Generalized projective geometries. Part II: General theory and equivalence with Jordan structures

Wolfgang Bertram

Abstract. In this work we introduce generalized projective geometries which are a natural generalization of projective geometry over a field or ring 𝕂 but also of other important geometries such as Grassmannian, Lagrangian or conformal geometry (see Part I). We also introduce the corresponding generalized polar geometries and associate to such a geometry a symmetric space over 𝕂. In the finite-dimensional over 𝕂 = ℝ, 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 to Jordan pairs, resp. Jordan triple systems over 𝕂 which 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 general base rings and in arbitrary dimension.

A.M.S classification: 17 C 30, 17 C 36, 17 C 37, 17 B 70, 51 A 05, 51 A 50, 53 C 35

Key words: projective geometry, polar geometry, symmetric space, Jordan pair, Jordan triple system

Vandœuvre-lès-Nancy, december 13, 2000

Wolfgang Bertram Institut Elie Cartan Université Nancy I, Faculté des Sciences B.P. 239 54506 Vandœuvre-lès-Nancy, Cedex France

email: bertram@iecn.u-nancy.fr

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, namely Jordan algebraic structures. On the one hand, projective spaces and projective geometry are not only central topics in mathematics but also play a foundational rôle in modern physics, see for example [V85]; on the other hand, Jordan algebras have been invented as a concept in the foundation of quantum mechanics, see [JvNW35]. 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 concerns 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 tangent object” (similar to the Lie algebra of a Lie group) allowing to transform geometric 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

V × V^{∗} × V → V, (x, φ, y) ↦ φ(x)y + φ(y)x

which is well-known to play an important rôle in projective differential 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 following, we describe our approach in some more detail.

0.2. Generalized projective geometries. The essential difference between our approach to projective geometry and the more traditional ones is that we do not try to base our theory on “incidence axioms” or other combinatorial structures but on algebraic laws. Our guiding model is here the theory of Lie groups (which is based on the group laws) and Loos’ theory of symmetric spaces [Lo69], 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: as explained in Part I of this work, it is convenient to consider a projective space X = ℙ(W) over a field or ring 𝕂 together with its dual space X′ = ℙ(W^{∗}) and to view elements a ∈ X′ as “affinizations” of X and vice versa. In general, a pair of spaces (X, X′), each of them parametrizing a family of affinizations on the other, is called an affine pair geometry over the base ring 𝕂 (Chapter 1). If such a structure is given, then for any fixed scalar r ∈ 𝕂 there is a natural ternary “multiplication map”

μ_{r} : X × X′ × X ⊃ D → X, (o, a, x) ↦ μ_{r}(o, a, x)

associating to an affinization a with affine part V_{a} ⊂ X and two points o, x ∈ V_{a} the product rx =: μ_{r}(o, a, x) in the 𝕂-module V_{a} with zero vector o. Exchanging the rôles of X and X′, a dual multiplication map μ′_{r} is defined. As usual, one defines left, right and middle multiplications associated to the ternary map μ = μ_{r} by

μ(x, a, y) = L_{x,a}(y) = M_{x,y}(a) = R_{a,y}(x).

Generalized projective geometries are now defined by requiring the following fundamental identities of projective geometry (Chapter 2):

(L_{x,a})^{t} = L_{a,x}, (R_{a,x})^{t} = R_{x,a}  (PG1)
(M_{x,y})^{t} = M_{y,x}, (M_{a,b})^{t} = M_{b,a},  (PG2)

complemented by a property (T) assuring the existence of translation groups. Here an identity of the type g^{t} = h means that the pair (g, h) behaves essentially like a projective map g and its transposed g^{t} in projective geometry, i.e. that the condition

g(μ_{s}(x, h(a), y)) = μ_{s}(g(x), a, g(y))

holds for all s ∈ 𝕂. In Part I we have shown that (PG1) and (PG2) hold for projective, Grassmannian, Lagrangian and conformal geometries. 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 automorphisms) or symmetric space laws (where left translations are automorphisms but right translations are not).

0.3. Tangent objects and “differential calculus”. Yet in another sense the identities (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 over 𝕂. The only assumption we need here is that (PG1) and (PG2) hold in all scalar extensions of 𝕂; we show that the scalar extension of a generalized projective geometry (X, X′) over 𝕂 by dual numbers over 𝕂 plays naturally the role of the tangent bundle (T X, T X′) (Chapter 7; the use of dual numbers in related contexts appears already in [F89, Chapter 4], [Lo79] and [Lo94]). Thus we have a sort of differential calculus on (X, X′), and in a similar way as one associates a Lie algebra to a Lie group or a Lie triple system to a symmetric space, derivations of the ternary maps μ_{r}, μ′_{r} at a base point (o, o′) give rise to a pair of ternary maps of tangent spaces,

T : V × V′ × V → V, T′ : V′ × V × V′ → V′,

V ≅ T_{o}X, V′ ≅ T_{o′}X′, satisfying the algebraic laws of a (linear) Jordan pair over 𝕂 (Theorem 9.5). Conversely, to any Jordan pair over 𝕂 we can construct a generalized projective geometry (X, X′) over 𝕂; these constructions are essentially inverse to each other and are functorial (Theorem 10.1). An important rôle in both constructions is played by the Lie algebra of derivations of (X, X′), known in Jordan theory as the associated Kantor-Koecher-Tits algebra (Chapter 9).

0.4. Polar and null geometries. Besides Jordan pairs, there are two other important algebraic categories in Jordan theory, namely Jordan triple systems and Jordan algebras. The former are the same as Jordan pairs with involution, see [Lo75, I.1]. Geometrically, they correspond to generalized polar geometries. Polarities can be defined in our general situation in the same way as in ordinary projective geometry: they are anti-automorphisms (p : X → X′, p′ : X′ → X) of order 2 which are not a null-system, i.e., not all x ∈ X are isotropic with respect to p (Chapter 3). The quadric of a polarity p is 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 to unital Jordan algebras is closely related to null systems and to inner 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 system T gives rise to a Lie triple system R = R_{T} defined by

R(X, Y)Z = −(T (X, Y, Z) − T (Y, X, Z));  (0.1)

the correspondence T ↦ R_{T} is called the Jordan-Lie functor (see [Be01]). In Chapter 4 we construct the corresponding functor on a geometric level: for any polar geometry (X, X′, p) the complement M^{(p)} of the associated quadric is a symmetric space over 𝕂 in an appropriate sense (Theorem 4.1). In the finite dimensional case over 𝕂 = ℝ we get a new and more conceptual construction of the geometric Jordan-Lie functor from [Be01], and for a general field 𝕂 we obtain a class of symmetric spaces which is algebraic over 𝕂 (cf. Section 11.2). In the real case, it is known by classification (work of E. Neher; cf. [Be01]) that all classical and about half of the exceptional simple symmetric 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 effective 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 conceptual 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 of exceptional Jordan structures, going back to the work of Freudenthal, Springer and others, can be found in [Io00]. Very closely related to our approach are the papers [Lo78], [Lo79] and [Lo95] by O. Loos and the papers [F89] and [F96] by J. Faulkner. Although the latter papers are placed in an incidence geometric context, our formalism is surprisingly close to the one developed there. Our presentation of the Kantor-Koecher-Tits algebra (Section 9) is motivated by Section 4 of [F89]. 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 [Lo79] (and also in our approach [Be01] 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 to get rid of regularity assumptions and thus to develop the theory in full generality, including the infinite dimensional case.

Organization of the paper. The contents is as follows:

1. Affine pair geometries

2. Generalized projective geometries

3. Generalized polar geometries

4. Associated symmetric spaces

5. Associated group actions

6. Bergman operator and quadratic maps

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. The first part [Be00] of this work is referred to as Part I.

Notation. 𝕂 denotes a commutative ring with unit 1 and 1 + 1 ∈ 𝕂^{∗}.

1. Affine pair geometries

1.1. A pair geometry is given by two sets X, X′ and a subset M ⊂ X × X′; if (x, a) ∈ M we say that x and a are remote or in general position. If (x, a) ∉ M, we say that (x, a) are neighboring. For o′ ∈ X′, respectively o ∈ X, we denote by

V_{o′} := {x ∈ X| (x, o′) ∈ M}, V′_{o} := {x′ ∈ X′| (o, x′) ∈ M}  (1.1)

the sets of objects remote from o′, respectively from o. We assume that the sets V_{o′}, o′ ∈ X′ (resp. V′_{o}, o ∈ X) cover X (resp. X′) or, equivalently, that all V_{o′} and all V′_{o} are non-empty. The case that X is covered already by one of the V_{o′} ’s is not excluded.

1.2. Homomorphisms of pair geometries are remoteness-preserving pairs of maps, that is, pairs g : X → Y , g′ : X′ → Y′ such that g(V_{o′}) ⊂ V_{g′(o′)} and g′(V′_{o}) ⊂ V′_{g(o)} for all o ∈ X, o′ ∈ X′. Local homomorphisms are defined by the same property; they are required to be defined at least on one pair of sets (V_{a}, V′_{x}) with (x, a) ∈ M.

1.3. If (X, X′, M) is a pair geometry, then so is (X′, X, M^{d}) with M^{d} = {(q, p)| (p, q) ∈ M}; 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. An affine pair geometry (over 𝕂) is a pair geometry (X, X′, M) such that for every element o′ ∈ X′ (resp. o ∈ X) a structure of an affine space over 𝕂 is defined on V_{o′} (resp. on V′_{o}). In other words, for each pair (o, o′) ∈ M there is a structure of a 𝕂module with zero vector o on V_{o′} and a 𝕂-module structure with zero vector o′ on V′_{o}. The affine parts V_{o′} ⊂ X, resp. V′_{o} ⊂ X′ are called affine cells of X, resp. of X′, and we say that X′ (resp. M) is the space of affinizations (resp. space of vectorializations) of X, and vice versa.

1.5. (The multiplication maps.) Given an affine pair geometry, we denote for any (o, o′) ∈ M and r ∈ 𝕂 by r_{o,o′} : V_{o′} → V_{o′} the multiplication by the scalar r in the 𝕂-module V_{o′} with zero vector o, and define dually r_{o′,o} : V′_{o} → V′_{o} as multiplication by r in the 𝕂-module V′_{o} with zero vector o′. Putting for a fixed r ∈ 𝕂 all these together, we define the multiplication maps associated to an affine pair geometry by

μ_{r} : X × X′ × X ⊃ D → X, (o, o′, x) ↦ μ_{r}(o, o′, x) := r_{o,o′}(x)  (1.2)

and dually

μ′_{r} : X′ × X × X′ ⊃ D′ → X′, (o′, o, x′) ↦ μ′_{r}(o′, o, x′) := r_{o′,o}(x′)  (1.3)

where

D = {(o, o′, x) ∈ X × X′ × X| (o, o′) ∈ M, (x, o′) ∈ M},  (1.4)

and dually for D′. Thus, if we fix the middle element, the partial maps

μ_{r}(·, o′, ·) : V_{o′} × V_{o′} → V_{o′}, μ′_{r}(·, o, ·) : V′_{o} × V′_{o} → V′_{o}  (1.5)

define on V_{o′} (resp. V′_{o}) the structure of an affine space over 𝕂 in the sense of Chapter 1 of Part I, and the concept of an affine pair geometry can be expressed by identities for the multiplication maps such as e.g. (Af3) which reads in this context

μ_{r}(p, a, q) = μ_{1−r}(q, a, p), μ′_{r}(a, p, b) = μ′_{1−r}(b, p, a).  (1.6)

1.6. Homomorphisms of affine pair geometries (X, X′), (Y, Y′) are remoteness-preserving pairs (g, g′) of maps g : X → Y , g′ : X′ → Y′ such that the multiplication maps are respected:

g(μ_{r}(o, o′, p)) = μ_{r}(g(o), g′(o′), g(o)), g′(μ′_{r}(o′, o, p′)) = μ′_{r}(g′(o′), g(o), g′(p′)). (1.7)

Equivalently, for all o′ ∈ X′ the restrictions

g : X ⊃ V_{o′} → V_{g′(o′)} ⊂ Y, g′ : X′ ⊃ V′_{o} → V′_{g(o)} ⊂ Y′

are affine maps. Local homomorphisms are local homomorphisms of pair geometries (in the sense of 1.2) respecting the multiplication maps. (Note that we do not require g′ to be determined uniquely by g or vice versa; however, in the “non-degenerate case” this property holds, see 2.8.) Isomorphisms are homomorphisms (g, g′) for which g and g′ are bijections. The composition of homomorphisms is again a homomorphism (with (gh)′ = g′h′); thus affine pair geometries form a category. In particular, we may speak of the automorphism group Aut(X, X′) of (X, X′).

1.7. A base point in (X, X′) is a fixed element (o, o′) ∈ M; a homomorphism of affine pair geometries with base point is a base point preserving homomorphism. The corresponding automorphism group Aut(X, X′; o, o′) = Aut(X, X′)_{o,o′} is called the structure group of (X, X′; o, o′); by definition of a homomorphism it acts linearly on V_{o′} × V′_{o}.

1.8. An adjoint pair of morphisms of affine pair geometries (X, X′), (Y, Y′) is a pair of maps g : X ⊃ U → Y , h : Y′ ⊃ U′ → X′ preserving remoteness in the sense that

g(V_{hp} ∩ U) ⊂ V_{p}, h(V′_{gq} ∩ U′) ⊂ V′_{q}

and verifying the relations

g(μ_{r}(o, h(o′), p) = μ_{r}(g(o), o′, g(p)), h(μ′_{r}(o′, g(o), p′)) = μ_{r}(h(o′), o, h(p′)). (1.8)

These relations can be rephrased by saying that g|_{V_{h(o′)}∩U} has a continuation by an affine map V_{h(o′)} → V_{o′}, and dually. We write h = g^{t} if (g, h) is an adjoint pair; this means just that (g, h) = (g, g^{t}) satisfies the relations (1.8) and shall not be interpreted in the sense that g^{t} be uniquely determined by g (although in the non-degenerate case it actually is, see 2.8 and 2.9). The conditions h = g^{t} for (X, X′), (Y, Y′) and g^{t} = h for the dual pairs (Y′, Y), (X′X) are equivalent. Clearly, if (g, g^{t}) and (f, f^{t}) are adjoint pairs such that g, f and f^{t}, g^{t} are composable, then (g ∘ f, f^{t} ∘ g^{t}) is an adjoint pair (this may be expressed by writing (gf)^{t} = f^{t}g^{t}). In particular, the affine pair geometries with base point form a category with respect to base-point preserving adjoint pairs of morphisms.

1.9. The preceding discussion shows that affine pair geometries can be turned into a category in two essentially different ways – in general, homomorphisms give not rise to adjoint pairs of morphisms or vice versa. However, if (g, g′) is an isomorphism, then clearly (g, g^{t}) with g^{t} = (g′)^{−1} 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 of 𝕂 we mean a unital commutative and associative 𝕂-algebra R. Let us denote by φ : 𝕂 → R, r ↦ r1_{R} the natural homomorphism (it need not be injective). A corresponding scalar extension of the affine pair geometry (X, X′) over 𝕂 is an affine pair geometry (X, X′)_{R} := (X_{R}, X′_{R}) over R together with a homomorphism of affine pair geometries over 𝕂

(Φ, Φ′) : (X, X′) → (X, X′)_{R}

such that, for all (x, a) ∈ M, the R-module (V_{Φ′(a)}, Φ(x)) (zero vector Φ(x)) is the usual scalar extension of the 𝕂-module (V_{a}, x), i.e.

(V_{Φ′(a)}, Φ(x)) ≅ (V_{a}, x) ⊗_{𝕂} R

as an R-module; more precisely, the following diagram shall commute:

\begin{aligned}(V_{a}, x) & = & (V_{a}, x) \\ ↓ & & ↓ \\ (V_{a}, x) ⊗_{𝕂} R & ≅ & (V_{Φ′(a)}, Φ(x)), \end{aligned}  (1.9)

where the first column is the natural map v ↦ v⊗_{𝕂}1_{R} and the second column is given by restriction of Φ. Dually, for (V′_{x}, a) a similar condition is required to hold. Moreover, we require (X, X′)_{R} to be minimal in the following sense: for every homomorphism (g, g′) of (X, X′) into a geometry (Y, Y′) defined over R, considered as a geometry over 𝕂, there exists a unique extension (g, g′)_{R} := (g_{R}, g′_{R}) to a homomorphism (X, X′)_{R} → (Y, Y′) defined over R. It is clear that this property determines (X, X′)_{R} up to isomorphy.

2. Generalized projective geometries

2.1. Assume (X, X′) is an affine pair geometry over 𝕂 with multiplication maps μ_{r}, μ′_{r}. As for any ternary map, we define right, left and middle multiplication associated to the ternary maps μ_{r} and μ′_{r} by

\begin{matrix}L_{x,a}(y) := R_{a,y}(x) := M_{x,y}(a) := & μ_{r}(x, a, y) \\ L_{a,x}(b) := R_{x,b}(a) := M_{a,b}(x) := & μ′_{r}(a, x, b) \end{matrix}  (2.1)

(where we add the superscript (r) if the dependence on r shall be indicated). Then (1.6) says that R_{a,y}^{(r)} = L_{y,a}^{(1−r)} , and thus we can transform left multiplications into right multiplications and vice versa. However, they cannot be transformed into middle multiplications since the latter exchange the partners X and X′ whereas 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 affine pair geometry satisfies the fundamental identities if, for all r ∈ 𝕂, the following holds:

(L_{x,a})^{t} = L_{a,x}, (R_{a,x})^{t} = R_{x,a},  (PG1)
(M_{x,y})^{t} = M_{y,x}, (M_{a,b})^{t} = M_{b,a},  (PG2)

where (x, a) ∈ M, and (x, y) ∈ X × X, (a, b) ∈ X′ × X′ are such that the middle multiplications are defined at at least one point. Since L_{x,a}^{(r)} = R_{a,x}^{(1−r)} , the first and the second condition in (PG1) are equivalent. Since g^{t} = h and h^{t} = g are equivalent, another equivalent formulation is (L_{a,x})^{t} = L_{x,a}. However, the two conditions in (PG2) are not equivalent: the first one says that (M_{x,y}, M_{y,x}) is an adjoint pair for (X′, X), (X, X′) and the second that (M_{a,b}, M_{b,a}) is an adjoint pair for (X, X′), (X′, X); both are self-dual. Using (1.8), the conditions (PG1) and (PG2) can be written more explicitely

\begin{aligned}L_{x,a}(μ_{s}(y, L_{a,x}(b), z)) & = μ_{s}(L_{x,a}(y), b, L_{x,a}(z)), \\ L_{a,x}(μ′_{s}(b, L_{x,a}(y), c)) & = μ′_{s}(L_{a,x}(b), y, L_{a,x}(c)), \\ M_{x,y}(μ′_{s}(a, M_{y,x}b, c)) & = μ_{s}(M_{x,y}(a), b, M_{x,y}(c)), \\ M_{a,b}(μ′_{s}(x, M_{b,a}y, z)) & = μ_{s}(M_{a,b}(x), y, M_{a,b}(z)). \end{aligned}

for all r, s ∈ 𝕂 and x, y, z ∈ X, a, b, c ∈ X′ where the expressions are defined. All variables written out, the preceding formulas read

\begin{aligned}μ_{r}(x, a, μ_{s}(y, μ′_{r}(a, x, b), z)) & = μ_{s}(μ_{r}(x, a, y), b, μ_{r}(x, a, z)), \\ μ′_{r}(a, x, μ′_{s}(b, μ_{r}(x, a, y), c)) & = μ′_{s}(μ′_{r}(a, x, b), y, μ′_{r}(a, x, c)), \\ μ_{r}(x, μ′_{s}(a, μ_{r}(y, bx), c), y) & = μ_{s}(μ_{r}(x, a, y), b, μ_{r}(x, c, y)), \\ μ′_{r}(a, μ′_{s}(x, μ_{r}(b, y, a), z), b) & = μ_{s}(μ_{r}(a, x, b), y, μ_{r}(a, z, b)). \end{aligned}

We also require that for all r ∈ 𝕂^{∗} the left multiplications L_{x,y}^{(r)} = r_{x,y} extend to bijections X → X such that all identities introduced so far still hold whenever all expressions are defined. This implies that, if both r and 1 − r belong to 𝕂^{∗}, the multiplication map μ_{r} is defined on the extended domain

D^{e} := {(x, a, y) ∈ X × X′ × X| (x, a) ∈ M or (y, a) ∈ M}.  (2.2)

Note that by (Af1) we have

r_{x,a} ∘ s_{x,a} = (rs)_{x,a}, 1_{x,a} = id_{V_{a}} .

Thus for r ∈ 𝕂^{∗} the condition (PG1) can be rephrased by saying that

(r_{x,a}, r_{a,x}^{−1}) ∈ Aut(X, X′).  (2.3)

2.3. (Translations.) Since by assumption the characteristic of 𝕂 is different from 2, we can express translations via major dilatations: for (o, o′) ∈ M,

τ_{v} := τ_{v}^{(o,o′)} := 2_{o,o′}2_{v,o′}^{−1}  (2.4)

is the translation by v in the 𝕂-module (V_{o′}, o). If the fundamental identity (PG1) in its version (2.3) holds, then the pair

(τ_{v}, τ̃_{v}) := (2_{o,o′}2_{v,o′}^{−1}, 2_{o′,o}^{−1}2_{o′,v})  (2.5)

belongs to Aut(X, X′); this can also be written

(τ̃_{v})^{t} = τ_{−v}.  (2.6)

The identities of affine geometry imply that τ_{v}τ_{w} = τ_{v+w} (sum in (V_{o′}, o)). If the transposed is unique (as in ordinary projective geometry), then we obtain by transposing:

(T)  τ̃_{v}τ̃_{w} = τ̃_{v+w}

which means that

τ_{V_{o′}} := {(τ_{v}, τ̃_{v})| v ∈ V_{o′}}  (2.7)

is a group, called the translation group with respect to the affinization o′; as a group, it is isomorphic to (V_{o′}, o, +). We say that our geometry satisfies the translation property if (T) holds together with its dual, for all (o, o′) ∈ M. The dual of (T) implies that

τ_{V′_{o}} := {(τ̃_{w}, τ_{w})| w ∈ V′_{o}}  (2.8)

with

τ̃_{w} := τ_{w}^{(o,o′)} := 2_{o,o′}^{−1}2_{o,w}  (2.9)

is an abelian group, isomorphic to (V′_{o}, o′, +). (Note that (T) is a consequence of the preceding identities if the transposed is unique; we do not know wether 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 [Lo75, Prop. 2.9]; if there were such a principle in the present context an identity such as τ_{v}τ_{w} = τ_{v+w} would imply its dual (T).)

2.4. A generalized projective geometry over 𝕂 is an affine pair geometry (X, X′) over 𝕂 in which the fundamental identities (PG1), (PG2) and the translation property (T) hold in all scalar extensions of 𝕂 – this means that, if φ : 𝕂 → R is any scalar extension, then there exists a scalar extension (X, X′)_{R} of (X, X′) in the sense of 1.10 verifying (PG1), (PG2) and (T) over R.

2.5. Homomorphisms and adjoint pairs of morphisms of generalized projective geometries are those of the underlying affine pair geometry. Therefore, generalized projective geometries can be turned into a category in two essentially different ways. The subgroup Int(X, X′) of Aut(X, X′) generated by all (r_{x,a}, r_{a,x}^{−1}) with (x, a) ∈ M, r ∈ 𝕂^{∗} is called the inner automorphism group.

2.6. For a definition of generalized projective geometries in characteristic 2, it will be necessary to add axiomatically the structure given by the following maps of four arguments:

\begin{matrix}X × X′ × X × X′ ⊃ W → X, & (o, o′, v, w) ↦ τ_{w}^{(o,o′)}(v), \\ X × X′ × X × X ⊃ W′ → X, & (o, o′, v, w) ↦ τ_{w}^{(o,o′)}(v) \end{matrix}  (2.10)

where W, W′ are 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 rôle played by the spaces X and X′ in our axioms is completely symmetric; thus (X′, X) with the space M_{d} = {(a, x)| (x, a) ∈ M} of vectorializations is again a generalized projective geometry, called the dual pair or dual geometry of (X, X′).

(2) (Direct products.) The direct product of (X, X′), (Y, Y′) is (X × Y, X′ × Y′) with remoteness given by V_{(o′,p′)} = V_{o′} × V_{p′}, and dually, which we require to carry the direct product structure V_{o′} × V_{p′} of affine spaces. It is then easily verified that (X × Y, X′ × Y′) is again a generalized projective geometry; the multiplication map is just the direct product of the ones of (X, X′) and (Y, Y′). In particular, we can define the direct product (X, X′) × (X′, X) = (X × X′, X′ × X) which will play an important role later on.

(3) (Subspaces.) These are subsets Y ⊂ X, Y′ ⊂ X′ such that (Y, Y′) is closed under all multiplications maps. Thus in the affinizations y′ ∈ Y′, Y is an affine subspace, and vice versa.

(4) (Inner ideals.) An inner ideal of X as a subset Y ⊂ X which is linear w.r.t. all possible affinizations, i.e. satisfying μ_{r}(Y, a, Y) ⊂ Y for all a ∈ X′; inner ideals of X′ are defined dually.

(5) (Congruences and quotient spaces.) A congruence is a subspace (R, R′) ⊂ (X × X, X′ ×X′) which is an equivalence relation – similarly as in the case of symmetric spaces ([Lo69, Chapter III]) one shows that (X, X′)/(R, R′) is again a generalized projective geometry.

(6) (Tangent bundle.) One can construct a tangent bundle (T X, T X′) of (X, X′) which is essentially scalar extension by dual numbers over 𝕂 - see Chapter 7.

(7) (Flat geometries.) The category of pairs of affine spaces over 𝕂 is imbedded in the category of generalized projective geometries as follows: let V, W be affine spaces in the usual sense; let X := V , X′ := W, M = V × W and let r_{o,o′} = r_{o} (i.e. all affine charts of X yield the same structure of affine space on V ), and dually r_{o′,o} = r_{o′}. 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 to a ∈ X′ the set V_{a} ⊂ X is injective, and dually.

(ii) We say that the generalized projective geometry is faithful if the map assigning to

a ∈ X′ the affine structure (V_{a}, A_{a} := μ_{·}(·, a, ·)) is injective. It is clear that a non-degenerate geometry is faithful; the converse is not true. If (g, g′) is an automorphism of a faithful geometry, then g determines g′ uniquely by the condition A_{g′(a)} = g_{∗}(A_{a}), where g_{∗} is the push-forward of the affine structure A_{a} by g, and conversely g is determined by g′.

2.9. (Conformal group.) The projective or conformal group of X is the group

Co(X) := {g : X → X | ∃g′ : X′ → X′ : (g, g′) ∈ Aut(X, X′)}.  (2.11)

The inner conformal group is the subgroup of Co(X) generated by the dilatations r_{x,a}, r ∈ 𝕂^{∗}, (x, a) ∈ M, and dually we define Co(X′) and its inner conformal group. If (X, X′) is faithful, then the surjective homomorphisms

Aut(X, X′) → Co(X), (g, g′) ↦ g, Aut(X, X′) → Co(X′), (g, g′) ↦ g′

are injective, hence Co(X) and Co(X′) are isomorphic to Aut(X, X′), and we have an isomorphism

Co(X) → Co(X′), g ↦ g′

and a canonical anti-isomorphism

Co(X) → Co(X′), g ↦ g^{t} = (g′)^{−1}.

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 (X, X′) is an isomorphism

(p, p′) : (X, X′) → (X′, X)

onto the dual pair. A correlation of (X, X′) is an antiautomorphism (p, p′) such that p′ = p^{−1} , or, equivalently, such that p^{t} = p. This in turn means that the identity

p(μ_{r}(x, p(y), z) = μ′_{r}(p(x), y, p(z))  (3.1)

and its dual hold.

3.2. With respect to a fixed correlation p, a point x ∈ X is called non-isotropic if x and p(x) are remote (i.e. (x, p(x)) ∈ M) and isotropic if x and p(x) are neighboring. A correlation is called a null-system if all points x ∈ M are isotropic and a polarity if there exist non-isotropic points. A generalized polar geometry is a generalized projective geometry (X, X′) together with a polarity p; homomorphisms of generalized polar geometries are homomorphisms (g, g′) of generalized projective geometries commuting with the respective polarities; in particular, the automorphism group Aut(X, X′, p) is the group of (g, g′) ∈ Aut(X, X′) such that g′ ∘ p = g ∘ p.

3.3. For a fixed polarity p, the set

Q^{(p)} := {x ∈ X| (x, p(x)) ∉ M}  (3.2)

of isotropic points is called the associated quadric, and its complement is denoted by

M^{(p)} = {x ∈ X| (x, p(x)) ∈ M}.  (3.3)

By the very definition of a polarity the set M^{(p)} is not empty. A polarity is called elliptic if the quadric is empty.

3.4. If a correlation p : X → X′ is fixed we will use it frequently to identify X with X′. Thus the product maps μ_{r} and μ′_{r} are both represented by a ternary map

μ̃_{r} : X × X × X ⊃ M × X → X, (x, y, z) ↦ μ_{r}(x, p(y), z)

satisfying identities which are obtained from (PG1) and (PG2) by simply forgetting the distinction between μ_{r} and μ′_{r}. Polar geometries are then characterized by the fact that some element of the diagonal in X × X belongs to M, 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 set X together with a subset of X × X containing some element of the diagonal and a family of ternary maps μ_{r} defined on an open subset of X × X × X and satisfying certain identities). Homomorphisms are then precisely the maps which are compatible with the multiplication maps μ̃_{r}. Null geometries can be axiomatically defined in a similar way.

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 important that one can associate to any generalized projective geometry a polar geometry in a canonical way:

Proposition 3.6. For any generalized projective geometry (X, X′), the generalized projective geometry (X ×X′, X′ ×X) admits a canonical polarity, given by the exchange map p(x, x′) = (x′, x). The corresponding space M^{(p)} ⊂ X × X′ is equal to the space M of vectorializations of X. Proof. It is clear that (x, x′) and p(x, x′) = (x′, x) are remote if and only if x and x′ are remote, whence the last claim. The first claim is proved by the following calculation:

pμ_{r}((x, x′), p(y, y′), (z, z′) = pμ_{r}((x, x′), (y′, y), (z, z′)) = p(μ_{r}(x, y′, z), μ′_{r}(x′, y, z′))
= (μ′_{r}(x′, y, z′), μ_{r}(x, y′, z)) = μ′_{r}((x′, x), (y, y′), (z′, z))
= μ_{r}(p(x, x′), (y, y′), p(z, z′)).

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 homomorphism (g, g′) we associate the pair (g × g′, g′ × g)). In Jordan theory the corresponding construction is given by imbedding the category of Jordan pairs into the category of Jordan triple systems by constructing the associated polarized Jordan triple system ([Lo75, p. 10]). In a closely related context, similar constructions have already been used by Rozenfeld, see [Fr65, p. 172].

4. Associated symmetric spaces over 𝕂

4.1. A symmetric space (in the sense of O. Loos [Lo69]) is a real manifold M with a smooth binary map μ : M × M → M, (x, y) ↦ μ(x, y) = σ_{x}(y) verifying

(M1) σ_{x}(x) = x (M2) σ_{x} ∘ σ_{x} = id_{M} (M3) σ_{x} ∈ Aut(μ), i.e. σ_{x}(μ(y, z)) = μ(σ_{x}(y), σ_{x}(z)) (M4) the fixed point x of σ_{x} is isolated. The automorphism σ_{x} is called the symmetry w.r.t. x, and the transvection group G(M) of a symmetric space is the group generated by all σ_{x}σ_{y} with x, y ∈ M. A connected symmetric space is homogeneous under the group G(M) and is of the form G/H where G = G(M) is a Lie group and H an open subgroup of a non-trivial involution of G; such spaces will be called homogeneous symmetric spaces. There exists a theory of symmetric k -varieties (see [He94]) 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 homogeneous” than the real or complex ones (see examples in Chapter 4 of Part I). We do not try to define here formally what a “symmetric space over 𝕂” should be, but the following theorem shows that any generalized polar geometry over 𝕂 defines a structure which certainly is one:

Theorem 4.2. Assume (X, X′, p) is a generalized polar geometry over 𝕂. Then the complement M^{(p)} of the associated quadric is stable under the binary map

μ(x, y) := μ_{−1}(x, p(x), y)

which satisfies the properties (M1) – (M3), and the symmetry with respect to a base point o ∈ X^{(p)} is σ_{o} = (−1)_{o,p(o)} (the negative of the identity of the vector space V_{p(o)} ).

Proof. If (g, g′) is an automorphisms of the polar geometry (X, X′, p), then g preserves the set M^{(p)} and is compatible with μ; in fact,

gμ(x, y) = gμ_{−1}(x, p(x), y)
= μ_{−1}(gx, g′p(x), gy)
= μ_{−1}(gx, p(gx), gy)
= μ(gx, gy).

Applying this to (g, g′) = ((−1)_{o,p(o)}, (−1)_{p(o),o}), we get the identity (M3). (Note that in general (g, g′) = (r_{x,p(x)}, r_{p(x),x}^{−1}) is an automorphism of the polar geometry if and only if r = r^{−1} ; thus r = −1 is the only non-trivial scalar for which the construction works.) The properties (M1) and (M2) are clear, and the last statement holds by the very definition of μ.

It is indeed reasonable to say that the fixed point o of σ_{o} = (−1)_{o,p(o)} is isolated: since 2 is invertible in 𝕂, we have, in every 𝕂-module, −x = x iff x = 0.

4.3. (The Jordan-Lie functor.) We say that (M^{(p)}, μ) is the symmetric space associated to the generalized polar geometry (X, X′, p). Homomorphisms of symmetric spaces are maps φ : M → N commuting with multiplication maps μ of M and N. The arguments given in the preceding proof show that then homomorphisms of polar geometries 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 the geometric Jordan-Lie functor; as mentioned in the introduction, it generalizes the geometric Jordan-Lie functor from the real finite dimensional case considered in [Be01].

Corollary 4.4. If (X, X′) is a generalized projective geometry, then the space M ⊂ X × X′ with the multiplication map

μ((o, o′), (x, x′)) = (μ_{−1}(o, o′, x), μ_{−1}(o′, o, x′))

is a symmetric space over 𝕂 in the sense of Theorem 4.2.

Proof. Apply Theorem 4.2 to the polarity of (X × X′, X′ × X) from Proposition 3.6.

4.5. The symmetric space M from the preceding corollary has as additional structure a double fibration over X and over X′ such that the fibers are affine spaces:

M = ⋃_{o∈X}{o} × V′_{o} = ⋃_{o′∈X′}{o′} × V_{o′}.  (4.1)

This can be seen as a sort of “polarization” on M; in fact these spaces generalize the para-Hermitian symmetric spaces introduced by Kozai and Kaneyuki ([KanKo85]). If (g, g′) is an automorphism of (X, X′), then, by the proof of Theorem 4.2, g×g′ preserves M and induces an automorphism of the symmetric space M which preserves the double fibration (4.1). Similarly it is verified that, if (g, g′) is an antiautomorphism of the generalized projective geometry (X, X′), then

g̃: X × X′ → X × X′, (x, a) ↦ (g′(a), g(x))

preserves M and induces an automorphism of the symmetric space M which exchanges 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 paraholomorphic”. The automorphism g̃ is involutive iff g′ = g^{−1} , that is, iff (g, g′) is a correlation, and it has a fixed point in M iff it is a polarity. Thus the polarities correspond precisely to the anti-paraholomorphic involutions having a fixed point in M. The whole fixed point set of a correlation p is

(X × X)_{p} = {(x, p(x))| x ∈ X} ⊂ X × X′;

if we identify X and X′ via p then this is just the diagonal in X × X. The intersection of this set with M is the fixed point set of p̃ in M; it is non-empty iff p is a polarity, and then

M^{(p)} → M_{p}, x ↦ (x, p(x))

is an isomorphism of symmetric spaces. Thus M^{(p)} is imbedded in M as a sort of “para-real form”.

5. Associated group actions

5.1. We have already defined the groups Aut(X, X′) and Int(X, X′) associated to a generalized projective geometry (X, X′). If (o, o′) ∈ M is a base point, we write V := V_{o′}, V′ := V′_{o} and consider the following stabilizer groups:

P := {(g, g′) ∈ Aut(X, X′)| g.o = o},
P′ := {(g, g′) ∈ Aut(X, X′)| g′.o′ = o′},
Str(V, V′) := P ∩ P′ = {(g, g′) ∈ Aut(X, X′)| g.o = o, g′.o′ = o′}.

(5.1)

The group Str(V, V′) is called the structure group of (V, V′); by the very definition of an automorphism, it is a subgroup of Gl(V) × Gl(V′). Note that, with τ̃_{w} defined by (2.9),

τ̃_{w}(o) = 2_{o,o′}^{−1}2_{o,w}(o) = o,  (5.2)

and thus τ_{V′} ⊂ P and, dually, τ_{V} ⊂ P′.

Lemma 5.2.

(i) P = Str(V, V′) · τ_{V′} (semidirect product) (ii) P′ = Str(V, V′) · τ_{V} (semidirect product)

Proof. (i) Since automorphisms are remoteness preserving we have g.o ∈ V iff (g.o, o′) ∈ M iff (g′)^{−1}.o′ ∈ V′. Therefore, if g.o = o, then w := (g′)^{−1}.o′ ∈ V′, and we let

(h, h′) := (g, g′) ∘ (τ̃_{w}, τ_{w}) = (g ∘ τ̃_{w}, g′ ∘ τ_{w}).

It follows that

h.o = g(τ̃_{w}.o) = g.o = o, (h′)^{−1}.o′ = τ_{−w}(g′)^{−1}.o′ = τ_{−w}(w) = o′,

whence also h′.o′ = o′, and therefore (h, h′) ∈ Str(V, V′). This proves existence of the decomposition. Let us prove uniqueness: if (g, g′) = (h, h′) ∘ (τ̃_{v}, τ_{v}) ∈ P, then (g′)^{−1}.o′ = (τ_{v})^{−1}.o′ = −v; thus v and hence also (h, h′) are uniquely determined. Part (ii) is the dual version of (i).

Let for a fixed base point (o, o′) ∈ M, Ω ⊂ Aut(X, X′) be the subset

Ω := {(g, g′) ∈ Aut(X, X′)| g.o ∈ V }.  (5.3)

Then, as remarked in the preceding proof,

Ω^{−1} = {(g, g′) ∈ Aut(X, X′)| g′.o′ ∈ V′}.  (5.4)

It is clear that P ⊂ Ω and P′ ⊂ Ω^{−1} .

Lemma 5.3. Ω = τ_{V} · P = τ_{V} · Str(V, V′) · τ_{V′} (unique decomposition)

Proof. Let (g, g′) ∈ Ω and put v := g.o. Then (τ_{−v}, τ̃_{−v}) ∘ (g, g′) ∈ P, and we can apply the preceding lemma. Conversely, if (g, g′) = (τ_{w}, τ̃_{w}) ∘ (p, p′) with w ∈ V , (p, p′) ∈ P, then g.o = w ∈ V , and moreover w and thus (p, p′) is uniquely determined.

5.4. The preceding decompositions do not extend to the whole group Aut(X, X′). However, in certain special cases we get the Harish-Chandra decomposition known from the theory of Hermitian symmetric spaces: let us say that a generalized projective geometry (X, X′) is stable if the intersection V_{a} ∩ V_{b} is non-empty for all a, b ∈ X′, and dually (cf. [Lo95, Prop. 3.3] for this terminology). In other words,

X = ⋃_{a∈V′} V_{a}, X′ = ⋃_{x∈V} V′_{x}.

Proposition 5.5. (“Harish-Chandra decomposition”) If (X, X′) is stable, then

Aut(X, X′) = Ω · τ_{V} = τ_{V} · Str(V, V′) · τ̃_{V} · τ_{V}

(non-unique decomposition).

Proof. Let (g, g′) ∈ Aut(X, X′); pick x ∈ g^{−1}(V) ∩ V = V_{g^{t}.o′} ∩ V_{o′}; then v := g(x) ∈ V . Thus (g, g′) ∘ (τ_{x}, τ̃_{x}) ∈ Ω, and we can apply the preceding lemma.

The assumption is fulfilled for instance in the finite dimensional case over a field (cf. Section 11.1) and for some infinite dimensional geometries modelled on complex or real Banach spaces.

5.6. (Connectedness.) We say that (x, a) and (y, b) ∈ M are connected if there is a sequence (p_{1}, q_{1}), …, (p_{n}, q_{n}) ∈ M such that (p_{1}, q_{1}) = (x, a), (p_{n}, q_{n}) = (y, b) and

(p_{j+1}, q_{j+1}) ∈ (V_{q_{j}}, V′_{p_{j}}), j = 1, …, n − 1.

It is clear that connected is an equivalence relation; thus we get a partition of M = ∪_{i∈I}M_{i} into connected components, and it is easily verified that all (X_{i}, X′_{i}, M_{i}) with X_{i} := pr_{1}(M_{i}), X′_{i} = pr_{2}(M_{i}) are subspaces of (X, X′, M), called connected components of (X, X′). (A natural example of a non-connected geometry is the Grassmannian (X, X) of all subspaces of a given 𝕂-module, cf. Chapter 3 of Part I.)

Theorem 5.7. Assume (X, X′) is connected and fix a base point (o, o′) ∈ M. Then the subgroup

G := PE(V, V′) :=< (r_{x,a}, r_{a,x}^{−1})| (x, a) ∈ M ∩ (V × V′), r ∈ 𝕂^{∗} >

of Int(X, X′) generated by all bijective major dilatations with center in M ∩ (V × V′) acts transitively on X, on X′ and on M, i.e.

X ≅ G/(P ∩ G), X′ ≅ G/(P′ ∩ G), M ≅ G/(Str(V, V′) ∩ G).

In particular, M is a homogeneous symmetric space in the sense that it is homogeneous under its automorphism group.

Proof. It is enough to show that, if (x, a) ∈ M ∩ (V × V′) is an arbitrary point, then there exists (g, g′) ∈ G such that (x, a) = (g.o, g′.o′); the claim then follows by a straightforward induction using connectedness. We can write (x, a) = (f.o, h′.o′) with (f, f′) = (τ_{x}, τ̃_{x}), (h, h′) = (τ̃_{a}, τ_{a}) ∈ G. Since (x, a) ∈ M, we have

(x, a) = (f.o, h′.o′) = (h, h′).(h^{−1}f.o, o′) = (h, h′).(τ_{h^{−1}f.o}.o, o′)

= (h, h′) ∘ (τ_{h^{−1}f.o}, τ_{h^{−1}f.o}).(o, o′),

= (τ̃_{a}, τ_{a}) ∘ (τ_{_{τ_{−a}}(x)}, τ̃_{_{τ_{−a}}(x)}).(o, o′),

thus we can take the element

(g, g′) = (τ̃_{a}, τ_{a}) ∘ (τ_{_{τ_{−a}}(x)}, τ̃_{_{τ_{−a}}(x)})  (5.5)

of G. The last statement follows now from the proof of Theorem 4.2 and Corollary 4.4 where it is shown that G acts as a group of automorphisms of the symmetric space M.

Transitivity of Aut(X, X′) on M means that up to isomorphism there is only one 𝕂module on which a connected geometry (X, X′) is 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

\begin{aligned}(x, a) & = (τ_{x}, τ̃_{x}).(o, τ̃_{−x}τ_{a}.o′) \\ & = (τ_{x}, τ̃_{x}) ∘ (τ̃_{_{τ_{−x}}(a)}, τ_{_{τ_{−x}}(a)}).(o, o′), \end{aligned}

and thus we could also have taken the element

(e, e′) := (τ_{x}, τ̃_{x}) ∘ (τ̃_{_{τ_{−x}}(a)}, τ_{_{τ_{−x}}(a)})  (5.6)

of G. These two possibilities correspond to two different ways of joining (o, o′) and (x, a) by using the double fibration of M mentioned in 4.5.

Corollary 5.8. With respect to a base point (o, o′) ∈ M, we have the following expression for the multiplication map μ_{r}: for all (a, x) ∈ M ∩ (V × V′) and r ∈ 𝕂,

r_{x,a} = τ_{x}τ̃_{_{τ_{−x}}(a)} r_{o,o′} (τ_{x}τ̃_{_{τ_{−x}}(a)})^{−1},

and dually. Proof. Just write (x, a) = (e.o, e′.o′) with (e, e′) given by (5.6) and use that r_{e.o,e′.o′} = e ∘ r_{o,o′} ∘ e^{−1} since (e, e′) is an automorphism.

5.9. (Non-homogeneous symmetric spaces.) If (X, X′, p) is a generalized polar geometry, then the associated symmetric space M^{(p)} is general not homogeneous under its automorphism group, even if (X, X′) is connected. The orbit structure of M^{(p)} under Aut(X, X′, p) is in general as complicated as the classification of non-degenerate quadratic forms over 𝕂 (which is a special case of our general set-up, see Section 5.3 of Part I). 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 point (o, o′) ∈ M and let V := V_{o′}, V′ := V′_{o}. Following a standard terminilogy in Jordan theory, the set of remote elements in V × V′,

M ∩ (V × V′) = {(x, a) ∈ M| (x, o′) ∈ M, (o, a) ∈ M}  (6.1)

is called the set of quasi-invertible pairs. From the definition it is clear that (x, a) is quasi-invertible in (X, X′; o, o′) iff so is (a, x) in (X′, X; o′, o). If (x, a) is quasiinvertible, then we can write (x, a) = (g, g′).(o, o′) = (e, e′).(o.o′) according to Equations (5.5) and (5.6). The element (g, g′)^{−1}∘(e, e′) ∈ Str(V, V′) defines the Bergman operator: for (x, a) quasi-invertible we define

\begin{aligned}B(x, a)^{−1} & := τ_{−_{τ_{−a}}(x)}τ̃_{−a}τ_{x}τ̃_{_{τ_{−x}}(a)}, \\ B(a, x)^{−1} & := τ_{−_{τ_{−x}}(a)}τ̃_{−x}τ_{a}τ̃_{_{τ_{−a}}(x)}. \end{aligned}  (6.2)

Then (B(x, a), B(a, x)^{−1}) is an automorphism of (X, X′) fixing (o, o′) and hence belongs to Str(V, V′).

6.2. From the definition it is easiliy deduced that the Bergman operator depends functorially on the geometry with base point: if (h, h′) is a base point preserving homomorphism, then

B(hx, h′a) ∘ h = h ∘ B(x, a),  (6.3)

and similarly for base point preserving adjoint pairs.

6.3. (The quadratic maps.) If (x, a) or (−x, a) belongs to M, then the following expressions are defined:

\begin{aligned}Q(x)a & := −M_{x,−x}^{( \frac{1}{2} )}(a) = −μ_{\frac{1}{2}} (x, a, −x), \\ Q(a)x & := −M_{a,−a}^{( \frac{1}{2} )}(x) = −μ′_{\frac{1}{2}} (a, x, −a). \end{aligned}  (6.4)

The first expression is linear in a; in fact, by (PG2) Q(x)^{t} = Q(x), i.e. (g, h) = (Q(x), Q(x)) is an adjoint pair which means that Q(x) : V′_{Q(x)o′} → V_{o′} is affine. Since Q(x)o′ = −μ_{\frac{1}{2}} (x, o′, −x) = o, it follows that Q(x) : V′_{o} → V_{o′} is linear with respect to the zero vectors o and o′. Next we show that Q(x)a is homogeneous quadratic in x: from the fact that (r_{x,a}, r_{a,x}) for r ∈ 𝕂 is an adjoint pair it follows that if (rx, a) is quasi-invertible, then so is (x, ra). Then we have by (PG1) and by linearity in a:

\begin{aligned}Q(rx)a & = −μ_{\frac{1}{2}} (rx, a, −rx) = −rμ_{\frac{1}{2}} (x, ra, −x) \\ & = −r^{2}μ_{\frac{1}{2}} (x, a, −x) = r^{2}Q(x)a. \end{aligned}

It will be shown in 8.7 that Q actually extends to a quadratic polynomial on V .

6.4. (The symmetry principle.) The expressions τ̃_{a}(x) and τ̃_{x}(a) cannot be directly compared since they belong to different spaces. However, we have for all a ∈ V′ and x ∈ V the following equality, called the symmetry principle:

τ_{x}Q(x)τ̃_{x}(a) = τ̃_{a}(x).  (6.5)

In fact, using (PG1) and the condition (Af3) from Chapter 1 of Part I we have

τ̃_{a}(x) = τ_{−a}^{−1}(x) = (2_{o,o′}^{−1}2_{o,−a})^{−1}(x) = 2_{o,−a}^{−1}2_{o,o′}(x)
= μ_{\frac{1}{2}} (o, −a, 2_{o,o′}(x)) = μ_{\frac{1}{2}} (o, −a, τ_{x}(x))
= τ_{x}μ_{\frac{1}{2}} (−x, τ̃_{−x}(−a), x) = τ_{x}(−Q(x)τ̃_{−x}(−a))
= τ_{x}Q(x)τ̃_{x}(a).

7. The tangent bundle

7.1. The tangent bundle of a generalized projective geometry (X, X′) defined over 𝕂 is the scalar extension

(Φ, Φ′) : (X, X′) → (T X, T X′) := (X_{𝕂(ε)}, X′_{𝕂(ε)})  (7.1)

by the ring 𝕂(ε) of dual numbers over 𝕂: 𝕂(ε) = 𝕂[x]/(x^{2}); a model of 𝕂(ε) is R = 𝕂 ⊕ 𝕂 with elements denoted by r + εs, r, s ∈ 𝕂 and multipliplied by the rule (r + εs)(r′ + εs′) = rr′ + ε(r′s + s′r). The scalar extension φ : 𝕂 → R is thus always injective, and so is, for any 𝕂-module V , the natural map

V → V_{𝕂(ε)} := V ⊗_{𝕂} R ≅ V ⊕ εV

given by restriction of Φ. If a base point (o, o′) is fixed, we will identify V with Φ(V) and V′ with Φ′(V′). Next we are going to show that (T X, T X′) is a fibered space with fibers carrying a natural 𝕂-module structure:

Proposition 7.2. Assume (Y, Y′) is a generalized projective geometry over 𝕂(ε), and let (o, a) ∈ M. Then the set

F_{o} := ε_{o,a}(V_{a}) ⊂ Y

as well as its affine structure over 𝕂 induced from V_{a} are independent of the point a ∈ V′_{o}. Proof. Let o′ ∈ V′_{o} be another affinization. We have to prove that

ε_{o,a}(V_{a}) = ε_{o,o′}(V_{o′})  (7.2)

as sets and as 𝕂-modules. Write a = τ_{a}^{(o′,o)}(o′) =: τ_{a}(o′) and note that, since (τ̃_{a}, τ_{a}) is an automorphism, τ̃_{a} : V_{o′} → V_{a} is a 𝕂-module isomorphism and ε_{o,a} = ε_{_{τ_{a}}(o),τ_{a}(o′)} = τ̃_{a} ∘ ε_{o,o′} ∘ τ̃_{−a}. It follows that

τ̃_{a} : ε_{o,o′}V_{o′} → ε_{o,a}V_{a}

is a 𝕂-module isomorphism. Thus (7.2) is proved if we can show that

τ̃_{a}(ε_{o,o′}z) = ε_{o,o′}z  (7.3)

for all z ∈ V_{o′}. In order to prove (7.3), we apply first the symmetry principle 6.4 with x = ε_{o,o′}z and then use that Q(x) is quadratic in x along with ε_{o,o′}^{2} = 0_{o,o′}:

τ̃_{a}(ε_{o,o′}z) = τ̃_{a}(x) = τ_{x}Q(x)τ̃_{x}(a)
= τ_{x}ε_{o,o′}^{2}Q(z)τ_{x}(a)
= τ_{x}(o) = x = ε_{o,o′}z

which had to be shown.

7.3. Equation (7.3) shows also that all pairs (ε_{o,o′}v, a) with v ∈ V , a ∈ V′, are quasi-invertible, and therefore also all pairs (v, ε_{o′,o}a) are quasi-invertible.

Corollary 7.4. Assume (Y, Y′) and (Z, Z′) are generalized projective geometries over 𝕂(ε).

(i) If (g, g′) : (Y, Y′) → (Z, Z′) is a homomorphism, then, for all o ∈ Y , g(F_{o}) ⊂ F_{g(o)} , and g : F_{o} → F_{g(o)} is 𝕂-affine.

(ii) If g : Y → Z, g^{t} : Z′ → Y′ is an adjoint pair of morphisms, then g(F_{o}) ⊂ F_{g(o)} for all o such that (o, g^{t}(b)) ∈ M for some b in the domain of definition of g^{t}, and g : F_{o} → F_{g(o)} is 𝕂-affine.

In particular, the relation

μ_{r}(F_{x}, F′_{a}, F_{y}) ⊂ F_{μ_{r}(x,a′,y)}  (7.4)

holds for all (x, a, y) ∈ D, and dually.

Proof. The first claim is proved by the following calculation:

g(F_{o}) = g(ε_{o,a}V_{a}) ⊂ ε_{g(o),g′(a)}V_{g′(a)} = F_{g(o)},

and since g : V_{a} → V_{g′(a)} is affine, so is is the map g : F_{o} → F_{g(o)} obtained by restriction to the affine subspace ε_{o,o′}V_{a} ⊂ V_{a}. For adjoint pairs, the calculation is similar, replacing a by g^{t}(b). Applying (ii) to the adjoint pairs given by right, left and middle multiplications, we get (7.4). (The assumption of (ii) is fulfilled since we assume μ_{r}(x, a′, y) to be defined.)

7.5. The sets F_{o} with o ∈ V_{o′} define a partition of V_{o′}, and since they do not depend on the affinization, the sets F_{o} with o ∈ Y define a partition of Y , and dually for Y′. Denote by (Y, Y′)/ ∼ the corresponding sets of equivalence classes; then relation (7.4) implies that there is a well-defined structure of generalized projective geometry over 𝕂 on (Y, Y′)/ ∼, and the projection

(π, π′) : (Y, Y′) → (Y, Y′)/ ∼  (7.5)

is a homomorphism.

7.6. We apply the preceding results to the tangent bundle (T X, T X′). The composition

(π, π′) ∘ (Φ, Φ′) : (X, X′) → (T X, T X′)/ ∼, (x, a) ↦ (Φ(x), Φ′(a))/ ∼  (7.6)

is a homomorphism which, in every affinization, is naturally identified with the identity map; one may say that it is a sort of “covering”. Using the minimality required in the definition of a scalar extension (see 1.10), one can show that (7.6) actually is a bijection; we omit the technicalities since in the sequel we will actually only need that (7.6) is a bijection when restricted to affine parts. For simplicity of notation we will consider (Φ, Φ′) to be an inclusion of (X, X′) in (T X, T X′) and (π, π′) to be a homomorphism onto (X, X′).

Using these conventions, we define the tangent space T_{o}X of X at o to be the 𝕂-module ε_{o,o′}V_{o′}, that is,

T_{o}X = π^{−1}(o) ⊂ T X,

and dually, we define the tangent space of X′ at o′. Thus the tangent bundle is a fibred space over the basis (X, X′):

T X = ⋃_{p∈X} T_{p}X, T X′ = ⋃_{a∈X′} T_{a}X′,

and we can write any element of T X as δ_{p} ∈ T_{p}X with a unique p ∈ X, and dually. If (g, g′) : (X, X′) → (Y, Y′) is a homomorphism, then we define its tangent map at o by

T_{o}g : T_{o}X → T_{g.o}Y, ε_{o,o′}v ↦ ε_{g.o,g′.o′}gv;

one easily verifies that this is well-defined and linear; the map (T g, T g′) : (T X, T X′) → (T X, T X′) thus obtained coinceds with the extension (g, g′)_{𝕂(ε)} over 𝕂(ε) of (g, g′) whose existence is required by 1.10. Similarly, for an adjoint pair (g, g^{t}) we let

T_{o}g : T_{o}X → T_{go}Y, ε_{o,g^{t}o′}v ↦ ε_{g.o,o′}gv

provided there exists o′ in the domain of g^{t} with (o, g^{t}(o′)) ∈ M.

8. Infinitesimal automorphisms

8.1. An infinitesimal automorphism of (T X, T X′) is an automorphism (g, g′) of (T X, T X′) preserving all tangent spaces, that is, we have π ∘ g = π, and dually. Since any automorphism of (T X, T X′) permutes the tangent spaces (Corollary 7.4), it is actually enough to require that g(0_{p}) ∈ T_{p}X, g′(0_{p′}) ∈ T_{p′}X′ for all p ∈ X, p′ ∈ X′ (where 0_{x} denotes the zero vector in a tangent space at x). The composition of (g, g′) with the zero section,

(ξ, ξ′) : (X, X′) → (T X, T X′), (p ↦ g(0_{p}), p′ ↦ g′(0_{p′}))

is a homomorphism; we call it the associated vector field.

Lemma 8.2. If (X, X′) is connected, then the following pairs (g, g′), defined with respect to a base point (o, o′) ∈ M, are infinitesimal automorphisms of (T X, T X′):

(i) (g, g′) = ((1 + ε)_{o,o′}, (1 − ε)_{o′,o}), (ii) (g, g′) = (τ̃_{εw}, τ_{εw}) with w ∈ V′, (iii) (g, g′) = (τ_{εv}, τ̃_{εv}) with v ∈ V . Proof. All three pairs are automorphisms of (T X, T X′): for the last two this follows directly from the definition of τ̃, and for the first one, this follows from the fact that (1 + ε)(1 − ε) = 1 − ε^{2} = 1.

In order to prove that (π, π′) ∘ (g, g′) = (π, π′), we verify this property first on the trivialization of (T X, T X′) given by (V, V′); then since (T X, T X′) is algebraically generated by (T V, T V′) (Theorem 5.7), both sides, being homomorphisms, must coincide on all of (T X, T X′). For the following calculations, note that x + εv ∈ T_{x}X; in fact

π(x + εv) = π((1 − ε)x + ε(x + v)) = π(ε_{x,o′}(x + v)) = x.  (8.1)

Thus for the pair from (i) we have

g(x) = (1 + ε)_{o,o′}(x) = x + εx ∈ T_{x}X  (8.2)

and dually. (The corresponding vector field is the Euler vector field corresponding to the trivialization (V, V′).) For the second pair we use the symmetry principle (6.3) twice (which is possible since all pairs (x, εw) are quasi-invertible, by 7.3).

\begin{aligned}g(x) = τ̃_{εw}(x) & = x + Q(x)τ̃_{x}(εw) \\ & = x + Q(x)εw = x + εQ(x)w ∈ T_{x}X \end{aligned}  (8.3)

and

g′(a) = τ_{εw}(a) = a + ε_{o′,o}w ∈ T_{a}X′.  (8.4)

For the third pair the same argument applies.

Proposition 8.3. If (g, g′) is an infinitesimal automorphism of (T X, T X′), then g and g′ act on all tangent spaces by translations, that is, with (ξ, ξ′) as in 8.1., we have

g(δ_{p}) = δ_{p} + ξ(p), g′(δ_{p′}) = δ_{p′} + ξ′(p′)

for all δ_{p} ∈ T_{p}X, δ_{p′} ∈ T_{p′}X′. In particular, the infinitesimal automorphisms form an abelian subgroup of Aut(T X, T X′).

Proof. Let o := p and choose o′ ∈ V′_{o} arbitrary; then since g(o) ∈ T_{o}X and T_{o}X ⊂ V_{o′} we can decompose, using Lemma 5.3,

(g, g′) = (τ_{g.o}, τ̃_{g.o}) ∘ (h, h′) ∘ (τ̃_{w}, τ_{w})  (8.5)

with w = (g′)^{−1}o′ ∈ T_{o′}X′. By Lemma 8.2, the first and the last and hence also the middle factor are then infinitesimal automorphisms. Since the first factor clearly acts by a translation on T_{o}X and the last factor acts trivially on T_{o}X (Equation (7.3)), it only remains to be shown that h acts trivially on T_{o}X. Now, writing h = ({a \atop c} {b \atop d}), ε = ({0 \atop 1} {0 \atop 0}) with respect to the decomposition T V = V ⊕ εV , the condition h ∘ ε = ε ∘ h yields

h = ( \begin{matrix}a & 0 \\ c & a \end{matrix} ) .  (8.6)

The condition π ∘ h = π implies x = π(h(x)) = a(x) and thus a = id_{V} , and h acts trivially on T_{o}X as had to be shown.

8.4. We denote by InfAut(T X, T X′) ⊂ Aut(X, X′) the subgroup of infinitesimal automorphisms; by the preceding proposition, its group law is simply given by addition in tangent spaces and will therefore be written additively. We let further, with respect to a base point (o, o′),

InfStr(T V, T V′) := Str(T V, T V′) ∩ InfAut(T X, T X′);  (8.7)

this is the subgroup of elements of Str(T V, T V′) of the form (8.6) with h_{1} = id. Then the decomposition (8.5) reads, additively,

InfAut(T X, T X′) ≅ V ⊕ InfStr(T V, T V′) ⊕ V′.  (8.8)

The infinitesimal automorphisms form a normal subgroup of Aut(T X, T X′); in particular, for all (h, h′) ∈ Int(T X, T X′) the conjugation

(h, h′)_{∗}(g, g′) := (hgh^{−1}, h′g′(h′)^{−1})

is an automorphism of InfAut(T X, T X′).

8.5. The following theorem is a main result of this chapter: we will prove that infinitesimal automorphisms are in an affinization represented by polynomial quadratic maps. Recall that (g, g′) ∈ InfAut(T X, T X′) is uniquely determined by its associated vector field (ξ(p), ξ′(a)) = (g(0_{p}), g′(0_{a})), and in (T V, T V′), identifying p with 0_{p} and a with 0_{a}, we can write

g(p) = p + εG(p), g′(q) = q + εG′(q)  (8.9)

with functions G : V → V , G′ : V′ → V′.

Theorem 8.6. For all (g, g′) ∈ InfAut(T X, T X′), the maps G : V → V , G′ : V′ → V′ are quadratic polynomial. More precisely, the map G is

(i) constant, if (g, g′) = (τ_{εv}, τ̃_{εv}) belongs to the first term in the decomposition (8.8), (ii) linear, if (g, g′) ∈ InfStr(T V, T V′), (iii) homogeneous quadratic polynomial, if (g, g′) = (τ̃_{εw}, τ_{εw}) with w ∈ V′ is an

element of the third term in (8.8).

Proof. It is enough to show that G is quadratic polynomial; by duality the corresponding statement for G′ then follows. Part (i) is clear since τ_{εv}(p) = p + εv and Part (ii) follows from (8.6) since h(p) = p + εc(p). Thus only Part (iii) remains to be proved. Comparing with Equation (8.3), we see that (iii) is equivalent to saying that Q(x)εw is quadratic polynomial in x. We have already seen (see 6.3) that this expression is homogeneous quadratic in x. In order to prove that the term

Q(x, z) := Q(x + z) − Q(x) − Q(z)  (8.10)

is linear in x and in z, we use the Bergman operator of (T X, T X′) associated to the base point (o, o′): we claim that (B(x, εa), B(εa, x)^{−1}) is an infinitesimal automorphism. In fact, replace a by εa in the definition of B(x, a) (Section 6.1) and recall that τ̃_{−x}(εa) = εa (Equation (7.3)); thus

\begin{aligned}B(x, εa)^{−1} & = τ_{−_{τ_{−εa}}(x)}τ̃_{−εa}τ_{x}τ̃_{_{τ_{−x}}(εa)} \\ & = τ_{−x}τ_{εQ(x)a}τ̃_{−εa}τ_{x}τ̃_{εa}, \\ B(εa, x)^{−1} & = τ_{−_{τ_{−x}}(εa)}τ̃_{−x}τ_{a}τ̃_{_{τ_{−εa}}(x)} \\ & = τ_{−εa}τ̃_{−x}τ_{εa}τ̃_{εQ(x)a}τ̃_{x}. \end{aligned}  (8.11)

These equations show that (B(x, εa), B(εa, x)^{−1}) is a composition of two infinitesimal automorphisms, where the first factor comes after a conjugation by (τ_{x}, τ̃_{x}). More explicitly, evaluating at a point z ∈ V , we get for the first component (note that if G is defined by by (8.9), then (τ_{−x}gτ_{x})(p) = p + εG(x + p))

\begin{aligned}B(x, εa)^{−1}z & = (τ_{−x}τ_{εQ(x)a}τ_{x})(τ_{−x}τ̃_{−εa}τ_{x})τ̃_{εa}(z) \\ & = z + ε(Q(x) + Q(z) − Q(z + x))(a) = z − εQ(x, z)a. \end{aligned}  (8.12)

This expression is linear in z since B(x, εa) is a linear operator, and since Q(x, z) is symmetric in x and z, it is also linear in x.

In the situation of the preceding proof, let

\begin{aligned}T (x, εa, z) & := T (x, εa)z := (Q(x + z) − Q(x) − Q(z))εa \\ T (a, εx, b) & := T (a, εx)b := (Q(a + b) − Q(a) − Q(b))εx. \end{aligned}  (8.13)

Equation (8.12) and its dual can be written in matrix form

B(x, εa)^{−1} = ( \begin{matrix}id & 0 \\ −T (x, εa) & id \end{matrix} ) , B(a, εx)^{−1} = ( \begin{matrix}id & 0 \\ −T (a, εx) & id \end{matrix} ) .

8.7. In the preceding proof, we have seen that V → Hom(V′, T_{o}X), x ↦ (a ↦ Q(x)εa = −μ_{\frac{1}{2}} (x, εa, −x)) is a homogeneous quadratic polynomial. Although this will not be used in the sequel, we remark here that one can deduce from (PG2) that this polynomial satisfies the identity Q(Q(x)y) = Q(x)Q(y)Q(x) known in Jordan theory as the “fundamental formula”.

8.8. (Derivations.) This section will not be needed in the sequel, but for the sake of completeness, we explain the relation between infinitesimal automorphisms and derivations: a derivation of (X, X′) is a homomorphism (ξ, ξ′) : (X, X′) → (T X, T X′) such that (π, π′) ∘ (ξ, ξ′) = id_{(X,X′)} . We have already seen that infinitesimal automorphisms are uniquely determined by the associated vector field which is a derivation. Conversely, we have:

Theorem 8.9. There is a canonical bijection between derivations and infinitesimal automorphisms, given by

ξ(x) := g(x), g(δ_{x}) := δ_{x} + ξ(x),

and dually. The derivations form a 𝕂-module w.r.t. pointwise addition and multiplication by scalars.

Proof. Using the infinitesimal automorphisms from Lemma 8.2, one proves first that for δ_{p} ∈ T_{p}X, δ_{q} ∈ T_{q}X, δ_{o′} ∈ T_{o′}X′,

\begin{aligned}μ_{r}(δ_{p}, δ_{o′}, δ_{q}) & = μ_{r}(δ_{p}, 0_{o′}, 0_{q}) + μ_{r}(0_{p}, δ_{o′}, 0_{q}) + μ_{r}(0_{p}, 0_{o′}, δ_{q}) \\ & = T_{p}(R_{o′,q}) · δ_{p} + T_{q}(L_{p,o}) · δ_{q} + T_{o′}(M_{p,q}) · δ_{o′}. \end{aligned}  (8.14)

Thus (ξ, ξ′) is a derivation iff

ξ(μ_{r}(p, a, q)) = T_{p}(R_{a,q}) · ξ(p) + T_{a}(M_{p.q}) · ξ′(a) + T_{q}(L_{p,a}) · ξ(q),  (8.15)

and dually. Since tangent maps are linear, this description shows that the derivations of (X, X′) form a 𝕂-module which we denote by Der(X, X′). The zero vector is the canonical imbedding (X, X′) → (T X, T X′). Moreover, a direct calculation now shows that if (ξ, ξ′) is a derivation, then the pair g := ξ + id : T X → T X, δ_{p} ↦ δ_{p} + ξ(p), g′ := ξ′ + id : T X′ → T X′, δ_{a} ↦ δ_{a} + ξ′(a) is an automorphism of (T X, T X′).

The preceding results on infinitesimal automorphisms can now be rewritten in terms of derivations; see [Lo69] for similar formulas in the case of symmetric spaces. In particular, an analogue of [Lo69, Theorem II.2.2] follows now from Theorem 8.6: (g, g′) = (− id_{V} , − id_{V′}) is an automorphism of (X, X′) which induces an involution of Der(X, X′); if we denote by Der(X, X′) = 𝔥 ⊕ 𝔮 the corresponding ±1-eigenspace decomposition, then the evaluation map at (o, o′) induces a bijection 𝔮 → V ⊕ V′, and 𝔥 is its kernel.

9. Kantor-Koecher-Tits algebra and Jordan functor

9.1. As in Lie theory, we have to derive twice in order to linearize identities which take account of non-commutativity – the reason is simply that the group of infinitesimal automorphisms of (T X, T X′) is abelian, and thus we lose all information on noncommutativity if we derive only once. Therefore we introduce the “double tangent bundle” (T T X, T T X′) := (T (T X), T (T X′)). It is isomorphic to the scalar extenstion of (X, X′) by

(𝕂(ε_{1}))(ε_{2}) = 𝕂 ⊕ ε_{1}𝕂 ⊕ ε_{2}𝕂 ⊕ ε_{3}𝕂  (9.1)

with ε_{3} = ε_{1}ε_{2} = ε_{2}ε_{1} and ε_{1}^{2} = ε_{2}^{2} = ε_{3}^{2} = 0. Fixing as usual a base point (o, o′) ∈ M, the corresponding linearization of (T T X, T T X′) is denoted by

T T V = V ⊕ ε_{1}V ⊕ ε_{2}V ⊕ ε_{3}V, T T V′ = V′ ⊕ ε_{1}V′ ⊕ ε_{2}V′ ⊕ ε_{3}V′.  (9.2)

The space (T T X, T T X′) is fibered over (X, X′), and every fiber decomposes in the way indicated by (9.2). Let π_{1} , π_{2} be the projections with kernels ε_{1}(V ⊕ ε_{2}V), resp. ε_{2}(V ⊕ ε_{1}V), and π_{3} = π_{1} ∘ π_{2} (kernel ε_{1}ε_{2}V ).

9.2. The group of infinitesimal automorphisms of (T X, T X′) can be injected into Aut(T T X, T T X′) in three different ways such that fibers over (X, X′) are preserved. However, it is no longer true that these actions commute – in fact, their commutators give rise to the Lie bracket, to be defined in the following theorem. Recall that, if the characteristic of 𝕂 is zero, the space Pol(V, V) of polynomial mappings from V to V is a Lie algebra over 𝕂 with respect to the Lie bracket (see e.g. [Koe69])

[X, Y](x) = DX(x) · Y (x) − DY (x) · X(x),  (9.3)

where DF is the ordinary total differential, defined in terms of dual numbers by the condition

F (x + εh) = F (x) + εDF (x) · h.  (9.4)

If the characteristic of 𝕂 is different from 2 or 3 (which we assume from now on), this formula still serves to define a bracket defined on quadratic polynomials having values in polynomials of degree at most 3.

Theorem 9.3. There is a unique structure of a Lie algebra over 𝕂 on the space InfAut(T X, T X′) such that, if elements are realized as quadratic polynomials according to Theorem 8.6, then their bracket is given by Formula (9.3). The group Aut(X, X′) acts by automorphisms of this Lie algebra structure.

Proof. The following definition of the Lie bracket follows the one given by Faulkner [F89] in a different context: let (g, g′), (h, h′) ∈ InfAut(T X, T X′). We realize these elements on (T T X, T T X′) in two different ways: we extend (g, g′) to an element (g_{1}, g′_{1}) ∈ Aut(T T X, T T X′) such that (π_{1}, π′_{1}) ∘ (g_{1}, g′_{1}) = (π_{1}, π′_{1}) and (h, h′) to an element (h_{2}, h′_{2}) such that that (π_{2}, π′_{2}) ∘ (h_{2}, h′_{2}) = (π_{2}, π′_{2}). Consider now the element

(f, f′) := (g_{1}h_{2}g_{1}^{−1}h_{2}^{−1}, (g_{1}h_{2}g_{1}^{−1}h_{2}^{−1})′)  (9.5)

of Aut(T T X, T T X′). Since infinitesimal automorphisms with respect to ε_{1} or with respect to ε_{2} form normal subgroups in Aut(T T X, T T X′), we see that (f, f′) belongs to both of them and hence satisfies (π_{3}, π′_{3}) ∘ (f, f′) = (π_{3}, π′_{3}). Therefore (f, f′) preserves fibers of the subbundle (X, X′)_{𝕂(ε_{3})} of (T T X, T T X′) and can thus be identified with an infinitesimal automorphism of this subbundle which we denote by ([g, h], [g′, h′]). It is clear that Aut(X, X′) acts by automorphisms of this bracket.

Next let us prove that with respect to an affinization the bracket is indeed described by formula (9.3): we write g_{1}(x) = x + ε_{1}G(x), h_{2}(x) = x + ε_{2}H(x) and apply (9.4) in the following calculation to G with respect to ε_{1} and to H with respect to ε_{2} :

g_{1}h_{2}g_{1}^{−1}h_{2}^{−1}(x) = ε_{1}G(x) + h_{2}g_{1}^{−1}(x − ε_{2}H(x))
= ε_{1}G(x) + h_{2}(x − ε_{2}H(x) − ε_{1}G(x − ε_{2}H(x)))
= ε_{1}G(x) + h_{2}(x − ε_{1}G(x) − ε_{2}(H(x) − ε_{1}DG(x) · H(x)))

= ε_{1}G(x) + (x − ε_{1}G(x)) + ε_{2}H(x − ε_{1}G(x)) − ε_{2}(x − ε_{1}DG(x) · H(x))

= x − ε_{1}ε_{2}(DH(x) · G(x) − DG(x) · H(x)),

and thus [g, h] is described by the polynomial [G, H] given by (9.3) (à priori, this polynomial is of degree at most 3, but Theorem 8.6 tells us that it is actually of degree at most 2). We conclude that InfAut(T X, T X′) is a Lie algebra: the defining identities of a Lie algebra are verified over the affinization (V, V′); every point is contained in some affinization; therefore they hold everywhere.

9.4. The Lie algebra defined in the preceding theorem is called the Kantor-Koecher-Tits algebra associated to (X, X′). This Lie algebra is 3-graded with grading 𝔤 = 𝔤_{1}⊕𝔤_{0}⊕𝔤_{−1} given by Equation (8.8), corresponding to the homogeneous polynomial components with respect to a fixed base point (o, o′): since the bracket of two homogeneous polynomials of degree r and s is of degree r + s − 1 and 𝔤 contains only quadratic polynomials, we have [𝔤_{1}, 𝔤_{1}] = 0 = [𝔤_{−1}, 𝔤_{−1}] and [𝔤_{1}, 𝔤_{−1}] ⊂ 𝔤_{0} . To any 3-graded Lie algebra over 𝕂, one associates two 𝕂-trilinear maps

T_{±} : 𝔤_{±1} × 𝔤_{∓1} × 𝔤_{±1} → 𝔤_{±1}, (x, a, z) ↦ T_{±}(x, a)z := [[x, a], z].  (9.6)

It is easily verified that then the following identities are satisfied:

(LJP1) T_{±}(x, a, z) = T_{±}(z, a, x) (LJP2) [T_{±}(x, a), T_{±}(y, b)] = T_{±}(T_{±}(x, a, y), b) + T_{±}(y, T_{∓}(a, x, b))

By definition, a linear Jordan pair over 𝕂 is a pair (V^{+}, V^{−}) of 𝕂-modules together with 𝕂-trilinear maps T_{±} such that the identities (LJP1) and (LJP2) hold (cf. [Lo75, Prop. 2.2. (b)]); equivalently, linear Jordan pairs can be defined as the polarized linear Jordan triple systems, see [Lo75, 1.12], or as the twisted polarized Lie triple systems, see [Be01, Ch. III.3] for the real case. Note that in the present case the maps T_{±} can be expressed by the quadratic maps Q: we write elements of 𝔤_{1} , respectively of 𝔤_{−1} , as

τ_{εv}(x) = x + εv =: x + εξ_{v}, τ̃_{εa}(x) = x + εQ(x)a =: x + εξ̃_{a};

then

\begin{aligned}[ξ_{v}, ξ̃_{a}](x) & = −εD(Q(·)a)(x)v = −2T (v, εa, x), \\ [[ξ_{v}, ξ̃_{a}], ξ_{w}] & = ξ_{−2T (v,εa,w)}. \end{aligned}

Thus the maps (T_{+}, T_{−}) are essentially given by (−2T (x, εa, y), −2T (a, εx, b)). We use the notation εT (x, a, y) := T (x, εa, y) (that is, we extend T to a 𝕂(ε)-trilinear map). Then we can summarize the preceding results:

Theorem 9.5. The pair of 𝕂-modules (V, V′) with the trilinear maps (x, a, y) ↦ T (x, a, y), (a, x, b) ↦ T (a, x, b) is a linear Jordan pair over 𝕂.

All linear Jordan pairs are obtained from 3-graded Lie algebras in the way just described; this will be used in the following chapter.

9.6. The Jordan pair structure can be written in terms of the Bergman operator of (T T X, T T X′): we rewrite (8.11) with ε_{1} := ε and x replaced by ε_{2}x:

\begin{aligned}B(ε_{2}x, ε_{1}a)^{−1} & = τ_{−ε_{2}x}τ̃_{−ε_{1}a}τ_{ε_{2}x}τ̃_{ε_{1}a} \\ B(ε_{1}a, ε_{2}x)^{−1} & = τ_{−ε_{1}a}τ̃_{−ε_{2}x}τ_{ε_{1}a}τ̃_{ε_{2}x}. \end{aligned}  (9.7)

These equations show that (B(ε_{2}x, ε_{1}a), B(ε_{1}a, ε_{2}x)^{−1}) is a commutator in the group Aut(T T X, T T X′) corresponding to a Lie bracket in InfAut(T X, T X′). From (8.12) we get

\begin{aligned}B(ε_{2}x, ε_{1}a)^{−1}z & = z − ε_{1}ε_{2}Q(x, z)a = z − ε_{1}ε_{2}T (x, a, z), \\ B(ε_{1}a, ε_{2}x)^{−1}b & = b − ε_{2}ε_{1}Q(a, b)x = b − ε_{1}ε_{2}T (a, x, b). \end{aligned}  (9.8)

Theorem 9.7. The associated Jordan pair depends functorially on the generalized projective geometry (X, X′) with base point (o, o′).

Proof. The Jordan pair is given by the Bergman operator, and by 6.2 the Bergman operator depends functorially on the pointed space.

Note that we do in general not get an induced homomorphism of the whole Kantor-Koecher-Tits algebra – this corresponds to the fact that homomorphisms of Lie triple systems do in general not induce homomorphisms of their standard imbeddings, cf. [Be01, Ch. V].

Theorem 9.8. The functor from Theorem 9.7 induces functors of the following kind:

(i) a functor associating to a generalized polar geometry over 𝕂 with base point a Jordan triple system over 𝕂,

(ii) a functor associating to a pointed symmetric space constructed from a polar geom-

etry a Lie triple system over 𝕂. Proof. (i) A polarity with p(o) = o′ defines an involution on the Jordan pair (V, V′) which is thus turned into a Jordan triple system.

(ii) Any Jordan triple system T defines by Formula (0.1) a Lie triple system R_{T} which by definition is the Lie triple system associated to the symmetric space M^{(p)} with base point o. (In the real finite dimensional case this is the curvature tensor of the symmetric space, see [Be01].)

10. Existence theorem

Theorem 10.1. For every Jordan pair (V^{+}, V^{−}) over a ring 𝕂 of characteristic different from 2 there exists a generalized projective geometry (X, X′) with base point (o, o′) whose associated Jordan pair is (V^{+}, V^{−}) and such that every Jordan pair homomorphism extends to a homomorphism of the associated generalized projective geometries. Proof. Given a Jordan pair (V^{+}, V^{−}) with trilinear maps T_{±} and quadratic maps Q_{±} over a ring 𝕂, let its Kantor-Koecher-Tits algebra

𝔤 := V^{+} ⊕ 𝔥 ⊕ V^{−}

be defined as in [Lo95, Section 1.1], with 𝔥 generated by the brackets

[v, w] = (−T_{+}(v, w), T_{−}(w, v)), v ∈ V^{+}, w ∈ V^{−},

and by the Euler operator ζ := (id_{V^{+}}, − id_{V^{−}}). We represent elements of End_{𝕂}(𝔤) by 3 × 3-matrices in the obvious way. The following elements (for v ∈ V^{+} , w ∈ V^{−}) are automorphisms of 𝔤 (see [Lo95, Lemma 1.2]):

exp_{+}(v) := e^{ad(v)} = ( \begin{matrix}1 & ad(v) & Q(v) \\ & 1 & ad(v) \\ & & 1 \end{matrix} ) , exp_{−}(w) := e^{ad(w)} = ( \begin{matrix}1 \\ ad(w) & 1 \\ Q(w) & ad(w) & 1 \end{matrix} ) ,

and exp_{±} : V^{±} → Aut(𝔤) are injective group homomorphisms onto groups called U^{+} and U^{−}. An automorphism h = (h^{+}, h^{−}) of (V^{+}, V^{−}) is identified with the element

h̃ := ( \begin{matrix}h_{+} \\ & h_{0} \\ & & h_{−} \end{matrix} )

of Aut(𝔤), with h_{0} given by conjugation with h. In particular, we have a homomorphism

𝕂^{∗} → Aut(𝔤), r ↦ r̃ id = ( \begin{matrix}r id \\ & id \\ & & r^{−1} id \end{matrix} ) .

Let G := PE(V^{+}, V^{−}) ⊂ Aut(𝔤) be the group generated by U^{+} and U^{−}, called the projective elementary group of (V^{+}, V^{−}). Set

H := G ∩ Aut(V^{+}, V^{−})

Because of h exp_{±}(x)h^{−1} = exp(h_{±}(x)), H normalizes U^{+} and U^{−}. Define ‘parabolic subgroups’ and the ‘big cell’ in G by

P^{±} = HU^{±} = U^{±}H, Ω = U^{−}HU^{+} = P^{−}P^{+}.

Now we are ready to define the generalized projective geometry (X, X′): let

X := X^{+} := G/P^{−}, X′ := X^{−} := G/P^{+};

denote by g.x := g^{+}(x), g′.x := g^{−}(x) the action of G on X, respectively on X′; let (o, o′) := (o_{+}, o_{−}) be the base point in X^{+} × X^{−} and define

M := G.(o_{+}, o_{−}) ≅ G/(P^{+} ∩ P^{−}) = G/H.

(Since (g, g′) fixes the Euler operator iff g and g′ are linear, a geometric model of M is the orbit of the Euler operator in 𝔤.) As usual we let V_{a} = {x ∈ X| (x, a) ∈ M}, V′_{x} = {a ∈ X′| (x, a) ∈ M}. Then the sets V_{o′} and V^{+} are naturally identified since

V_{o_{−}} = {g.o^{+}| g ∈ G, g.o^{−} = o^{−}} = P^{+}.o^{+} = U^{+}.o^{+} ≅ V^{+},

and dually. Therefore, by transport of structure, the spaces V_{o′} and V′_{o} carry natural 𝕂-module structures. Since, by its definition, H acts linearly on V^{+} × V^{−}, these 𝕂module structures are invariant under the stabilizer group H of the origin in M = G/H, and thus we can transport them in a well-defined way to any point (x, a) ∈ M. Summing up, we have defined the multiplication map μ_{r} : X × X′ × X ⊃ D → X and its dual, with D given by Equation (1.4). Moreover, by construction G acts as a group of automorphisms of (μ_{r}, μ′_{r}); in particular, all (r_{x,a}, r_{a,x}^{−1}) with r ∈ 𝕂^{∗} are of the form (g, g′) with g ∈ G and thus define automorphisms. This means that the identity (PG1) holds for all r ∈ 𝕂^{∗}.

Let us show that (PG1) holds in fact for all scalars r ∈ 𝕂. We will need the notion of the quasi-inverse in a Jordan pair: in [Lo95, Theorem 1.4] it is shown that, for (x, y) ∈ V^{+} × V^{−}, the condition (x, y) ∈ M is equivalent to the quasi-invertibility of (x, y), and then

exp_{−}(y) exp_{+}(x).o_{+} = exp_{+}(x^{y}).o_{+} = x^{y}

is the usual quasi-inverse in a Jordan pair. This means that the operator τ̃_{y} defined by Equation (2.9) is, in terms of the Jordan pair (V^{+}, V^{−}), given by the formula

τ̃_{y}(x) = x^{y}.

Since (τ̃_{y}, τ_{y}) is an automorphism, we see that

τ̃_{y} : V_{o′} → V_{y}  (10.1)

is a 𝕂-module isomorphism. Before proving the general case of (PG1), note that the identity (T) from 2.3 holds (see [Lo75, Theorem 3.7]). Now let us show that, for (x, a) ∈ M and r ∈ 𝕂,

r_{x,a} : V_{r_{a,x}b} → V_{b}

is 𝕂-linear for all b ∈ V′_{x}. We may assume that (x, a) = (o, o′). Now, it is easily verified that for all r ∈ 𝕂 the following relation holds:

(rx)^{y} = r(x^{ry})  (10.2)

(see [Lo94, Prop. 1.2]) which together with (10.1) implies that

r_{o,o′} : V_{rb} → V_{b}

is a 𝕂-module homomorphism. We have proved (PG1).

Next we verify (PG2). Let r ∈ 𝕂 and assume x, y ∈ V_{c}, c ∈ X′. We choose notation such that c = o′ and o = μ_{r}(y, o′, x) = M_{y,x}(o′), i.e.

(1 − r)y + rx = 0.  (10.3)

Then we have to show that

M_{x,y} : V′_{o} ⊃ U → V_{o′}, a ↦ M_{x,y}(a) = μ_{r}(x, a, y),

where U = {a ∈ V′_{o}| (x, a) ∈ M} = V′_{o} ∩V′_{x}, has an extension to an affine map V′_{o} → V_{o′}. Let us first prove the claim for the special case r = \frac{1}{2} ; in this case y = −x, M_{x,y}(o′) = o, and thus we have to prove that M_{x,y} is linear. In fact,

μ_{\frac{1}{2}} (x, a, −x) = 2_{x,a}^{−1}(−1)_{o,o′}(x) = 2_{x,a}^{−1}2_{x,o′}(o) = τ_{o′}^{a,x}(o).

Since (x, a) ∈ M, we can write (x, a) = (g.o, g′.o′) with g = τ_{x}τ̃_{_{τ_{−x}}(a)} (Equation (5.6)). Then by invariance under the projective group,

τ_{o′}^{a,x}(o) = τ_{o′}^{g′o′,go}(o) = gτ_{(g′)^{−1}o′}(g^{−1}o)

= τ_{x}τ̃_{_{τ_{−x}}(a)}τ̃_{−_{τ_{−x}}(a)}τ̃_{−_{τ_{−x}}(a)}(−x)

= x + τ̃_{−_{τ_{−x}}(a)}(−x)

= −Q(x)a

where for the last equality we have used the symmetry principle ([Lo75, Prop. 3.3])

τ̃_{y}(z) = z + Q(z)τ̃_{z}(y)

for a = τ̃_{z}(y), z = −x. This proves the equation

Mx,−x^{\frac{1}{2}}a = −Q(x)a;

the expression clearly is affine in a, proving the claim for r = \frac{1}{2} . Now we are going to prove (PG2) for general r ∈ 𝕂. We need the following expression for μ_{r}(x, a, y): for (a, x) ∈ M ∩ (V × V′),

r_{x,a} = τ_{x}τ̃_{_{τ_{−x}}(a)} r_{o,o′} (τ_{x}τ̃_{_{τ_{−x}}(a)})^{−1}.  (10.5)

In fact, this is proved in the same way as Corollary 5.8 (its proof uses only (PG1) which has already been established.) We evaluate at y and mind Equation (10.2):

r_{x,a}(y) = x + rτ̃_{(r−1)_{τ_{−x}}(a)}(y − x).  (10.6)

Next, using that τ̃_{u}(v) = B(v, u)^{−1}(v − Q(v)u),

r_{x,a}(y) = x + rB(y − x, (1 − r)τ̃_{−x}(a))^{−1} (y − x − Q(y − x)(r − 1)τ̃_{−x}(a)). (10.7)

We transform the B -operator appearing in this expression using first the following identity JP35 from [Lo75]: B(v, u)^{−1} = B(−v, τ̃_{v}(u)), and then taking account of (10.3):

B(y − x, (1 − r)τ̃_{−x}(a))^{−1} = B(y − x, τ̃_{y−x}((r − 1)τ̃_{−x}(a)))
= B(y − x, (r − 1)τ̃_{(r−1)(y−x)−x}(a))
= B(y − x, (r − 1)a)
= B((r − 1)(y − x), a) = B(−x, a).

Using this, (10.7) reads

r_{x,a}(y) = x + rB(−x, a)(y − x) − r(r − 1) B(−x, a)Q(y − x)τ̃_{−x}(a).  (10.8)

The second term of the right hand side of (10.8) equals

(id −T (−x, a) + Q(x)Q(a))(r(y − x)) = y + T (x, a)y + Q(x)Q(a)y,

where we use that r(y − x) = y. In order to calculate the third term we use that, by the identity JP23 of [Lo75],

B(−x, a)Q(y − x) = B(x − y, (r − 1)a)Q(y − x)
= Q(y − x)B((r − 1)a, x − y)
= Q(y − x)B(a, −x),

and we obtain for the third term in (10.8):

r(r − 1) B(−x, a)Q(y − x)τ̃_{−x}(a) =
r(r − 1)Q(y − x)B(a, −x)B(a, −x)^{−1}(a − Q(a)(−x)) =
r(r − 1)Q(y − x)a + (r − 1)Q(y − x)Q(a)rx =
r(r − 1)Q(y − x)a + (r − 1)^{2}Q(y − x)Q(a)y =
r(r − 1)Q(y − x)a + Q(x)Q(a)y

Thus (10.8) finally gives

r_{x,a}(y) = x + y + T (x, a)y − r(r − 1)Q(y − x)a.

This is clearly affine in a, as has to be shown.

Let us prove now that a Jordan pair homomorphism g^{±} : V^{±} → W^{±} extends to a homomorphism of the associated pointed generalized projective geometries. First of all, (g^{+}, g^{−}) defines on the affine parts belonging to the base points indeed a local homomorphism of the generalized projective geometry: this follows by using the explicit formula (10.5) for the multiplication maps together with the relation

g^{+}(x^{y}) = g^{+}(x)^{g^{−}(y)}  (10.9)

(see [Lo94, Eqn. I.1.(7)]). Since (X^{+}, X^{−}) is connected and thus algebraically generated by (V^{+}, V^{−}), there is at most one prolongation to a global homomorphism. We have to prove existence of such a prolongation. For simplicity, let us first assume that (V^{+}, V^{−}) is stable in the sense of 5.4. This means that

V^{−} × V^{+} → X^{+}, (a, y) ↦ τ̃_{a}τ_{y}.o

is surjective, and dually. The fibers of this map define an equivalence relation on V^{−} × V^{+} , called projective equivalence, and X^{+} = X(V) is called the projective space of V (see [Lo94]). From (10.9) it can be deduced that g^{+} × g^{−} passes to the quotient as a well-defined map X(g) : X(V) → X(W), see [Lo94, Section 1.3], which is then a homomorphism of generalized projective geometries. In the general non-stable case essentially the same argument applies: since (V, V′) is generating, there are surjective maps

V × V′ × V × … × V → X_{n}, (x, a, y, …, z) ↦ τ_{x}τ̃_{a}τ_{y} … τ_{x}(o),

(2n factors) such that ∪_{n}X_{n} = X, and dually. The fibers of these maps define equivalence relations which can be explicitly described in terms of the Jordan pair, see [F95, Theorem 1]. The explicit formula is fairly complicated, but one can conclude as above that ×^{n}(g^{+} ×g^{−}) passes to the quotient as a well-defined map X(g) : X(V) → X(W) which then is a homomorphism of generalized projective geometries.

Finally, if V is a Jordan pair over 𝕂 and φ : 𝕂 → R is a scalar exension, then V ⊗_{𝕂} R is a Jordan pair over R, and the associated space (X_{R}, X′_{R}) is a scalar extension of (X, X′) in the sense of 1.10: in fact, it is clear on the level of Jordan pairs that homomorphisms defined over 𝕂 extend to homomorphisms defined over R, and by the preceding arguments, this carries over to the level of spaces. By connectedness the extension thus obtained is unique.

Summing up, (X, X′) is a connected generalized projective geometry over 𝕂. Its assocated Jordan pair is nothing but the Jordan pair we started with; in fact, the operators B(x, y) and T (x, y) we introduced in Chapters 6 and 8 are precisely the operators associated to the given Jordan pair.

10.2. If one wants to announce Theorem 10.1 in the form of an equivalence of categories, then one has to introduce a notion of “simply connectedness” for generalized projective geometries. More precisely, if (X, X, o, o′) is a connected generalized projective geometry with base point, V = (V^{+}, V^{−}) the associated Jordan pair and X(V) the geometry associated to V , then the identity map of V^{+} × V^{−} is a local homomorphism X(V) → (X, X′). By the preceding argument, it extends to a homomorphism X(V) → (X, X′) which is surjective since (X, X′) is connected. Thus it is a covering in a sense extending the corresponding notion of [F89, Chapter 3]; we do not know wether it has to be always injective. (Rationality arguments show that this is so in the finite dimensional case over a field.)

10.3. An analogue of Theorem 10.1 in the category given by generalized projective geometries and adjoint pairs of morphisms is true: an adjoint pair preserving base points defines a so-called structural transformation of the associated Jordan pairs; conversely, an analogue of (10.9) holds for structural transformations (see [Lo95, Prop. 1.2. (e)]), which implies that structural transformations are adjoint pairs of morphisms of the associated geometry, defined on the affine part (V, V′). They do in general not extend to the whole of (X, X′), as shows already the example of ordinary projective geometry over 𝕂 where any non-zero linear map together with its transposed defines an adjoint pair on the quotient.

10.4. Since Jordan triple systems are the same as Jordan pairs with involutions, Theorem 10.1 implies that we can associate to every Jordan triple system a generalized polar geometry in a functorial way; taking the associated symmetric space, we get a functor from Jordan triple systems over 𝕂 into symmetric spaces over 𝕂.

11. Problems and further results

11.1. (Algebraic equations of hyperplanes and quadrics.) In ordinary projective geometry over 𝕂, the hyperplanes H_{a} = X \ V_{a} and the various quadrics are given by algebraic equations. The analogue of these equations in our setting is as follows: one proves that the Bergman operator extends to a biquadratic map, given by the formula

B(x, a) = id −T (x, a) + Q(x)Q(a)  (11.1)

(cf. [Lo75, I.2.11]). Then, with respect to a fixed base point (o, o′), the complements of hyperplanes and of quadrics are sets defined by a non-degeneracy condition on the Bergman operator; more precisely, one can show (cf. [Lo95, Theorem 1.4]):

(i) V_{a} ∩ V_{o′} = {x ∈ V | B(x, a) ∈ Gl(V)}, (ii) M ∩ (V × V′) = {(x, a) ∈ V × V′| B(x, a) ∈ Gl(V)}, (iii) for a given polarity, M^{(p)} ∩ V = {x ∈ V | B(x, p(x)) ∈ Gl(V)}.

These conditions can be formulated more intrinsically in terms of sections of certain vector bundles over (X, X′); see [Be01] for the real finite dimensional case. Note that in the finite dimensional case over a field the conditions from (i) – (iii) are polynomial since they can be written in terms of det B(x, a). More difficult is the task to find the equations of hyperplanes and quadrics passing through the origin (“parabolic realization”); it is closely related to determining the incidence structure of (X, X′), see below.

Here, in the finite dimensional case, the rank of the quadratic operators Q(a) plays an important rôle.

11.2. (“Jordan theoretic Campbell-Hausdorff formula”) One can show that the operators τ̃_{a} are given by the usual formula for the quasi-inverse in Jordan theory (see [Lo75, I.3]),

τ̃_{a}(x) = x^{a} = B(x, a)^{−1}(x − Q(x)a);  (11.2)

more generally, all automorphisms g can be written g(x) = d_{g}(x)^{−1}n_{g}(x) (x ∈ V ∩ g^{−1}(V)) with a quadratic denominator d_{g} and a quadratic numerator n_{g}(x) (see [Be01] for the finite dimensional real case). Together with Corollary 5.8 this gives an explicit formula for the maps μ_{r} in terms of the associated Jordan pair. In the finite dimensional case over a field the inverse in Gl(V) is rational and thus also our formulas are rational; thus also the associated symmetric spaces will be “algebraic over 𝕂” – see [Be01, Section X.3] for the real case. Since the explicit formula describes the multiplication maps in a canonical chart, we may consider it as a Jordan analogue of the Campbell-Hausdorff formula.

11.3. (Jordan algebras.) We have described the geometric objects associated to Jordan pairs and Jordan triple systems, but not yet the geometric object corresponding to (unital) Jordan algebras. It is known that unital Jordan algebras are the same as Jordan pairs containing invertible elements (see [Lo75, I.1.6]). This property can be translated to our context by requiring the existence of inner polarities, but it is also closely related to the existence of canonical null-systems which explains the somewhat special rôle of the Jordan inverse in a Jordan algebra (work in progress; see also Chapter 4 of Part I).

11.4. (Incidence structure.) There are two structures associated to a generalized projective geometry which are related to what one might call the associated incidence or remoteness structure: one the one hand, we have the distribution of the “hyperplanes” H_{a}, a ∈ X′ (see 11.1); on the other hand, there exist subspaces which appear linearly in every affinization (inner ideals, see 2.7 (4)). In ordinary projective geometry these two structures are almost the same; in general, the situation is much more complicated, and one would like to have a good Jordan theoretic description. In the finite-dimensional and non-degenerate case over a field these structures seem to be related to buildings in the sense of J. Tits; therefore a general theory of the correspondence between algebra and incidence structure is an important topic for further investigations. In a final step one has to study the action of a polarity on these structures and to describe the new structure on the associated symmetric spaces thus obtained – here one will get a vast generalization of the concept of a generalized conformal structure proposed by S. Gindikin, S. Kaneyuki and others (cf. [GiKa98]).

References

[Be00] Bertram, W., “Generalized projective geometries. Part I: From linear algebra via affine algebra to projective algebra”, preprint, Nancy 2000 (referred to as “Part I”).

[Be01] Bertram, W., The Geometry of Jordan- and Lie Structures, Springer Lecture Notes in Mathematics vol. 1754, to appear, Berlin 2001.

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