(de Gruyter 2002
Generalized projective geometries: general theory and equivalence with Jordan structures
Wolfgang Bertram
(Communicated by R. Lo¨wen)
Abstract. In this work we introduce generalized projective geometries which are a natural generalization of projective geometries over a field or ringKbut also of other important geo- metries such as Grassmannian, Lagrangian or conformal geometry (see [3]). We also introduce the correspondinggeneralized polar geometriesand associate to such a geometry asymmetric space overK. In the finite-dimensional case overK¼R, all classical and many exceptional symmetric spaces are obtained in this way. We prove that generalized projective and polar geometries are essentially equivalent to Jordan algebraic structures, namely toJordan pairs, respectively toJordan triple systemsoverKwhich are obtained as a linearized tangent version of the geometries in a similar way as a Lie group is linearized by its Lie algebra. In contrast to the case of Lie theory, the construction of the ‘‘Jordan functor’’ works equally well over gen- eral base rings and in arbitrary dimension.
Key words.projective geometry, polar geometry, symmetric space, Jordan pair, Jordan triple system.
2000 Mathematics Subject Classification. 17C30, 17C36, 17C37, 17B70, 51A05, 51A50, 53C35
0 Introduction
0.1 Geometry and algebra. The aim of this work is to bring together two topics, a geometric one, namely projective geometry, and an algebraic one, namelyJordan algebraic structures. On the one hand, projective spaces and projective geometry are not only central topics in mathematics but also play a foundational roˆle in modern physics, see for example [22]; on the other hand, Jordan algebras have been invented as a concept in the foundation of quantum mechanics, see [13]. Thus it seems not to be without interest to find a geometric concept unifying these two theories; such a concept, called generalized projective geometries, is proposed in the present work. In fact, many authors have already remarked that there are important relations between the two topics mentioned (see point 0.6. below); however, most of the literature con- cerns special cases, and it seems that the problem to establish a general equivalence between the two categories in question has not been raised. As for any equivalence,
the problem has two aspects: coming from geometry, we want to find a ‘‘linear tan- gent object’’ (similar to the Lie algebra of a Lie group) allowing to transform geo- metric problems into linear algebra. This is achieved in Chapter 9 (Theorems 9.5 and 9.8) in a very general context. In the special case of projective geometry, the tangent object is a trilinear composition of the kind
VVV !V; ðx;f;yÞ 7!fðxÞyþfðyÞx
which is well-known to play an important roˆle in projective di¤erential geometry.
Conversely, coming from algebra, we want to ‘‘integrate’’ our algebraic structure to a global geometric object; in other words, we are looking for a Jordan analogue of Lie’s third theorem (this is achieved in Chapter 10, see Theorem 10.1). In the fol- lowing, we describe our approach in some more detail.
0.2 Generalized projective geometries. The essential di¤erence between our approach to projective geometry and the more traditional ones is that we donottry to base our theory on ‘‘incidence axioms’’ or other combinatorial structures but on algebraic laws. Our guiding model here is the theory of Lie groups (which is based on the group laws) and Loos’ theory of symmetric spaces [16], which is based on a set of
‘‘non-associative’’ algebraic identities. Let us briefly describe the main features of the algebraic identities we have in mind: it is convenient to consider a projective space X¼PðWÞover a field or ringKtogether with its dual spaceX0¼PðWÞand to view elements aAX0 as ‘‘a‰nizations’’ of Xand vice versa. In general, a pair of spaces ðX;X0Þ, each of them parametrizing a family of a‰nizations on the other, is called ana‰ne pair geometryover the base ringK(Chapter 1). If such a structure is given, then for any fixed scalarrAKthere is a natural ternary ‘‘multiplication map’’
mr:XX0XID!X; ðo;a;xÞ 7!mrðo;a;xÞ
associating to an a‰nization a with a‰ne part VaHX and two points o;xAVa the productrx¼:mrðo;a;xÞin theK-moduleVawith zero vector o. Exchanging the roˆles ofXandX0, a dual multiplication mapm0ris defined. In [3] we give an explicit formula for the multiplication map for the case of projective and Grassmannian ge- ometries and derive its most important properties by elementary linear algebra. As usual, one definesleft,rightandmiddle multiplicationsassociated to the ternary map m¼mrby
mðx;a;yÞ ¼Lx;aðyÞ ¼Mx;yðaÞ ¼Ra;yðxÞ:
Generalized projective geometries are now defined by requiring the following funda- mental identities of projective geometry(Chapter 2):
ðLx;aÞt¼La;x; ðRa;xÞt¼Rx;a ðPG1Þ ðMx;yÞt¼My;x; ðMa;bÞt¼Mb;a; ðPG2Þ
complemented by a property (T) assuring the existence oftranslation groups. Here an identity of the typegt ¼hmeans that the pairðg;hÞbehaves essentially like a projec- tive mapgand its transposedgtin projective geometry, i.e. that the condition
gðmsðx;hðaÞ;yÞÞ ¼msðgðxÞ;a;gðyÞÞ
holds for allsAK. In [3] we have shown that (PG1) and (PG2) hold for the ‘‘classical geometries’’: projective, Grassmannian, Lagrangian and conformal geometry. For several reasons we believe that these identities are indeed a good starting point for an axiomatic theory: they are simple, highly symmetric and ‘‘complete’’ in the sense that all partial operators obtained by fixing two elements have a good functorial relation with the whole structure—in this sense our equations are the best one could expect and behave nicer than group laws (where left and right translations are not auto- morphisms) or symmetric space laws (where left translations are automorphisms but right translations are not).
0.3 Tangent objects and ‘‘di¤erential calculus’’. Yet in another sense the iden- tities (PG1) and (PG2) behave nicer than group or symmetric space laws: they imply a strong regularity of the multiplication maps even in the infinite-dimensional case in the sense that generically everything can be expressed by (quadratic) polynomials combined with inversions in some general linear group—in the finite-dimensional case over a field everything is thus rational overK. The only assumption we need here is that (PG1) and (PG2) holdin all scalar extensions ofK; we show that the scalar ex- tension of a generalized projective geometryðX;X0ÞoverKbydual numbers overK plays naturally the role of thetangent bundleðTX;TX0Þ(Chapter 7; the use of dual numbers in related contexts appears already in [6, Chapter 4], [19] and [20]). Thus we have a sort of di¤erential calculus onðX;X0Þ, and in a way similar to the way that one associates a Lie algebra to a Lie group or a Lie triple system to a symmetric space, derivations of the ternary mapsmr,m0rat a base pointðo;o0Þgive rise to a pair of ternary maps of tangent spaces,
T :VV0V!V; T0:V0VV0!V0;
VGToX,V0GTo0X0, satisfying the algebraic laws of a (linear) Jordan pair overK (Theorem 9.5). Conversely, to any Jordan pair overKwe can construct a generalized projective geometryðX;X0ÞoverK; these constructions are essentially inverse to each other and are functorial (Theorem 10.1). An important roˆle in both constructions is played by the Lie algebra ofderivations of ðX;X0Þ, known in Jordan theory as the associatedKantor–Koecher–Tits algebra(Chapter 9).
0.4 Polar and null geometries.Besides Jordan pairs, there are two other important algebraic categories in Jordan theory, namelyJordan triple systemsandJordan alge- bras. The former are the same asJordan pairs with involution, see [17, I.1]. Geometri- cally, they correspond togeneralized polar geometries. Polarities can be defined in our general situation in the same way as in ordinary projective geometry: they are anti-
automorphisms ðp:X !X0;p0:X0!XÞ of order 2 which are not a null-system, i.e., not allxAX areisotropicwith respect to p(Chapter 3). Thequadricof a polarity pis the set of isotropic points of p; it can be described by an algebraic condition in terms of the associated Jordan triple system (see Section 11.1).
The geometric object corresponding tounital Jordan algebras is closely related to null systems and toinner polarities; we will come back to this point in subsequent work (cf. Section 11.3).
0.5 Symmetric spaces and Jordan–Lie functor. It is well-known that any Jordan triple systemTgives rise to a Lie triple system R¼RT defined by
RðX;YÞZ¼ ðTðX;Y;ZÞ TðY;X;ZÞÞ; ð0:1Þ the correspondence T7!RT is called the Jordan–Lie functor(see [5]). In Chapter 4 we construct the corresponding functor on a geometric level: for any polar geometry ðX;X0;pÞthe complementMðpÞ of the associated quadric is a symmetric space over Kin an appropriate sense (Theorem 4.1). In the finite-dimensional case overK¼R we get a new and more conceptual construction of thegeometric Jordan–Lie functor from [5], and for a general field K we obtain a class of symmetric spaces which is algebraic overK(cf. Section 11.2). In the real case, it is known by classification (work of E. Neher; cf. [5]) that all classical and about half of the exceptional simple sym- metric spaces are obtained by our construction; therefore one may conjecture that also in the general case our construction yields an important part of the finite- and even infinite-dimensional symmetric spaces over a general base field or ring. The Jordan algebraic description is a very e¤ective and powerful tool in the study of such spaces.
It seems to be a rather deep problem to understand the Jordan–Lie functor in a con- ceptual way: which intrinsic property of a symmetric space makes it associated to one or several generalized polar geometries? We hope that the approach presented here will help to solve this problem.
0.6 Related work.As already mentioned, the relation between geometry and Jordan structures has attracted the attention of many authors. A quite extensive bibliography on the geometry ofexceptionalJordan structures, going back to the work of Freuden- thal, Springer and others, can be found in [12]. Very closely related to our approach are the papers [18], [19] and [21] by O. Loos and the papers [6] and [8] by J. Faulkner.
Although the latter papers are placed in an incidence geometric context, our formal- ism is surprisingly close to the one developed there. Our presentation of the Kantor–
Koecher–Tits algebra (Section 9) is motivated by Section 4 of [6]. Comparing with the above mentioned papers by O. Loos, the reader will find that our identity (PG1) for invertible scalars is in fact implicitly contained in [19] (and also in our approach [5] to the real case), whereas the identity (PG2) seems to be completely new—in fact, the discovery of the identity (PG2) was a big surprise to us; since right multiplications in symmetric spaces have no known functorial interpretation, we did not expect the situation in Jordan theory to be that much better. Correspondingly, the central part in the proof of the existence theorem for generalized projective geometries (Theorem
10.1) is the verification of (PG2). It is precisely the identity (PG2) that allows one to get rid of regularity assumptions and thus to develop the theory in full generality, in- cluding the infinite-dimensional case.
Organization of the paper.The contents is as follows:
1. A‰ne pair geometries
2. Generalized projective geometries 3. Generalized polar geometries 4. Associated symmetric spaces 5. Associated group actions
6. Bergman operator and quadratic map 7. The tangent bundle
8. Infinitesimal automorphisms
9. Kantor–Koecher–Tits algebra and the Jordan functor 10. Existence theorem
11. Problems and further topics
The main results can be found in Chapters 9 and 10. In Chapter 11 we mention some further topics and problems which we hope to investigate in subsequent work.
Examples and motivation for the axiomatic approach presented here are given in our paper [3]. I am grateful to John R. Faulkner and to Ottmar Loos for helpful com- ments during the 2000 Oberwolfach conference on Jordan algebras which lead to the general approach, including the case of baserings, which is presented here.
Notation.Kdenotes a commutative ring with unit 1 and 1þ1AK. 1 A‰ne pair geometries
1.1. A pair geometry is given by two sets X, X0 and a subset MHXX0; if ðx;aÞAM we say that x and a are remote orin general position. If ðx;aÞBM, we say thatðx;aÞareneighboring. Foro0AX0, respectivelyoAX, we denote by
Vo0 :¼ fxAXj ðx;o0ÞAMg; Vo0:¼ fx0AX0j ðo;x0ÞAMg ð1:1Þ the sets of objects remote from o0, respectively fromo. We assume that the setsVo0, o0AX0(resp.Vo0,oAX) coverX(resp.X0) or, equivalently, that allVo0and allVo0are non-empty. The case thatXis covered already by one of theVo0’s is not excluded.
1.2. Homomorphisms of pair geometries are remoteness-preserving pairs of maps, that is, pairs g:X !Y, g0:X0!Y0 such thatgðVo0ÞHVg0ðo0Þ and g0ðVo0ÞHVgðoÞ0
for all oAX, o0AX0.Local homomorphismsare defined by the same property; they are required to be defined at least on one pair of setsðVa;Vx0Þwithðx;aÞAM.
1.3. If ðX;X0;MÞ is a pair geometry, then so is ðX0;X;MdÞ with Md ¼ fðq;pÞ j ðp;qÞAMg; we call it the dual pair geometry. All axioms we are going to add will appear together with their dual version, thus assuring the existence of dual objects.
1.4. Ana‰ne pair geometry (overK) is a pair geometryðX;X0;MÞsuch that for every element o0AX0 (resp.oAX) a structure of an a‰ne space over K is defined onVo0 (resp. onVo0). In other words, for each pairðo;o0ÞAMthere is a structure of a K-module with zero vectoroonVo0 and aK-module structure with zero vectoro0 onVo0. The a‰ne partsVo0HX, resp.Vo0HX0are calleda‰ne cellsofX, resp. ofX0, and we say thatX0 (resp.M) is thespace of a‰nizations(resp.space of vectorializa- tions) ofX, and vice versa.
1.5 The multiplication maps. Given an a‰ne pair geometry, we denote for any ðo;o0ÞAMandrAKbyro;o0:Vo0 !Vo0 the multiplication by the scalarrin theK- moduleVo0 with zero vectoro, and define duallyro0;o:Vo0!Vo0as multiplication by rin theK-moduleVo0with zero vectoro0. Putting for a fixedrAKall these together, we define themultiplication maps associated to an a‰ne pair geometryby
mr:XX0XID!X; ðo;o0;xÞ 7!mrðo;o0;xÞ:¼ro;o0ðxÞ ð1:2Þ and dually
mr0:X0XX0ID0!X0; ðo0;o;x0Þ 7!m0rðo0;o;x0Þ:¼ro0;oðx0Þ ð1:3Þ where
D¼ fðo;o0;xÞAXX0Xj ðo;o0ÞAM;ðx;o0ÞAMg; ð1:4Þ and dually forD0. Thus, if we fix the middle elemento0, the partial maps
mrð;o0;Þ:Vo0Vo0 !Vo0; m0rð;o;Þ:Vo0Vo0!Vo0 ð1:5Þ define onVo0 (resp.Vo0) the structure of an a‰ne space overKin the sense of Chapter 1 of [3], and the concept of an a‰ne pair geometry can be expressed by several iden- tities for the multiplication maps which are denoted by (Af1)–(Af5) in [3] and from which one can recover the whole of a‰ne geometry over K([3, Theorem 1.1]). For instance, the identity (Af3) reads in the notation used here
mrðp;a;qÞ ¼m1rðq;a;pÞ; m0rða;p;bÞ ¼m1r0 ðb;p;aÞ: ð1:6Þ We will not need the explicit form of the other identities in the sequel; it su‰ces here
to remark that a‰ne maps are precisely the algebraic homomorphisms of the multi- plication maps and that translations can be recovered from the multiplication maps via Formula (2.4) given below.
1.6. Homomorphisms of a‰ne pair geometries ðX;X0Þ, ðY;Y0Þ are remoteness- preserving pairsðg;g0Þof mapsg:X !Y,g0:X0!Y0such that the multiplication maps are respected:
gðmrðo;o0;pÞÞ ¼mrðgðoÞ;g0ðo0Þ;gðoÞÞ;
g0ðm0rðo0;o;p0ÞÞ ¼m0rðg0ðo0Þ;gðoÞ;g0ðp0ÞÞ: ð1:7Þ Equivalently, for allo0AX0the restrictions
g:XIVo0!Vg0ðo0ÞHY; g0:X0IVo0!VgðoÞ0 HY0
are a‰ne maps.Local homomorphismsare local homomorphisms of pair geometries (in the sense of 1.2) respecting the multiplication maps. (Note that we do not requireg0 to be determined uniquely bygor vice versa; however, in the ‘‘non-degenerate case’’
this property holds, see 2.8.) Isomorphisms are homomorphisms ðg;g0Þfor which g andg0are bijections. The composition of homomorphisms is again a homomorphism (withðghÞ0¼g0h0); thus a‰ne pair geometries form a category. In particular, we may speak of theautomorphism groupAutðX;X0ÞofðX;X0Þ.
1.7. A base point in ðX;X0Þ is a fixed element ðo;o0ÞAM; a homomorphism of a‰ne pair geometries with base pointis a base point preserving homomorphism. The corresponding automorphism group AutðX;X0;o;o0Þ ¼AutðX;X0Þo;o0 is called the structure group ofðX;X0;o;o0Þ; by definition of a homomorphism it acts linearly on Vo0Vo0.
1.8. Anadjoint pair of morphismsof a‰ne pair geometriesðX;X0Þ,ðY;Y0Þis a pair of mapsg:XIU!Y,h:Y0IU0!X0preserving remoteness in the sense that
gðVhðpÞVUÞHVp; hðVgðqÞ0 VU0ÞHVq0
(where the subsetsUHX,U0HX0shall contain at least one a‰ne cell) and satisfy- ing the relations
gðmrðo;hðo0Þ;pÞ ¼mrðgðoÞ;o0;gðpÞÞ;
hðm0rðo0;gðoÞ;p0ÞÞ ¼mrðhðo0Þ;o;hðp0ÞÞ: ð1:8Þ These relations can be rephrased by saying that gjVhðo0ÞVU has an extension by an a‰ne map Vhðo0Þ!Vo0, and dually. We write h¼gt ifðg;hÞis an adjoint pair; this means just thatðg;hÞ ¼ ðg;gtÞsatisfies the relations (1.8) and shall not be interpreted in the sense thatgtbe uniquely determined byg(although in the non-degenerate case
it actually is, see 2.8 and 2.9). The conditionsh¼gtforðX;X0Þ,ðY;Y0Þandgt¼h for the dual pairsðY0;YÞ,ðX0XÞare equivalent. Clearly, ifðg;gtÞandðf;ftÞare ad- joint pairs such thatg, f and ft,gtare composable, thenðgf;ftgtÞis an adjoint pair (this may be expressed by writingðgfÞt ¼ftgt). In particular, the a‰ne pair ge- ometries with base point form a category with respect to base-point preserving ad- joint pairs of morphisms.
1.9. The preceding discussion shows that a‰ne pair geometries can be turned into a category in two essentially di¤erent ways—in general, homomorphisms do not give rise to adjoint pairs of morphisms or vice versa. (This is very well known from pro- jective geometry: some authors require homomorphisms of projective spaces to be in- duced by injectivelinear maps, whereas others allow morphisms to be possibly de- fined only on some a‰ne part, see e.g. [1]; algebraically, this corresponds exactly to the distinction of two categories made here. Since the Jordan pair of ordinary projec- tive geometry is simple, homomorphisms have to be injective or trivial.) However, ifðg;g0Þis an isomorphism, then clearlyðg;gtÞwithgt¼ ðg0Þ1 is an adjoint pair. In general, categorial notions will refer to the category defined in 1.6 and not to the category defined in 1.8.
1.10 Scalar extensions.By a scalar extension ofKwe mean a unital commutative and associative K-algebra R. Let us denote byf:K!R,r7!r1R the natural ho- momorphism (it need not be injective). A correspondingscalar extension of the a‰ne pair geometryðX;X0ÞoverKis an a‰ne pair geometryðX;X0ÞR:¼ ðXR;XR0ÞoverR together with a homomorphism of a‰ne pair geometries overK
ðF;F0Þ:ðX;X0Þ ! ðX;X0ÞR
such that, for all ðx;aÞAM, theR-module ðVF0ðaÞ;FðxÞÞ(zero vector FðxÞ) is the usual scalar extension of theK-moduleðVa;xÞ, i.e.
ðVF0ðaÞ;FðxÞÞGðVa;xÞn
KR
as anR-module; more precisely, the following diagram shall commute:
ðVa;xÞ ðVa;xÞ
??
?y
??
?y ðVa;xÞn
KR G ðVF0ðaÞ;FðxÞÞ;
ð1:9Þ
where the first column is the natural map v7!vn
K1R and the second column is given by restriction ofF. Dually, forðVx0;aÞa similar condition is required to hold.
Moreover, we require ðX;X0ÞR to be minimal in the following sense: for every ho- momorphismðg;g0ÞofðX;X0Þinto a geometry ðY;Y0Þdefined over R, considered as a geometry overK, there exists a unique extensionðg;g0ÞR:¼ ðgR;gR0Þto a homo-
morphismðX;X0ÞR! ðY;Y0Þdefined overR. It is clear that this property determines ðX;X0ÞRup to isomorphisms. (One might be tempted to define scalar extensions only by this universal property; however, it seems not to be possible to deduce from it the isomorphism (1.9). Note that we do not make here any claims of existence of scalar extensions for general a‰ne pair geometries.)
2 Generalized projective geometries
2.1. AssumeðX;X0Þis an a‰ne pair geometry overKwith multiplication mapsmr, mr0. As for any ternary map, we defineright,leftandmiddle multiplicationassociated to the ternary mapsmrandm0rby
Lx;aðyÞ:¼Ra;yðxÞ:¼Mx;yðaÞ:¼mrðx;a;yÞ
La;xðbÞ:¼Rx;bðaÞ:¼Ma;bðxÞ:¼m0rða;x;bÞ ð2:1Þ (where we add the superscriptðrÞif the dependence onrshall be indicated). Then (1.6) says thatRðrÞa;y¼Lð1rÞy;a , and thus we can transform left multiplications into right mul- tiplications and vice versa. However, they cannot be transformed into middle multi- plications since the latter exchange the partnersXandX0whereas the former preserve them. We call operators of the type of left, right or middle multiplications altogether interior operators of the geometry.
2.2 The fundamental identities. We say that an a‰ne pair geometry satisfies the fundamental identitiesif, for allrAK, the following holds:
ðLx;aÞt¼La;x; ðRa;xÞt ¼Rx;a; ðPG1Þ ðMx;yÞt¼My;x; ðMa;bÞt¼Mb;a; ðPG2Þ where ðx;aÞAM, and ðx;yÞAXX, ða;bÞAX0X0 are such that the middle multiplications are defined at at least one point. SinceLðrÞx;a¼Rð1rÞa;x , the first and the second condition in (PG1) are equivalent. Sincegt ¼handht ¼gare equivalent, an- other equivalent formulation isðLa;xÞt¼Lx;a. However, the two conditions in (PG2) are not equivalent: the first one says thatðMx;y;My;xÞis an adjoint pair forðX0;XÞ, ðX;X0Þand the second thatðMa;b;Mb;aÞis an adjoint pair forðX;X0Þ,ðX0;XÞ; both are self-dual. Using (1.8), the conditions (PG1) and (PG2) can be written more ex- plicitly
Lx;aðmsðy;La;xðbÞ;zÞÞ ¼msðLx;aðyÞ;b;Lx;aðzÞÞ; La;xðms0ðb;Lx;aðyÞ;cÞÞ ¼ms0ðLa;xðbÞ;y;La;xðcÞÞ; Mx;yðms0ða;My;xb;cÞÞ ¼msðMx;yðaÞ;b;Mx;yðcÞÞ;
Ma;bðms0ðx;Mb;ay;zÞÞ ¼msðMa;bðxÞ;y;Ma;bðzÞÞ:
for allr;sAKandx;y;zAX,a;b;cAX0where the expressions are defined. With all variables included, the preceding formulas read
mrðx;a;msðy;m0rða;x;bÞ;zÞÞ ¼msðmrðx;a;yÞ;b;mrðx;a;zÞÞ;
m0rða;x;ms0ðb;mrðx;a;yÞ;cÞÞ ¼ms0ðm0rða;x;bÞ;y;mr0ða;x;cÞÞ;
mrðx;ms0ða;mrðy;bxÞ;cÞ;yÞ ¼msðmrðx;a;yÞ;b;mrðx;c;yÞÞ; mr0ða;ms0ðx;mrðb;y;aÞ;zÞ;bÞ ¼msðmrða;x;bÞ;y;mrða;z;bÞÞ:
We also require that for all rAK the left multiplications LðrÞx;y¼rx;y extend to bi- jections X!X such that all identities introduced so far still hold whenever all ex- pressions are defined. This implies that, if both rand 1rbelong to K, the multi- plication mapmr is defined on theextended domain
De:¼ fðx;a;yÞAXX0Xj ðx;aÞAMorðy;aÞAMg: ð2:2Þ By elementary properties of a‰ne geometry we have (cf. the identity (Af1) from [3]),
rx;asx;a¼ ðrsÞx;a; 1x;a¼idVa:
Thus forrAK the condition (PG1) can be rephrased by saying that
ðrx;a;r1a;xÞAAutðX;X0Þ: ð2:3Þ The automorphismsðrx;a;r1a;xÞ ððx;aÞAM;rAKÞwill also be calledinner automor- phismsormajor dilatations, and the subgroup IntðX;X0Þof AutðX;X0Þgenerated by them is called theinner automorphism group.
2.3 Translations. Since by assumption 2 is invertible in K, we can express trans- lationsvia major dilatations: forðo;o0ÞAM,
tv:¼tðo;v o0Þ:¼2o;o021v;o0 ð2:4Þ is the translation byvin theK-moduleðVo0;oÞ. If the fundamental identity (PG1) in its version (2.3) holds, then the pair
ðtv;tt~vÞ:¼ ð2o;o021v;o0;21o0;o2o0;vÞ ð2:5Þ belongs to AutðX;X0Þ; this can also be written
ð~ttvÞt¼tv: ð2:6Þ
The identities of a‰ne geometry imply thattvtw¼tvþw(sum inðVo0;oÞ). If the trans- pose is unique (as in ordinary projective geometry), then we obtain by transposing:
~ttv~ttw¼~ttvþw ðTÞ
which means that
tVo0 :¼ fðtv;~ttvÞ jvAVo0g ð2:7Þ is a group, called thetranslation group with respect to the a‰nization o0; as a group, it is isomorphic to ðVo0;o;þÞ. We say that our geometrysatisfies the translation prop- erty if (T) holds together with its dual, for all ðo;o0ÞAM. The dual of (T) implies that
~ttVo0 :¼ fð~ttw;twÞ jwAVo0g ð2:8Þ with
~ttw:¼tt~wðo;o0Þ:¼21o;o02o;w ð2:9Þ is an abelian group, isomorphic toðVo0;o0;þÞ. The mapstt~w:X !X are calleddual translations. Our notational convention is such that groups denoted by tVaðaAX0Þ act by usual translations on thefirst factor of the geometry ðX;X0Þand groups de- noted by tt~Vx0 ðxAXÞ act by usual translations on the second factor. Thus, for in- stance, the group denoted bytVx0 ðxAXÞacts by usual translations on the first factor of the dual geometryðX0;XÞ; as a group, it is of course isomorphic to~ttVx0. (Note that (T) is a consequence of the preceding identities if the transpose is unique; we do not know whether this is true also in the general case. It may be conjectured that this is indeed so since for Jordan pairs a duality principle holds, see [17, Proposition 2.9], which is used in the proof of the addition formula [17, Theorem 3.7] that corresponds to our formula (T).)
2.4. A generalized projective geometry over K is an a‰ne pair geometry ðX;X0Þ overK in which the fundamental identities (PG1), (PG2) and the translation prop- erty (T) hold in all scalar extensions ofK—this means that, iff:K!Ris any scalar extension, then there exists a scalar extensionðX;X0ÞRofðX;X0Þin the sense of 1.10 satisfying (PG1) (also in its extended version, if the scalar is invertible), (PG2) and (T) (together with its dual) overR.
2.5. Homomorphisms andadjoint pairs of morphismsof generalized projective geo- metries are those of the underlying a‰ne pair geometry. Therefore, generalized pro- jective geometries can be turned into a category in two essentially di¤erent ways.
2.6. For a definition of generalized projective geometries if 2BK, it will be necessary to add axiomatically the structure given by the following maps of four arguments:
XX0XX0IW!X; ðo;o0;v;wÞ 7!~ttwðo;o0ÞðvÞ;
XX0XXIW0!X; ðo;o0;v;wÞ 7!tðo;ow 0ÞðvÞ
ð2:10Þ
whereW;W0are defined by conditions similar to (2.2). This idea will be taken up in subsequent work.
2.7 Categorial constructions.Since the category of generalized projective geometries is essentially defined by algebraic laws, it behaves fairly well with respect to some standard categorial constructions:
(1) (Duality.) The roˆle played by the spacesX andX0 in our axioms is completely symmetric; thusðX0;XÞwith the spaceMd ¼ fða;xÞ j ðx;aÞAMgof vectorializa- tions is again a generalized projective geometry, called thedual pair ordual ge- ometryofðX;X0Þ.
(2) (Direct products.) Thedirect productofðX;X0Þ,ðY;Y0ÞisðXY;X0Y0Þwith remoteness given by Vðo0;p0Þ¼Vo0Vp0, and dually, which we require to carry the direct product structureVo0Vp0 of a‰ne spaces. It is then easily verified that ðXY;X0Y0Þis again a generalized projective geometry; the multiplication map is just the direct product of the ones ofðX;X0ÞandðY;Y0Þ. In particular, we can define the direct productðX;X0Þ ðX0;XÞ ¼ ðXX0;X0XÞwhich will play an important role later on.
(3) (Subspaces.) These are subsetsYHX,Y0HX0such thatðY;Y0Þis closed under all multiplications maps. Thus in the a‰nizationsy0AY0,Yis an a‰ne subspace, and vice versa.
(4) (Inner ideals.) An inner ideal of Xas a subset YHX which is linear w.r.t. all possible a‰nizations, i.e. satisfyingmrðY;a;YÞHY for allaAX0; inner ideals of X0are defined dually.
(5) (Congruences and quotient spaces.) A congruence is a subspace ðR;R0ÞH ðXX;X0X0Þwhich is an equivalence relation—similar to the case of sym- metric spaces ([16, Chapter III]) one shows thatðX;X0Þ=ðR;R0Þis again a gener- alized projective geometry.
(6) (Tangent bundle.) One can construct a tangent bundle ðTX;TX0Þ of ðX;X0Þ which is essentially scalar extension by dual numbers overK—see Chapter 7.
(7) (Flat geometries.) The category of pairs of a‰ne spaces over Kis imbedded in the category of generalized projective geometries as follows: let V;W be a‰ne spaces in the usual sense; let X:¼V, X0:¼W, M¼VW and let ro;o0 ¼ro (i.e. all a‰ne charts of X yield the same structure of a‰ne space on V), and duallyro0;o¼ro0. The axioms are easily verified.
2.8 Faithful and non-degenerate geometries.
(i) We say that the generalized projective geometry is non-degenerate if the map assigning toaAX0the setVaHX is injective, and dually.
(ii) We say that the generalized projective geometry isfaithfulif the map assigning to aAX0the a‰ne structureðVa;Aa:¼mð;a;ÞÞis injective, and dually.
It is clear that a non-degenerate geometry is faithful; the converse is not true. If ðg;g0Þis an automorphism of a faithful geometry, then gdetermines g0 uniquely by the conditionAg0ðaÞ¼gðAaÞ, wheregis the push-forward of the a‰ne structureAa byg, and converselygis determined byg0.
2.9 Conformal group.Theprojectiveorconformal group of Xis the group
CoðXÞ:¼ fg:X !Xjbg0:X0!X0:ðg;g0ÞAAutðX;X0Þg: ð2:11Þ The inner conformal group is the subgroup of CoðXÞ generated by the dilatations rx;a,rAK,ðx;aÞAM, and dually we define CoðX0Þand its inner conformal group.
IfðX;X0Þis faithful, then the surjective homomorphisms
AutðX;X0Þ !CoðXÞ; ðg;g0Þ 7!g; AutðX;X0Þ !CoðX0Þ; ðg;g0Þ 7!g0 are injective, hence CoðXÞand CoðX0Þare isomorphic to AutðX;X0Þ, and we have an isomorphism
CoðXÞ !CoðX0Þ; g7!g0 and a canonical anti-isomorphism
CoðXÞ !CoðX0Þ; g7!gt¼ ðg0Þ1:
For instance, this is the case in ordinary projective geometry (which is non- degenerate).
3 Generalized polar geometries
3.1. An antiautomorphism of a generalized projective geometry ðX;X0Þis an iso- morphism
ðp;p0Þ:ðX;X0Þ ! ðX0;XÞ
onto the dual pair. AcorrelationofðX;X0Þis an antiautomorphismðp;p0Þsuch that p0¼p1, or, equivalently, such that pt ¼p. This in turn means that the identity
pðmrðx;pðyÞ;zÞÞ ¼mr0ðpðxÞ;y;pðzÞÞ ð3:1Þ and its dual hold.
3.2. With respect to a fixed correlation p, a pointxAX is called non-isotropicifx and pðxÞare remote (i.e. ðx;pðxÞÞAM) andisotropicifxand pðxÞare neighboring.
A correlation is called a null-systemif all pointsxAX are isotropic and apolarityif there exist non-isotropic points. Ageneralized polar geometryis a generalized projec- tive geometryðX;X0Þtogether with a polarityp;homomorphisms of generalized polar geometriesare homomorphismsðg;g0Þof generalized projective geometries commut- ing with the respective polarities; in particular, theautomorphism groupAutðX;X0;pÞ is the group of all elementsðg;g0ÞAAutðX;X0Þsuch thatg0p¼pg.
3.3. For a fixed polarity p, the set
QðpÞ:¼ fxAXj ðx;pðxÞÞBMg ð3:2Þ
of isotropic points is called theassociated quadric, and its complement is denoted by MðpÞ¼ fxAXj ðx;pðxÞÞAMg: ð3:3Þ By the very definition of a polarity the set MðpÞ is not empty. A polarity is called ellipticif the quadric is empty.
3.4. If a correlation p:X!X0is fixed we will use it frequently to identifyXwith X0. Thus the product mapsmrandmr0 are both represented by a ternary map
m~
mr:XXXID!X; ðx;y;zÞ 7!mrðx;pðyÞ;zÞ
satisfying identities which are obtained from (PG1) and (PG2) by simply forgetting the distinction betweenmrandm0r. Polar geometries are then characterized by the fact that some element of the diagonal inXXbelongs toM, whereas for null geometries this is not the case. One may use these properties for an axiomatic definition of a generalized polar geometry (which is thus a setX together with a subset ofXX containing some element of the diagonal and a family of ternary mapsmrdefined on a subset of XXX and satisfying certain identities). Homomorphisms are then precisely the maps which are compatible with the multiplication mapsmm~r.
3.5. Not every generalized projective geometry does admit a polarity—take e.g. the flat case given a by a pair of non-isomorphic vector spaces. It is all the more impor- tant that one can associate to any generalized projective geometry a polar geometry in a canonical way:
Proposition 3.6.For any generalized projective geometryðX;X0Þ,the generalized pro- jective geometryðXX0;X0XÞadmits a canonical polarity,given by the exchange map pðx;x0Þ ¼ ðx0;xÞ.The corresponding space MðpÞHXX0is equal to the space Mof vectorializations of X.
Proof. It is clear thatðx;x0Þand pðx;x0Þ ¼ ðx0;xÞare remote if and only ifxandx0 are remote, whence the last claim. The first claim is proved by the following calcula- tion:
pmrððx;x0Þ;pðy;y0Þ;ðz;z0Þ ¼pmrððx;x0Þ;ðy0;yÞ;ðz;z0ÞÞ ¼pðmrðx;y0;zÞ;mr0ðx0;y;z0ÞÞ
¼ ðm0rðx0;y;z0Þ;mrðx;y0;zÞÞ ¼mr0ððx0;xÞ;ðy;y0Þ;ðz0;zÞÞ
¼mrðpðx;x0Þ;ðy;y0Þ;pðz;z0ÞÞ: r
3.7. The preceding proposition yields an imbedding of the category of generalized projective geometries into the category of generalized polar geometries (if to a ho-
momorphismðg;g0Þwe associate the pairðgg0;g0gÞ). In Jordan theory the cor- responding construction is given by imbedding the category of Jordan pairs into the category of Jordan triple systems by constructing theassociated polarized Jordan tri- ple system ([17, p. 10]). In a somewhat related context, similar constructions have already been used by Rozenfeld, see [9, p. 172].
4 Associated symmetric spaces overK
4.1. Asymmetric space(in the sense of O. Loos [16]) is a real smooth manifold M with a smooth binary mapm:MM!M,ðx;yÞ 7!mðx;yÞ ¼sxðyÞsatisfying (M1) sxðxÞ ¼x
(M2) sxsx¼idM
(M3) sxAAutðmÞ, i.e.sxðmðy;zÞÞ ¼mðsxðyÞ;sxðzÞÞ (M4) the fixed pointxofsxis isolated.
The automorphism sx is called thesymmetry w.r.t. x, and the transvection group GðMÞof a symmetric space is the group generated by allsxsywithx;yAM. Acon- nected symmetric space is homogeneous under the groupGðMÞand is of the form G=H whereG¼GðMÞis a Lie group andHan open subgroup of the group of fixed points of a non-trivial involution ofG; such spaces will be calledhomogeneous sym- metric spaces. There exists a theory of symmetric k-varieties (see [11]) over general base fields, but not of general symmetric spaces in the sense of Loos (possibly infinite- dimensional and defined over rings)—one reason for this is certainly that symmetric spaces (in the sense of Loos) over a general base field or ring will be ‘‘less homoge- neous’’ than the real or complex ones (see examples in Chapter 4 of [3]). We do not try to define here formally what a ‘‘symmetric space over K’’ should be, but the fol- lowing theorem shows that any generalized polar geometry overKdefines a structure which certainly is one:
Theorem 4.2. Assume ðX;X0;pÞ is a generalized polar geometry over K. Then the complement MðpÞ of the associated quadric is stable under the binary map
mðx;yÞ:¼m1ðx;pðxÞ;yÞ
which satisfies the properties (M1)–(M3), and the symmetry with respect to a base point oAMðpÞisso ¼ ð1Þo;pðoÞ(the negative of the identity of theK-module VpðoÞ).
Proof. If ðg;g0Þis an automorphism of the polar geometry ðX;X0;pÞ, then g pre- serves the setMðpÞand is compatible withm; in fact,
gmðx;yÞ ¼gm1ðx;pðxÞ;yÞ
¼m1ðgx;g0pðxÞ;gyÞ
¼m1ðgx;pðgxÞ;gyÞ
¼mðgx;gyÞ:
Applying this to ðg;g0Þ ¼ ðð1Þo;pðoÞ;ð1ÞpðoÞ;oÞ, we get the identity (M3). (Note that in generalðg;g0Þ ¼ ðrx;pðxÞ;r1pðxÞ;xÞis an automorphism of the polar geometry if and only if r¼r1; thus the construction works precisely for the scalarsrsuch that r2¼1.) The properties (M1) and (M2) are clear, and the last statement holds by the
very definition ofm. r
It is indeed reasonable to say that the fixed pointo of so¼ ð1Þo;pðoÞ is isolated:
since 2 is invertible inK, we have, in everyK-module,x¼xi¤x¼0.
4.3 The Jordan–Lie functor. We say that ðMðpÞ;mÞ is the symmetric space as- sociated to the generalized polar geometry ðX;X0;pÞ.Homomorphisms of symmetric spacesare mapsf:M!Ncommuting with multiplication mapsmofMandN. The arguments given in the preceding proof show that then homomorphisms of polar geo- metries induce homomorphisms of the associated symmetric spaces. Thus we have defined a covariant functor from generalized polar geometries over K into spaces having the properties from Theorem 4.2. This functor is called thegeometric Jordan–
Lie functor; as mentioned in the introduction, it generalizes the geometric Jordan–Lie functor from the real finite-dimensional case considered in [5].
Corollary 4.4. If ðX;X0Þ is a generalized projective geometry, then the space MH XX0with the multiplication map
mððo;o0Þ;ðx;x0ÞÞ ¼ ðm1ðo;o0;xÞ;m1ðo0;o;x0ÞÞ is a symmetric space overKin the sense ofTheorem 4.2.
Proof. Apply Theorem 4.2 to the polarity of ðXX0;X0XÞ from Proposition
3.6. r
4.5. The symmetric space Mfrom the preceding corollary has as additional struc- ture adouble fibrationoverXand overX0such that the fibers are a‰ne spaces:
M¼ 6
oAX
fog Vo0¼ 6
o0AX0
fo0g Vo0: ð4:1Þ
This can be seen as a sort of ‘‘polarization’’ onM; in fact these spaces generalize the para-Hermitian symmetric spacesintroduced by Kozai and Kaneyuki ([14]). Ifðg;g0Þ is an automorphism ofðX;X0Þ, then, by the proof of Theorem 4.2,gg0preservesM and induces an automorphism of the symmetric spaceMwhich preserves the double fibration (4.1). Similarly, it can be verified that, if ðg;g0Þis an antiautomorphism of the generalized projective geometryðX;X0Þ, then
g~
g:XX0!XX0; ðx;aÞ 7! ðg0ðaÞ;gðxÞÞ
preserves M and induces an automorphism of the symmetric space M which ex-
changes the fibers of the double fibration (4.1). Automorphisms of the first type can be considered as ‘‘para-holomorphic’’ whereas the ones of the second type are sort of
‘‘anti para-holomorphic’’. The automorphism gg~is involutive i¤ g0¼g1, that is, i¤
ðg;g0Þis a correlation, and it has a fixed point inMi¤ it is a polarity. Thus the po- larities correspond precisely to the anti-paraholomorphic involutions having a fixed point inM. The whole fixed point set of a correlation pis
ðXXÞ~pp¼ fðx;pðxÞÞ jxAXgHXX0;
if we identifyXandX0via pthen this is just the diagonal inXX. The intersection of this set withMis the fixed point set of ~ppinM; it is non-empty i¤ pis a polarity, and then
MðpÞ!M~pp; x7! ðx;pðxÞÞ
is an isomorphism of symmetric spaces. Thus MðpÞ is imbedded in M as a sort of
‘‘para-real form’’.
5 Associated group actions
5.1. We have already defined the groups AutðX;X0Þand IntðX;X0Þassociated to a generalized projective geometryðX;X0Þ(Sections 1.6 and 2.2). Ifðo;o0ÞAMis a base point, we writeV :¼Vo0,V0:¼Vo0and consider the following stabilizer groups:
P:¼ fðg;g0ÞAAutðX;X0Þ jg:o¼og;
P0:¼ fðg;g0ÞAAutðX;X0Þ jg0:o0¼o0g;
StrðV;V0Þ:¼PVP0¼ fðg;g0ÞAAutðX;X0Þ jg:o¼o;g0:o0¼o0g:
ð5:1Þ
The group StrðV;V0Þis called the structure group ofðV;V0Þ; by the very definition of an automorphism, it acts as a subgroup of GlðVÞ GlðV0Þ. Note that, with ~ttw defined by (2.9),
~ttwðoÞ ¼21o;o02o;wðoÞ ¼o; ð5:2Þ and thus~ttV0HPand, dually,tVHP0.
Lemma 5.2.(i)P¼StrðV;V0Þ ~ttV0 (semidirect product) (ii)P0¼StrðV;V0Þ tV(semidirect product)
Proof. (i) Since automorphisms are remoteness preserving we have g:oAV i¤
ðg:o;o0ÞAMi¤ðg0Þ1:o0AV0. Therefore, ifg:o¼o, thenw:¼ ðg0Þ1:o0AV0, and we let
ðh;h0Þ:¼ ðg;g0Þ ðtt~w;twÞ ¼ ðgtt~w;g0twÞ:
It follows that
h:o¼gð~ttw:oÞ ¼g:o¼o; h0:o0¼g0tw:o0¼g0ðwÞ ¼o0;
and therefore ðh;h0ÞAStrðV;V0Þ. This proves existence of the decomposition. Let us prove uniqueness: ifðg;g0Þ ¼ ðh;h0Þ ð~ttv;tvÞAP, thenðg0Þ1:o0¼ ðtvÞ1:o0¼ v;
thusvand hence alsoðh;h0Þare uniquely determined. Part (ii) is the dual version of
(i). r
For a fixed base pointðo;o0ÞAM, letWHAutðX;X0Þbe the subset
W:¼ fðg;g0ÞAAutðX;X0Þ jg:oAVg: ð5:3Þ
Then, as remarked in the preceding proof,
W1¼ fðg;g0ÞAAutðX;X0Þ jg0:o0AV0g: ð5:4Þ It is clear thatPHWandP0HW1.
Lemma 5.3.W¼tVP¼tVStrðV;V0Þ ~ttV0 (unique decomposition)
Proof.Letðg;g0ÞAWand putv:¼g:o. Thenðtv;~ttvÞ ðg;g0ÞAP, and we can apply the preceding lemma. Conversely, ifðg;g0Þ ¼ ðtw;~ttwÞ ðp;p0ÞwithwAV,ðp;p0ÞAP, theng:o¼wAV, and moreoverwand thusðp;p0Þis uniquely determined. r 5.4. The preceding decompositions do not extend to the whole group AutðX;X0Þ.
However, in certain special cases we get the Harish–Chandra decompositionknown from the theory of Hermitian symmetric spaces: let us say that a generalized projec- tive geometryðX;X0Þisstableif the intersectionVaVVbis non-empty for alla;bAX0, and dually (cf. [21, Proposition 3.2] for this terminology). As a consequence,Xis then already covered by theVa withaAV0, and dually:
X¼ 6
aAV0
Va; X0¼ 6
xAV
Vx0:
Proposition 5.5(‘‘Harish–Chandra decomposition’’).IfðX;X0Þis stable,then AutðX;X0Þ ¼WtV ¼tVStrðV;V0Þ ~ttV0tV
(non-unique decomposition).
Proof.Letðg;g0ÞAAutðX;X0Þ; pickxAg1ðVÞVV¼Vgt:o0VVo0; thenv:¼gðxÞAV.
Thusðg;g0Þ ðtx;~ttxÞAW, and we can apply the preceding lemma. r It can be shown that the stability condition is fulfilled for instance in the finite-
dimensional case over a field (cf. [19], [21]) and for some infinite-dimensional geo- metries modelled on complex or real Banach spaces.
5.6 Connectedness.We say thatðx;aÞandðy;bÞAM areconnectedif there is a se- quenceðp1;q1Þ;. . .;ðpn;qnÞAM such thatðp1;q1Þ ¼ ðx;aÞ,ðpn;qnÞ ¼ ðy;bÞand
ðpjþ1;qjþ1ÞAðVqj;Vp0jÞ; j¼1;. . .;n1:
It is clear that connectedness is an equivalence relation; thus we get a partition of M¼6
iAIMiinto connected components, and it is easily verified that allðXi;Xi0;MiÞ withXi:¼pr1ðMiÞ,Xi0¼pr2ðMiÞare subspaces ofðX;X0;MÞ, calledconnected com- ponents of ðX;X0Þ. (A natural example of a non-connected geometry is the Grass- mannianðX;XÞofallsubspaces of a given K-module, cf. Chapter 2 of [3].) It is an easy exercise to show that a stable geometry is connected (but the converse is not true).
Theorem 5.7. AssumeðX;X0Þis connected and fix a base pointðo;o0ÞAM.Then the subgroup
G:¼htVUtt~V0i
of IntðX;X0Þgenerated by the translation group and the dual translation group, acts transitively on X,on X0and on M,i.e.
XGG=ðPVGÞ; X0GG=ðP0VGÞ; MGG=ðStrðV;V0ÞVGÞ:
In particular,X,X0 and Mare homogeneous under the action ofAutðX;X0Þ,and M is a homogeneous symmetric space in the sense that it is homogeneous under its auto- morphism group.
Proof. It is enough to show that, if ðx;aÞAMVðVV0Þ is an arbitrary point, then there existsðg;g0ÞAGsuch thatðx;aÞ ¼ ðg:o;g0:o0Þ; the claim then follows by a straightforward induction using connectedness. We can writeðx;aÞ ¼ ðf:o;h0:o0Þwith ðf;f0Þ ¼ ðtx;~ttxÞ,ðh;h0Þ ¼ ð~tta;taÞAG. Sinceðx;aÞAM, we have
ðx;aÞ ¼ ðtt~a;taÞ:ð~ttaðxÞ;o0Þ
¼ ðtt~a;taÞ ðt~ttaðxÞ;~tt~ttaðxÞÞ:ðo;o0Þ; thus we haveðx;aÞ ¼ ðg;g0Þ:ðo;o0Þwith the element
ðg;g0Þ ¼ ð~tta;taÞ ðt~ttaðxÞ;~tt~ttaðxÞÞ ð5:5Þ of G. We have proved that G, and hence also AutðX;X0Þ, act transitively onM, X andX0. Since AutðX;X0Þacts by automorphisms of the symmetric spaceM(cf. Sec-
tion 3.7, the proof of Theorem 4.2 and Corollary 4.4),Mis homogeneous under its
automorphism group. r
In [21], the group G is called the projective elementary group ofðV;V0Þ. Transi- tivity of AutðX;X0ÞonMmeans that up to isomorphism there is onlyoneK-module on which a connected geometryðX;X0Þis modelled, namely the equivalence class of all vectorializations under the automorphism group. For non-connected geometries this is in general no longer true. Note that in the preceding proof we could also have written
ðx;aÞ ¼ ðtx;~ttxÞ:ðo;~ttxta:o0Þ
¼ ðtx;~ttxÞ ð~tt~ttxðaÞ;t~ttxðaÞÞ:ðo;o0Þ;
and thus we could also have taken the element
ðe;e0Þ:¼ ðtx;~ttxÞ ðtt~~ttxðaÞ;t~ttxðaÞÞ ð5:6Þ ofG. These two possibilities correspond to two di¤erent ways of joining ðo;o0Þand ðx;aÞby using the double fibration ofMmentioned in 4.5.
Corollary 5.8.With respect to a base pointðo;o0ÞAM,we have the following expres- sion for the multiplication mapmr:for allða;xÞAMVðVV0Þand rAK,
rx;a¼tx~tt~ttxðaÞro;o0ðtx~tt~ttxðaÞÞ1; and dually.
Proof.Just writeðx;aÞ ¼ ðe:o;e0:o0Þwithðe;e0Þgiven by (5.6) and use thatre:o;e0:o0 ¼
ero;o0e1sinceðe;e0Þis an automorphism. r
5.9 Non-homogeneous symmetric spaces.IfðX;X0;pÞis a generalized polar geome- try, then the associated symmetric spaceMðpÞis general nothomogeneous under its automorphism group, even ifðX;X0Þis connected. The orbit structure ofMðpÞunder AutðX;X0;pÞ is in general as complicated as the classification of non-degenerate quadratic forms overK(which is a special case of our general set-up, see Section 5.3 of [3]). However, there may exist polarities ‘‘of symplectic type’’; for those, the orbit structure is as simple as the classification of symplectic forms (which also is a special case of our general set-up). Theorem 5.7 shows that the ‘‘exchange polarity’’ from Proposition 3.6 is of the latter type.
6 Bergman operator and quadratic map
6.1. In this section we fix a base pointðo;o0ÞAM and letV:¼Vo0,V0:¼Vo0. Fol- lowing a standard terminology in Jordan theory, the set of remote elements inVV0, MVðVV0Þ ¼ fðx;aÞAMj ðx;o0ÞAM;ðo;aÞAMg ð6:1Þ
