The following notations, deﬁni-tions and explanations are mostly taken from [15], chapter 8. Deﬁnition 2.3.1.LetPbe a projective space andπ⊂P×Pbe a symmetric correspondence. Letx⊥πydenote (x, y)∈π. For a subsetX⊂Pwe deﬁne X⊥(π):={y∈P|x⊥πyfor allx∈X}. πis apolarityinPif for allx∈Pthe setx⊥(π)is eitherPitself or a hyperplane ofP. LetP the orthogonality relation with respect Thenbe a projective space. to a reﬂexive sesquilinear formfinVinduces a polarityπinP. A polarity is
2 BASIC DEFINITIONS
6
called oftrace typeif it is represented by a trace-valued form. Now letPbe a projective space of a vector spaceV. A proportionality class κof quadratic forms inVis called aprojective quadratic forminP. Any element qof the class is said to representκ. The Witt index and defect ofκare those of q form. Theκisnon-defectiveifqis. The polarity represented byβqis called thepolarity associated toκ. 2.4 Algebras In this thesis we need some kind of algebras, namelyquaternion algebras,bi-quaternion algebrasandCliﬀord algebras. Before explaining them we need some basic algebra facts aboutpolynomialsandpolynomial identities: Deﬁnition 2.4.1.LetA Abe an associative algebra.polynomialfinAis an element of the free associative algebraF(A).fis called apolynomial identity onAiff6≡0 andf(a1, . . . , an) = 0 for alla1, . . . an∈A. Ais called aPI-algebraif it has a polynomial identityfonA. If (A, σ) is an associative algebra with involution, we say that (A, σ) satisﬁes a polynomial identity if we have a polynomialf∈ F(A) such that for allai∈A: f(a1, . . . , an, σ(ai), . . . , σ(an)) = 0. The following theorems are taken from [14], p.276 and p.36: Theorem 2.4.2.(Amitsur)Let(A, σ)be an associative algebra with involution. If(A, σ)satisﬁes a polynomial identity, thenAsatisﬁes a polynomial identity as well. Theorem 2.4.3.(Kaplansky)LetAbe a division algebra which satisﬁes a poly-nomial identityf. Then[A:Z(A)]<∞.
Now, we turn to the particular algebras and start with the quaternion alge-bras which appear in the cases of Moufang sets of the polar line and hermitian Moufang sets. They are mostly the counterexamples or cases which we have to analyze separately: LetEbe a ﬁeld,σan automorphism onEwithσ2= id and letK= FixE(σ) be the ﬁxed point set underσ. We will writex¯ :=xσ. E/Kseparable quadratic extension and by choosingis a β∈K∗, we construct a subringQ:= (E/K, β) ofM(2, E) consisting of the matrices aβb¯a¯b wherea, b∈E. We can identifyEwith its image inQunder the mapa7→ diag(a,¯a). Then obviouslyK=Z(Q) and dimKQ= 4. Deﬁne e2:01β0 = Then every element ofQcan be written asa+e2bwitha, b∈E, and this description is unique. We deﬁne a multiplication onQby the usual rules onE ¯ and viae22:=βandae2:=e2a.
2 BASIC DEFINITIONS
7
Deﬁnition 2.4.4.Aquaternion algebrais any algebra isomorphic toQ= (E/K, β) for some separable quadratic extensionE/Kand someβ∈K∗as deﬁned above. A quaternion algebra has astandard involution, denoted by x7→x¯ and given by a+b= ¯2b e2a−e We will now give some facts about quaternion algebras. In the following, letQ= (E/K, β) be a quaternion algebra of characteristic 2. Letσ:x7→x¯ denote the standard involution. Sincea+e2b= ¯a+e2bin charK= 2, we have FixQ(σ) ={d+e2a|d∈K, a∈E} all squares of ﬁxed points. Obviously, (d+e2a)2=d2+βa¯apoints, and they lie in the centerare again ﬁxed Z(Q) =K sincea¯a∈FixE=F=Z(Q). Apparently, this only holds for the standard involution. If we have a quaternion algebra with non-standard involutionτwe can always ﬁnd an element ﬁxed by τwhose square does not lie in the center. Moreover, every non-standard in-volution in a quaternion algebra is of the formτ=Int(u)◦σ, whereσis the standard involution anduσ+u= 0 (see [9], Proposition (2.21)). taking a By separable quadratic subﬁeldE/kwhich is orthogonal tou, the involutionτcan be written as (x+uy)τ=x+uyσwherex, y∈E. Next, we look at the biquaternion algebras. We just give their deﬁnition since they have only a small importance for this thesis. Deﬁnition 2.4.5.Abiquaternion algebrais an algebra which is the tensor product of two quaternion algebras. Biquaternion algebras are the central simple algebras of degree 4 and expo-nent 2 or 1. They are explained in§16 of [9]. Finally, we come to the Cliﬀord algebras. These are important examples for the Jordan algebras which we deﬁne later on. Here we distinguish between the Cliﬀord algebraand theCliﬀord algebra with basepointwhich is introduced by N. Jacobson and K. McCrimmon in [6]. Letkbe a ﬁeld andVa vector space overK. Thetensor algebraT(V) is deﬁned as T(V) :=k⊕V⊕(V⊗kV)⊕(V⊗kV⊗kV)⊕ ∙ ∙ ∙ Deﬁnition 2.4.6.1. Let (k, V, q) be a quadratic space,T(V) the tensor algebra as deﬁned above andI(V, q) the ideal ofT(V) given byI(V, q) := hu⊗u−q(u)∙1|u∈Vi. Then theCliﬀord algebra ofqis deﬁned as C(V, q) :=C(q) :=T(V)/I(V, q). 2. Let (k, V, q) as above, 1∈Vbe a basepoint, that meansq(1) = 1k. Let T(V) be again the tensor algebra andIB(q) an ideal ofT(V) given by IB(q) :=h1k−1, x⊗x−f(1, x)x+q(x)1|x∈Viwherefis the bilinear form associated toq the. ThenCliﬀord algebra with basepoint1 is deﬁned asC(q,1) :=T(V)/IB(q). 2.5 Moufang sets The most important structures in this thesis are the Moufang sets. They are introduced by Jacques Tits in [16]. Moufang sets are the rank-one-case of Mo-ufang buildings. In a group theoretic sense, Moufang sets are just thesplit
2 BASIC DEFINITIONS
8
BN-pairs of rank Here,1 (see for example [8] for the proof). we only state the deﬁnition and some basic facts: Deﬁnition 2.5.1.AMoufang setis a pair (X,(Ux)x∈X), whereXis a set andUxis a subgroup of Sym(X) for allx∈Xsuch that the following holds: (M1)Uxstabilizesxand operates sharply transitively onX\{x}for allx∈X (M2) For allx, y∈Xandα∈Uywe haveαUxα−1=Uα(x) The groupsUxare calledroot groups. The condition (M1) is satisﬁed if for all ﬁxedx0∈Xthe following three identities hold: (M1a) For ally∈X\{x0}:{u∈Ux0|u(y) =y}={idX} (M1b) For allz, y∈X\{x0}exists au∈Ux0such thatu(z) =y (M1c) For allu∈Ux0:u(x0) =x0 We will give examples of Moufang sets later on in section 3, where we explain several kinds of Moufang sets. More examples can be looked up in [8]. An important property of all Moufang sets is theµ-action: Given two elements 0,∞ ∈X, there exist unique elementsu∈U0andu0, u00∈ U∞such thatu0uu00interchanges 0 and∞ By(see for example [8], (4.4)). (M1b) there exists an elementx∈Xsuch thatu(x) =∞. Therefore we put µ(x) :=u0uu00and we call this thesimpleµ-action now(see also [1, p.3]). If x0∈X\{0,∞}, the actionµ(x, x0) :=µx,x0:=µ(x)−1µ(x0) ﬁxes both 0 and∞ and is called thedoubleµ-action. In this thesis, we often sayµ-multiplicationwhen we mean the doubleµ-action. In some cases we write for theµ-multiplication justµxinstead ofµ1,x. In [18, Chapter 33], theµ-multiplication is given for all Moufang sets we are interested in. Note that [18] only deals with Moufang polygons, but since all the Moufang sets we are considering arise as residues of Moufang polygons, the formulas can be found there. Deﬁnition 2.5.2.Let (X,(Ux)x∈X) and (Y,(Uy)y∈Y) be two Moufang sets. Anisomorphismbetween (X,(Ux)x∈X) and (Y,(Uy)y∈Y) (or justMoufang iso-morphismbetweenXandY) is a bijectionβ:X→Ysuch that for allx∈X the mapux7→βuxβ−1deﬁnes a group isomorphism fromUxontoUβ(x). It is well known that a Moufang isomorphism preserves theµ-multiplication: β(µa(b)) =µ0β(a)(β(b)) for theµ-multiplicationsµandµ0in the corresponding Moufang sets. We will also need the notion of asub Moufang set, in particular for the Moufang sets of the polar line. It is deﬁned as follows: Deﬁnition 2.5.3.Let (X,(Ux)x∈X) be a Moufang set. A pair (X0,(Ux00)x0∈X0) is called asub Moufang setifX0⊂Xand U0x0={u∈Ux0| ∀y∈X0:u(y)∈X0} In particular, a sub Moufang set is again a Moufang set. Note that a Moufang set just consists of an arbitrary setXwhich has not to be a ring or something else. As we will see in the next section, we only deal with
2 BASIC DEFINITIONS
9
Moufang sets whose given setXis a structure over a skew ﬁeldKas a vector space, ﬁxed point set or projective line. For these Moufang sets, the following holds: Theorem 2.5.4.LetM1, M2be two Moufang sets with elementary abelian root groups which are given over (skew) ﬁeldsK1, K2. LetT1:=hµa,b|a, b∈M1i andT2:=hµa,b|a, b∈M2idenote the torus ofM1resp.M2, and letM1∼=M2. Then the following holds: 1.charK1= charK2. 2. Ifϕ:M1→M2is an isomorphism, thenϕmapsZ(T1)ontoZ(T2). Proof.1. Letp= charK1andq= charK2 the root groups are. Since elementary abelianp-groups (resp.q-groups) and a Moufang isomorphism induces an isomorphism on the root groups, we getp=q. 2. Choosea, b∈M1such thatµa,b∈Z(T1). Then for allc, d∈M1 we haveϕ(µa,bµc,d) =ϕ(µc,dµa,b) and since theµ-multiplication is pre-served under the Moufang isomorphism, we haveµϕ(a),ϕ(b)µϕ(c),ϕ(d)= µϕ(c),ϕ(d)µϕ(a)ϕ(b)and henceϕ(µa,b)∈Z(T2). So each element ofZ(T1) is mapped onto an element ofZ(T2). By looking atϕ−1we get the converse as well.
2.6 Jordan algebras In the literature you can ﬁnd several deﬁnitions of Jordan algebras. The notion of a quadratic Jordan algebra was introduced by K. McCrimmon in [10] in 1966. In this thesis we follow the deﬁnition of McCrimmon and Zelmanov as in [13]. As we will see, Jordan (division) algebras and Moufang sets are closely related, see also [3] and [8]. We start with the general deﬁnition and will take a look at the Jordan algebras we are interested in afterwards. LetRbe a ring,JanR-module with an element 1∈J∗. Suppose that for every elementx∈Jthere exists a linear mapUx:J→Jwhich maps an element y∈JontoUxysuch thatU:J→End(J), x7→Uxisquadratic: For allλ∈R andx∈Jwe haveUλx=λ2Ux, and the map (x, y)7→Ux,y:=Ux+y−U−Uy x is bilinear forx, y∈J. Deﬁnition 2.6.1.1. Aunital (quadratic) Jordan algebraJ:= (J, U,1) over a ringRis anR-moduleJwith an element 1∈J∗and a quadratic mapU:J→EndR(J) deﬁned as above, such that in all scalar extensions JRofJthe following holds: (JA1)U1= idJ (JA2)UxVy,x=Vx,yUx=UU(x)y,x (JA3)UUy=UxUyUx whereVx,y(z) :={xyz}:=Ux,z(y) andx, y, z∈J.