Generalized projective geometries. Part I:

From linear algebra via affine algebra to projective algebra

Wolfgang Bertram

Abstract. We introduce an algebraic formalism, called “affine algebra”, which corresponds to affine geometry over a field or ring 𝕂 in a similar way as linear algebra corresponds to affine geometry with respect to a fixed base point. In a second step, we describe projective geometry over 𝕂 by a similar formalism, called “projective algebra”. We observe that this formalism not only applies to ordinary projective geometry, but also to several other geometries such as, e.g., Grassmannian geometry, Lagrangian geometry and conformal geometry. These are examples of “generalized projective geometries”. The axiomatic definition and general theory of such geometries is given in Part II of this work.

A.M.S classification: 17 C 37, 51 A 05, 51 A 50, 51 N 25, 53 C 35

Key words: affine geometry, projective geometry, polar geometry, Grassmannian, Lagrangian, symmetric spaces

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. Synthetic versus analytic geometry. In this work we present an approach to affine and projective geometry over a field or ring 𝕂 which is in a sense situated in between of the approaches usually considered “analytic”, respectively “synthetic” or “axiomatic”. It is analytic in the sense that we accept the base field or ring 𝕂 to be given à priori, and it is axiomatic in the sense that we describe affine and projective spaces by intrinsic properties and do not define them, as usual in analytic geometry, by some construction using linear algebra over 𝕂. The intrinsic properties we are interested in are algebraic identities which in a sense encode the whole geometry. The basic idea is very simple: the algebraic laws of affine geometry are obtained by taking the algebraic identities of a vector space (or a 𝕂-module) and forgetting the base point, and similarly for projective geometry (see below). Of course, there are several ways to do this; the interesting problem is then to understand the algebraic nature of these identities and to find a most “natural” set of identities defining affine geometry. Pushing this idea one step further to projective geometry, we arrive naturally at two identities (PG1) and (PG2) which we call “fundamental identities of projective geometry”. In Part II of this work we will show that they are indeed fundamental in the sense that it is possible to use them as starting point of an axiomatic theory of “generalized projective geometries”. The remarkable feature of the identities (PG1) and (PG2) is that their algebraic structure is equivalent to the foundational identities of Jordan algebraic structures (see Part II). In fact, at the origin of our research in this domain was precisely the problem to find geometric structures corresponding to Jordan algebras, -triple systems and -pairs and in this way to develop a geometric approach to Jordan theory. This was partially achieved in [Be01] for the real finite-dimensional case; the approach presented here uses entirely different methods and is much more general. However, the motivation by the real finitedimensional case is still visible in the present work – cf. Chapters VI and XIII of [Be01]. Let us now describe in some more detail what we mean by the terms “affine algebra” and “projective algebra”.

0.2. Affine algebra. The motto of affine geometry is: “All points are equal.” If we take this serious, then linear algebra leads automatically to “affine algebra”. In order to explain this, let us look at the major dilatations of a vector space V over 𝕂: for r ∈ 𝕂 and x ∈ V let

r_{x} : V → V, y ↦ r_{x}(y) = x + r(y − x) = (1 − r)x + ry  (0.1)

be the dilatation with center x and ratio r. Since all points are equal, we consider all dilatations together for r fixed and x running through V , that is, we look at the “product maps”

π_{r} : V × V → V, (x, y) ↦ π_{r}(x, y) = r_{x}(y).  (0.2)

Of course, these products are in general neither associative nor commutative, but they satisfy some other and fairly “geometric” laws which we call (Af1) – (Af4) (Chapter 1).

If the characteristic of 𝕂 is different from 2, then the vector addition can be recoverd from these product maps: π_{\frac{1}{2}} (x, y) = \frac{x+y}{2} is just the midpoint of x and y and thus

x + y = 2π_{\frac{1}{2}} (x, y) = π_{2}(0, π_{\frac{1}{2}} (x, y)).  (0.3)

We have chosen our axioms (Af1) – (Af4) such that they imply in turn the vector space axioms if one defines the sum by (0.3), and this gives us an equivalence of categories between affine spaces and sets V equipped with product maps (π_{r})_{r∈𝕂} satisfying (Af1)

– (Af4) (Theorem 1.1). Slightly modifying these ideas, we may also treat the case of characteristic 2 and even the case of a base ring. In any case, all categorial notions are completely natural in this approach, and in our opinion the presentation of all standard topics of affine geometry over 𝕂 is now much more pleasant – we give a short outline in Chapter 1, with the emphasis on the universal space, which is, in the usual approach, according to M. Berger ([B94, p.67]), a “rather technical chapter” and thus very well suited to illustrate the naturality of our method. Another illustration will be given in Chapter 6 where we consider in a similar way affine spaces equipped with a quadratic form: the affine point of view leads to a natural construction generalizing the conformal compactification of Euclidean space (Theorem 6.2).

0.3. Projective algebra. In order to describe projective geometry by algebraic identities, we have to introduce one more variable, that is, we look at ternary maps instead of binary maps. The additional variable is taken from the dual projective space X′ = P(W^{∗}) of hyperplanes in the given projective space X = P(W), where for simplicity we assume here W ≅ 𝕂^{n+1} to be finite-dimensional over a field 𝕂 of characteristic different from 2 (in the main text we deal with the general case). Every element [λ] ∈ X′ defines a natural affine structure on V_{[λ]} = X \ [ker λ]; if π_{r}^{[λ]} is the corresponding multiplication map in the sense explained above, putting these together, we get a ternary map

μ_{r} : X × X′ × X ⊃ D → X, ([x], [λ], [z]) ↦ π_{r}^{[λ]}([x], [z]),  (0.4)

where the (Zariski-dense) subset D is defined by the conditions λ(x) ≠ 0, λ(z) ≠ 0. Dually, we get a ternary map μ′_{r} defined on a subset of X′ × X × X′. It is easy to derive explicit formulae for these maps (Section 2.4); however, our aim is not to work with these formulae but with their algebraic properties. By the very definition it is clear that, if the middle argument [λ] is fixed, the resulting partial map satisfies the identites (Af1) – (Af4). There are two more identities, which we call the “fundamental identities of projective geometry”, denoted by (PG1) and (PG2). Roughly speaking, they describe the behaviour of “Left, Right and Middle multiplications” with respect to the whole structure: the ternary maps μ, μ′ give rise, by fixing two arguments, to partial maps defined via

\begin{aligned}μ(x, a, z) & = L_{x,a}(z) = M_{x,z}(a) = R_{a,z}(x), \\ μ′(a, x, b) & = L_{a,x}(b) = M_{a,b}(x) = R_{x,b}(a); \end{aligned}  (0.5)

in our case, left and right multiplications L_{x,a}, R_{a,x} are operators on X (related to each other just by changing the scalar r to 1 − r), whereas the middle multiplication M_{x,y} maps X′ to X. Now (PG1) says essentially that L_{x,a} is an “adjoint operator” of L_{a,x} and (PG2) that M_{x,y} is an “adjoint operator” of M_{y,x}.

In Chapters 3, 5 and 7 we show that the identities (PG1) and (PG2) hold also in the case of Grassmannian geometry X = Gras_{p,p+q}(𝕂), X′ = Gras_{q,p+q}(𝕂), in the case of a Lagrangian geometry (X the space of maximal isotropic subspaces of a certain form, and X′ = X) and in the conformal geometry of projective quadric X with X′ = X; a natural approach to the latter case is given via a “universal space associated to a quadratic affine space” (Chapter 6).

0.4. Polar algebra and symmetric spaces. Maybe even more interesting than the projective geometries are the polar geometries, that is, the projective geometries (X, X′) together with an identification of the partners X and X′ given by a polarity p : X → X′. The interesting new feature (which was in fact one of the starting points of the author’s research in this domain, see [Be01]) is that to any generalized polar geometry in this sense we can associate a symmetric space; this construction is very general and defines finite or infinite-dimensional symmetric spaces over fields or rings with the only restriction that 1 + 1 is invertible in 𝕂. These spaces are algebraic over 𝕂 in the finite-dimensional case over a field. The general idea is explained in Chapter 4 of Part II; here we describe some typical examples such as the projective spaces itself: if 𝕂 = ℝ or 𝕂 = ℂ, then the projective spaces are well-known to be (compact) symmetric spaces, ℝP^{n} = O(n + 1)/(O(n) × O(1)), resp. ℂP^{n} = U(n + 1)/(U(n) × U(1)) (see e.g. [Hel78]). For other base fields such as 𝕂 = ℚ such a description is in general no longer possible. However, our formalism still permits to describe spaces such as ℚP^{n} as symmetric spaces in a sense close to Loos’ approach [Lo69]. They are in general no longer homogeneous, but in Part II it is shown that Jordan- and hence Lie-theoretic methods still apply; if one is interested in analysis on such spaces, this is certainly a major progress and a strong argument for using Jordan algebraic methods since Lie theory alone does not suffice here. The list of examples of symmetric spaces to which our methods apply is very long – in fact, in the simple finite-dimensional case over 𝕂 = ℝ a classification is possible (due to work of E. Neher); it reveals the surprising fact that all classical and about half of the exceptional symmetric spaces are given by our construction (see [Be01]). Therefore one may conjecture that also in the general case over a base field or ring, possibly infinite-dimensional, an important part of “symmetric spaces over 𝕂” is given by the methods introduced in this work.

Organization of the paper. The contents of the paper is as follows:

1. Affine algebra

2. Projective algebra: the classical case

3. Projective algebra: the Grassmannian case

4. Projective algebra + polarity = polar algebra

5. Projective algebra: the Lagrangian case

6. Affine metric algebra

7. Projective algebra: the conformal case

In Part II of this work we present the axiomatic theory of generalized projective and polar geometries and, in particular, their equivalence with Jordan pairs and Jordan triple systems. Both parts are fairly independent of each other. The first part is adressed not only to specialists but also to the general mathematician who finds it interesting to present basic mathematical structures in a new form, maybe even in his teaching – the author has used some of the material from Chapter 1 in an undergraduate course in linear algebra at the Technische Universität Clausthal and he would like to thank the students for their patience.

1. Affine algebra

Let V be a vector space over a commutative field 𝕂. We consider the major dilatations r_{x} (with center x ∈ V and ratio r ∈ 𝕂) defined by Equation (0.1) and the “product maps” π_{r} : V × V → V defined by Equation (0.2). Note that r_{0} is just usual multiplication by the scalar r in the vector space V . The maps π_{r}, r ∈ 𝕂, have the following properties: (Af1) The map r ↦ r_{x} is a homomorphism of the group 𝕂^{∗} into the group of bijections

of V fixing x, that is,

π_{1}(x, y) = y, π_{r}(x, π_{x}(x, y)) = π_{rs}(x, y), π_{r}(x, x) = x.

(Af2) For all r ∈ 𝕂, the map r_{x} is an endomorphism of π_{s}, s ∈ 𝕂:

π_{r}(x, π_{s}(y, z)) = π_{s}(π_{r}(x, y), π_{r}(x, z));
this can for r ≠ 0 also be written r_{x}s_{y}r_{x}^{−1} = s_{r_{x}y}.

(Af3) The “barycentric condition”: r_{x}y = (1 − r)_{y}x, that is, for r, s ∈ 𝕂^{∗},

π_{r}(x, y) = π_{1−r}(y, x).

(Af4) The group generated by the r_{x}r_{y}^{−1} (r ∈ 𝕂^{∗}, x, y ∈ V ) is abelian, that is

r_{x}r_{y}^{−1}s_{u}s_{v}^{−1} = s_{u}s_{v}^{−1}r_{x}r_{y}^{−1}
for all r, s ∈ 𝕂^{∗}.

In fact, (Af1) and (Af3) are immediate from the definition; for (Af2) note that gπ_{s}(y, z) = π_{s}(gy, gz) for every affine map g. Since 𝕂 is commutative, g := r_{x} (r ∈ 𝕂) is affine, and we get (Af2). Finally, in order to prove (Af4), we use the relation r_{x}s_{y}(z) = (1 − r)x + r(1 − s)y + rsz which shows that

r_{x}r_{y}^{−1} = τ_{(1−r)x+(r−1)y} = τ_{(1−r)(x−y)}

is a translation and hence commutes with all s_{x}s_{y}^{−1} , whence (Af4). Note that (Af2) replaces in a sense the law of associativity and (Af3) replaces the law of commutativity.

If 1 + 1 ≠ 0 in 𝕂, then the vector space addition can be recovered from the maps π_{r} since \frac{1}{2}(x + y) = 2_{x}^{−1}(y) and thus x + y = π_{2}(0, π_{\frac{1}{2}} (x, y)).

Theorem 1.1. The category of affine spaces over a field 𝕂 of characteristic different from 2 is equivalent to the category of sets V equipped with a family π_{r}, r ∈ 𝕂, of “product maps”

π_{r} : V × V → V, (x, y) ↦ π_{r}(x, y) =: r_{x}(y)

satisfying the properties (Af1) - (Af4). More precisely, if product maps with the properties (Af1) - (Af4) are given and o ∈ V is an arbitrary point, then

\begin{aligned}x + y & := x +_{o} y := π_{2}(o, π_{\frac{1}{2}} (x, y)) = 2_{o}2_{x}^{−1}(y), \\ rx & := π_{r}(o, x) = r_{o}(x) \end{aligned}

defines on V the structure of a vector space over 𝕂 with zero vector o, and this construction is inverse to the preceding one. Affine maps g : V → V′ in the usual sense are precisely the homomorphisms of product maps, that is, maps g : V → V′ such that gπ_{r}(x, y) = π′_{r}(gx, gy) for all x, y ∈ V , r ∈ 𝕂.

Proof. The verification of the vector space axioms for the structure defined in the theorem is straightforward – except for the second distributive law (r + s)x = rx + sx which we prove now: if s = 0, this is clear. If s ≠ 0, multiplying by s^{−1} and using the first distributive law we may reduce to s = 1. Now, using (Af1), (Af3) and (Af4) we get

rx + x = x + r_{o}x = 2_{o}2_{x}^{−1}r_{o}x
= (2_{o}2_{x}^{−1})(r_{o}r_{x}^{−1})x
= (r_{o}r_{x}^{−1})(2_{o}2_{x}^{−1})x = r_{o}r_{x}^{−1}2_{o}x
= r_{o}r_{x}^{−1}(−1)_{x}o = r_{o}(−r)_{x}^{−1}o
= r_{o}(1 + r^{−1})_{o}x = (r + 1)_{o}x = (r + 1)x.

The verification of the remaining statements is again straightforward.

As usual, the vector space (V, +_{o}, ·_{o}) is called the linearization or vectorialization of V with respect to the base point o. If we take another base point a, then it is easily verified that

x +_{a} y = x +_{o} y −_{o} a.  (1.1)

Let us give some short comments on generalizations of the preceding theorem:

(a) If 𝕂 is assumed to be a skew-field with 1 + 1 ≠ 0, then everything goes through if we require (Af1) to hold only if rs = sr in 𝕂.

(b) If 𝕂 is assumed to be a ring with unit 1 and such that 1 + 1 ∈ 𝕂^{∗}, then (V, +, ·) has all properties of a module over 𝕂 with the exception of the second distributive law – our proof of (r + s)x = rx + sx holds only for invertible r, s. We can add the second distributive law by brute force: incorporating all additive groups of vectorializations in a single map, we define

ρ : V × V × V → V, (x, a, y) ↦ 2_{a}2_{x}^{−1}(y) = x +_{a} y = x +_{o} y −_{o} a.

Now we require in addition to (Af1) – (Af4) the following property (Af5) to hold:

(Af5)  π_{r+s}(x, y) = ρ(π_{r}(x, y), x, π_{s}(x, y))

which just says that (r + s)y = ry + sy holds with respect to any point x.

(c) The case of characteristic 2: in this case the vector addition cannot be recovered from the major dilatations in a reasonable way. We have to add a new structure feature, namely a ternary map

ρ : V × V × V → V, (x, a, y) ↦ ρ(x, a, y)

such that for fixed a, ρ(·, a, ·) defines an abelian group with neutral element a; such that all r_{x} (r ∈ 𝕂, x ∈ V ) are endomorphisms of ρ in the obvious sense, and finally such that (Af5) holds. It is trivial that then V with x+y := ρ(x, o, y), rx := π_{r}(o, x) is a 𝕂-module.

For geometry, it is rather a gain than a loss that the axiomatics has to be changed according to the cases (a), (b), (c) since these disctinctions correspond to important differences in the geometry of the spaces. In particular, from a Jordan theoretic point of view it is rather satisfying that already at this elementary stage a ternary map shows up which reminds similar compositions for Jordan and associative algebras incorporating all “homotopes” or “mutations”.

The preceding discussion shows that one can develop the whole theory of affine spaces over 𝕂 starting with the axioms (Af1) - (Af4), possibly complemented by (Af5). As mentioned in the introduction, this approach has some advantages compared with the usual definition via a simply transitive vector space action. In the following we give a short outline of how one can develop the theory, with the aim to introduce the universal space which naturally leads to our next topic, projective geometry. Since the proofs are elementary, we leave them as an easy exercise to the reader or give some short comments. For simplicity, we assume in the following that 𝕂 is a commutative ring of characteristic different from 2.

1.2. Affine subspaces are subsets closed under all π_{r} ’s. These are precisely the usual affine subspaces.

1.3. We say that the ratio of (a, b, π_{r}(a, b)) is r. Then a map is affine if and only if it preserves collinearity and all ratios. 1.4. Let M be set and W an affine space over 𝕂. Then the set Fun(M, W) of maps from M to W becomes an affine space by defining

(π_{r}(f, g))(x) := π_{r}(f(x), g(x)),

i.e. by requiring that all evaluation maps are homomorphisms. If M also is an affine space, then the space Aff(M, W) given by all affine maps from M to W is an affine subspace of Fun(M, W). In particular, we get the spaces Aff(𝕂, V) and Aff(V, 𝕂) of “geodesics”, respectively of “affine functions”. 1.5. The space of complements is an example that will be important later on: let E ⊂ V be an affine subspace and put

C_{E,V} := {f ∈ Aff(V, V)| f^{2} = f, Im(f) = E}.

One verifies that C_{E,V} is closed under all π_{r} ’s, i.e. it is an affine subspace. If we fix a base point o ∈ E, then C_{E,V} can be interpreted as the space of complementary submodules of E by identifying f with ker(f). Affinely, C_{E,V} is interpreted as the set of equivalence classes of maximal affine subspaces that intersect E transversally.

1.6. The affine spaces Aff(V, V) and Fun(V, V) are equipped with a natural base point, namely the identity map id_{V} . Thus they carry natural 𝕂-module structures with zero vector id_{V} . (At a first glance this looks surprising: the zero vector is not the “zero map” in the usual sense!) Define

Tran(V) := {r_{x}r_{y}^{−1}| x, y ∈ V, r ∈ 𝕂^{∗}},
Dil(V) := Tran(V) ∪ {r_{x}| x ∈ V, r ∈ 𝕂}.

One shows that Tran(V) and Dil(V) are affine subspaces of Aff(V, V) containing the identity and hence are submodules (in fact, using a base point one sees that Tran(V) is the usual group of translations, and Dil(V) consists of all maps of the form x ↦ ax + b, a ∈ 𝕂, b ∈ V ). If 𝕂 is a field, then we have the additional description

Dil(V) = {r_{x} ∘ s_{y}| x, y ∈ V; r, s ∈ 𝕂};

this follows from the formula (with respect to a base point o ∈ V )

r_{x} ∘ s_{y}(z) = ((1 − r)x + r(1 − s)y) + (rs)z

which implies that for rs = 1, r_{x}s_{y} is a translation and otherwise it is a major dilatation (with center \frac{(1−r)x+r(1−s)y}{1−rs} and ratio rs). However, in the case of a general base ring, the composition of two dilatations is in general no longer a dilatation.

1.7. (The universal space.) For any r ∈ 𝕂, the map

u_{r} : V → Dil(V), x ↦ r_{x}

is affine. It is injective if 1 − r is invertible in 𝕂. This is the case for r = 0; the imbedding u_{0} is called the canonical imbedding of V into its universal space, and barycentric calculus is linear algebra in u_{0}(V). (If 𝕂 is a skew-field, everything goes through except that Dil(V) is no longer contained in Aff(V, V).)

1.8. The construction of the universal space is functorial: for any affine map α : V → W there are unique linear maps

T α : Tran(V) → Tran(W),  T̃α : Dil(V) → Dil(W),

such that

(T α(f))(α(x)) = α(f(x)), (T̃α(f))(α(x)) = α(f(x))

for every f ∈ Tran(V) (resp. f ∈ Dil(V)) and every x ∈ V . (One has to define

(T α)(r_{x}r_{y}^{−1}) := r_{α(x)}r_{α(y)}^{−1}, (T̃α)(r_{x}) := r_{α(x)};

this is well-defined since, after linearization of α, it amounts to sending x ↦ ax + b to y ↦ ay + α(b).) Let us add the remark that also for symmetric spaces one proceeds in a similar way: given a homomorphism α : M → M′ of symmetric spaces, one tries to define a homomorphism of the corresponding “transvection groups” by α̃(s_{x}s_{y}) := s_{α(x)}s_{α(y)} (where s_{z} is the geodesic symmetry w.r.t. z); however, this is well-defined only in very “regular” situations and gives in the general case rise to rather difficult problems.

1.9. Polynomial mappings of degree at most k between affine spaces V and W are defined as maps f : V → W such that, for all fixed y ∈ V , the maps from V to W defined by

x ↦ f(π_{r}(x, y)) and x ↦ π_{r^{k}}(f(x), f(y))

are congruent modulo a polynomial of degree at most k − 1, the space of polynomials of degree zero being defined as the space of constant maps. This coincides with the usual definition (under some restrictions on the characteristic of 𝕂).

1.10. The tensor product of affine spaces can be described via a universal property with respect to biaffine maps.

2. Projective algebra: the classical case

2.1. From now on we assume that 𝕂 is a commutative ring with unit 1 and 1 + 1 ∈ 𝕂^{∗} (for a first reading, one should think of 𝕂 as a field) and W some fixed 𝕂-module. A line in W is the image of an injective homomorphism v : 𝕂 → W (i.e. if x := φ(1) ∈ W, then 𝕂x ≅ 𝕂 as a submodule), and the projective space over W is the space of lines,

X := P(W) := {Im(v)| v : 𝕂 → W injective }.

Note that v : 𝕂 → W and w : 𝕂 → W have the same image iff v(s) = w(rs) for some r ∈ 𝕂^{∗}; therefore P(W) is the set of equivalence classes of injections 𝕂 → W under the relation

v ∼ w ⇔ ∃r ∈ GL(1, 𝕂) : v = w ∘ r.

A hyperplane in W is the kernel of a surjective homomorphism λ : W → 𝕂, and the dual projective space over W is the space of hyperplanes:

X′ := P(W)′ := {ker(λ)| λ : W → 𝕂 surjective }.

Note that α : W → 𝕂 and β : W → 𝕂 have the same kernel iff α(x) = rβ(x) for some r ∈ 𝕂^{∗}; therefore P(W)′ is the set of equivalence classes of surjections W → 𝕂 under the relation

α ∼ β ⇔ ∃r ∈ GL(1, 𝕂) : α = r ∘ β.

It is clear that, for a general 𝕂-module W, one or both of the spaces X and X′ may be empty. In the sequel we will assume that both are non-empty; in the case of a base field this is always the case.

2.2. We say that [v] ∈ P(W) and [λ] ∈ P(W)′ are incident if λ ∘ v = 0, i.e. if λ(v(1)) = 0, and we say that [v] ∈ P(W) and [λ] ∈ P(W)′ are remote if λ ∘ v : 𝕂 → 𝕂 is bijective, i.e. if λ(v(1)) ∈ 𝕂^{∗}. If 𝕂 is a field, then remoteness is the same as nonincidence; in general it is not. Let

\begin{aligned}Q & := \{([v], [λ]) ∈ X × X′| λ ∘ v = 0\}, \\ M & := \{([v], [λ]) ∈ X × X′| λ(v(1)) ∈ 𝕂^{∗}\} \end{aligned}  (2.1)

be the sets of incident, respectively remote pairs. Note that the sum Im(v) + ker(λ) is direct if and only if the pair ([v], [λ]) is remote; therefore M can be interpreted as the space of direct sum decompositions W = L ⊕ H into a line L and a hyperplane H. For any fixed element [v] ∈ X, resp. [λ] ∈ X′, we denote by

\begin{aligned}V_{[λ]} & := \{[v] ∈ X| ([v], [λ]) ∈ M\}, \\ V′_{[v]} & := \{[λ] ∈ X′| ([v], [λ]) ∈ M\} \end{aligned}  (2.2)

the spaces of complements of [v], respectively of [λ]. For a general 𝕂-module, one or both of the sets Q and M may be empty. We will assume that M is not empty, and moreover that X is covered by the sets V_{[λ]} and X′ by the sets V′_{[v]} ; in other words, we assume that every line and every hyperplane admit a complement. Under these assumptions the map

P(W)′ → P(W^{∗}), [λ] ↦ [λ^{∗} : 𝕂^{∗} = 𝕂 → W^{∗}] = [r ↦ λ ∘ r]

is a well-defined bijection. In particular, all this holds if 𝕂 is a field. We will say that (X, X′) with the preceding assumptions is a projective geometry over the ring 𝕂.

2.3. According to 1.5, the spaces V_{[λ]} and V′_{[v]} carry natural structures of affine spaces over 𝕂. Put in another way: for fixed ([v], [λ]) ∈ M, V_{[λ]} carries a natural 𝕂-module structure with zero vector [v], and dually. (Therefore we will sometimes denote elements of M by ([o], [o′]).) As in the preceding chapter, we will put all these structures (for a fixed scalar r ∈ 𝕂) together: for ([v], [λ]) ∈ M and [w] ∈ V_{[λ]} let

μ_{r}([v], [λ], [w]) := r_{[v],[λ]}([w]) := π_{r}([v], [w]) = rw

where the last two expressions are taken in the 𝕂-module V_{[λ]} with zero vector [v]. In other words, we consider the ternary map

μ_{r} : X × X′ × X ⊃ D → X, ([v], [λ], [w]) ↦ μ_{r}([v], [λ], [w]),

where

D = {([v], [λ], [w]) ∈ X × X′ × X| ([v], [λ]) ∈ M, ([w], [λ]) ∈ M},

and dually we define a map

μ′_{r} : X′ × X × X′ ⊃ D′ → X′, ([λ], [x], [ν]) ↦ μ′_{r}([λ], [x], [ν]) =: r_{[λ],[x]}([ν]).

2.4. It is easy to derive an explicit formula for μ_{r}, μ′_{r}:

\begin{aligned}μ_{r}([v], [λ], [w]) & = [(1 − r)v ∘ λ ∘ w + rw ∘ λ ∘ v], \\ μ′_{r}([λ], [x], [ν]) & = [(1 − r)λ ∘ x ∘ μ + rμ ∘ x ∘ λ]. \end{aligned}  (2.3)

In fact, the right-hand side of the first equation is well-defined (it is independent of the scaling of v, w and λ, and the term in brackets is again an injection 𝕂 → W). Now affinization with respect to ker(λ) just means to normalize v and w such that λ∘v = λ∘w = id_{𝕂}, and we get in the affininization with respect to [λ], μ_{r}([v], [λ], [w]) = [(1 − r)v + rw] which coincides with the map π_{r} from the preceding chapter. Similar arguments prove the second equation. – The first formula from (2.3) can be rewritten

r_{[v],[λ]} = P((1 − r)λ ⊗ v + rλ(v) id_{W})  (2.4)

where (λ ⊗ v)(x) := λ(x)v and P in front of a linear map means that we pass to the quotient.

For r = \frac{1}{2} we get a particularly pleasant formula: (2.3) can be written

μ_{\frac{1}{2}} ([x], [λ], [z]) = [λ(z)x + λ(x)z].  (2.5)

The right hand side is known to be a Jordan triple product on the space W. Its appearance at this point is rather surprising since in general Jordan triple products describe “infinitesimal” objects and not global ones as is the case here – in fact, already in the case of higher Grassmannians the analog of (2.5) is more complicated and shows a less direct relation with a Jordan triple product (cf. Equation (3.1)).

2.5. Next we want to find algebraic identities for the maps μ_{r} in the spirit of the preceding chapter. It is clear by the very definition of the μ_{r} that, for a fixed [λ] ∈ X′, the partial maps μ_{r}(·, [λ], ·) satisfy the algebraic identities (Af1) – (Af5) of the maps π_{r} from the preceding chapter. There are two more sets of remarkable identitities; we will call them the fundamental equations of projective geometry, (PG1) and (PG2).

2.6. Our construction of the maps μ_{r}, μ′_{r} in the given context is completely natural, that is, invariant under all relevant group actions. More precisely, this means that, if for g ∈ PGL(W) we denote by

g′ = (g^{t})^{−1} : X′ → X′, [λ] ↦ [λ ∘ g^{−1}]

the natural dual action, then the equivariance property

g(μ_{r}([x], [a], [y])) = μ_{r}(g[x], g′[a], g[y])  (2.6)

holds. Put another way, we have

g(μ_{r}([x], g^{t}[a], [y])) = μ_{r}(g[x], a, g[y])  (2.7)

(which can also be seen directly from (2.3)), and (2.7) holds even if g : W → U is a general linear map (different from the zero map), i.e. g passes to the quotient as a not everywhere defined map; it is a morphism of projective spaces in the sense of [B94, p. 96]. Geometrically, (2.7) means that the restriction

V_{g^{t}[a]} → V_{a}, [z] ↦ g.[z]  (2.8)

is affine. Now we are going to apply (2.7) to g = s_{[x],[λ]} (for s ∈ 𝕂^{∗} this is a bijection, else it is just a morphism). Using (2.4) we calculate its transposed:

(s_{[x],[λ]})^{t} = P((1 − s)λ ⊗ x + sλ(x) id_{W})^{∗} = P((1 − s)x ⊗ λ + sλ(x) id_{W^{∗}}) = s_{[λ],[x]},

i.e.

(s_{[x],[a]})^{t} = s_{[a],[x]} : X′ → X′.  (2.9)

Combining (2.7) and (2.9), and omitting brackets in the notation of elements of X and X′, we get the following identity

s_{x,a}(μ_{r}(y, s_{a,x}b, z)) = μ_{r}(s_{x,a}(y), b, s_{x,a}(z)),

which can also be written

μ_{s}(x, a, μ_{r}(y, μ_{s}(a, x, b), z)) = μ_{r}(μ_{s}(x, a, y), b, μ_{s}(x, a, z)).  (PG1)

A dual identity holds with the rôles of X and X′ reversed.

2.7. For any ternary map μ one gets, by fixing two elements, operators of right, middle and left multiplication, defined by R_{b,c} = μ(·, b, c), M_{a,b} = μ(a, ·, b), L_{a,b} = μ(a, b, ·). The identity (PG1) describes the functorial properties of left multiplications for μ_{r}: the pair (r_{x,a}, r_{a,x}^{−1}) is for r ∈ 𝕂^{∗} an automorphism of (X, X′). Right multiplications can be transformed into left multiplications because of the identity (Af3): μ_{r}(a, b, ·) = μ_{1−r}(·, b, a). The middle multiplications

M_{x,y} := M_{x,y}^{(r)} : X′ ⊃ (V′_{x} ∩ V′_{y}) → X, a ↦ μ_{r}(x, a, y)

cannot be reduced to right or left multiplications since they exchange the partners X and X′. An explicit formula for the middle multiplication is obtained from (2.3) by eliminating the variable λ:

M_{x,z} = P((1 − r)z^{∗} ⊗ x + rx^{∗} ⊗ z)  (2.10)

with transposed

(M_{x,z})^{t} = P((1 − r)x^{∗} ⊗ z + rz^{∗} ⊗ x) = M_{z,x}.  (2.11)

(The reader should convince himself that only an inclusion, but no identification of W with its double dual is used in this formula; here we omit the details since the calculation is given in more detail for the more general Grassmannian case, see 3.6.) Now we apply (2.7) to the pair g : W^{∗} → W, g^{t} : W^{∗} → W ⊂ (W^{∗})^{∗} inducing M_{x,z} : X′ → X, M_{z,x} : X′ → X; we obtain:

M_{x,z}^{(r)}μ′_{s}(a, M_{z,x}^{(r)}b, c) = μ_{s}(M_{x,z}^{(r)}a, b, M_{x,z}^{(r)}c)

which can also be written

μ_{r}(x, μ′_{s}(a, μ_{r}(z, b, x), c), z) = μ_{s}(μ_{r}(x, a, z), b, μ_{r}(x, c, z)).  (PG2)

As for (PG1), a dual identity of (PG2) holds.

2.8. In the case of characteristic 2 the maps μ_{r}, μ′_{r} are, similarly as in the preceding chapter, not sufficient for an algebraic description of projective geometries. We have to add a new structure feature by incorporating all the maps ρ (see point (c) after Theorem 1.1) belonging to various affinizations into a single map which now depends on four arguments: for a linearization ([v], [λ]) ∈ M and [w] ∈ V_{[λ]} let τ_{[w]} := τ_{[w]}^{[v],[λ]} be the translation by [w] in the 𝕂-module corresponding to the given linearization. This map is induced by an invertible operator on W and acts therefore on X and on X′ by bijections; we denote the dual action on X′ by

τ̃_{[w]} := (τ_{[w]})′ = ((τ_{[w]})^{t})^{−1} : X′ → X′.  (2.12)

Taking the transposed inverse of the identity τ_{[w]}τ_{[u]} = τ_{[w]+[u]} (sum in the 𝕂-module V_{[λ]}, [v]), we get

τ̃_{[w]}τ̃_{[u]} = τ̃_{[w]+[u]}  (T)

which just reflects the fact that we have indeed an action of translation groups on the dual space. As in 2.4 one can give explicit formulae for these actions:

\begin{aligned}τ_{[y]}^{[x],[a]}([z]) & = [a(y)a(z)x − a(x)a(z)y + a(y)a(x)z], \\ τ_{[y]}^{[x],[a]} & = P(a(y)a ⊗ x − a(x)a ⊗ y + a(y)a(x) id), \\ τ_{[y]}^{[x],[a]}([b]) & = P(a(y)x ⊗ a − a(x)y ⊗ a + a(y)a(x) id) · [b] \\ & = [a(y)b(x)a − a(x)b(y)a + a(y)a(x)b], \end{aligned}  (2.13)

where [z] ∈ X, [b] ∈ X′, and a(y) is a shorthand for a(y(1)), etc. The first formula is proved in the same way as (2.3) (if we normalize w.r.t. a, then only the term x − y + x remains which is in keeping with (1.1)); the second follows by eliminating the variable z and the third follows by transposing. – Summing up our discussion of projective geometry, we have

Theorem 2.9. The multiplication maps μ_{r}, μ′_{r} of a projective space (P(W), P(W)′) over 𝕂 satisfy the following identities: the partial maps obtained by fixing the middle element satisfy the identities (Af1) - (Af5), the left, resp. middle multiplications satisfy the relations (PG1), resp. (PG2), and the translation property (T) holds.

2.10. It is not possible to reconstruct a projective geometry from the properties of the preceding theorem – the reason is that they hold for a much bigger class of geometries. In fact, we will encounter in the next chapter geometries for which the middle multiplications M_{x,y} are (say, for r = \frac{1}{2} ) bijections. In the present case, however,

the middle multiplication M_{x,y}^{( \frac{1}{2} )} is the “midpoint map” associating to x, y ∈ X the midpoint as a function of the affinization; the image of the midpoint map always lies on the projective line determined by x and y if x ≠ y, and therefore M_{x,y} is not a bijection unless our projective space is of dimension 1.

In fact, one can show that the projective spaces are characterized among the more general geometries by the property that lines are independent of the affinization and thus have an invariant geometric meaning; but this is not a property which can be expressed by algebraic identitities.

2.11. (M as a symmetric space.) If 𝕂 is a field, then the action of the automorphism group PGL(W) is transitive on the space M of vectorializations – in fact, if W = L ⊕ H = L_{1} ⊕ H_{1} are two decompositions in lines and hyperplanes, then H and H_{1} are isomorphic as vector spaces, and by taking the direct sum of this isomorphism with an isomorphism of L onto L_{1} we get an element of GL(W) transforming one decomposition into the other. The stabilizer group of one decomposition is isomorphic to P(GL(1, 𝕂) × GL_{𝕂}(W/𝕂)), and thus

M ≅ P(GL_{𝕂}(W))/P(GL(1, 𝕂) × GL_{𝕂}(W/𝕂))

as a homogeneous space. If W is a finite dimensional vector space over 𝕂 = ℝ or ℂ, then M = GL(n + 1, 𝕂)/(GL(1, 𝕂) × GL(n, 𝕂)) is well-known to be a symmetric space in the differential geometric or group theoretic sense.

For a general base ring the situation is slightly more complicated since the complements H and H_{1} need no longer be isomorphic to each other (the isomorphism class depends on the complementary embedding of 𝕂 into W). Using the terminology of Part II, this means that the geometry (X, X′) is in general not connected. The preceding arguments remain valid if we restrict attention to subsets of X′ given by complements which are isomorphic among each other.

3. Projective algebra: the Grassmannian case

3.1. As in the preceding chapter, let W be a module over a commutative ring 𝕂 with unit 1 and 1 + 1 ∈ 𝕂^{∗}. We fix a submodule L ⊂ W different from 0 and W and define the Grassmannian variety of submodules of W of type L by

X := Gras_{L}(W) := {E ⊂ W | E submodule isomorphic to L}.

We also assume that L admits a complementary submodule H and let

X′ := Gras_{H}(W).

(We could also work with the bigger space of all submodules P such that W/P ≅ L, but this space is in general non-connected in the sense explained in 2.11.) We identify elements of X with injections v : L → W modulo the equivalence relation given by

v ∼ w :⇔ ∃g ∈ GL(L) : w = v ∘ g

and elements of X′ with surjections λ : W → L such that ker(λ) is isomorphic to H and modulo the equivalence relation

λ ∼ ψ :⇔ ∃g ∈ GL(L) : ψ = g ∘ λ.

3.2. We say that [v] ∈ Gras_{L}(W) and [λ] ∈ Gras_{H}(W) are remote if λ ∘ v : L → L is a bijection; we do not define incidence (which in this case would be more difficult to distinguish from non-remoteness). Let

M := {([v], [λ]) ∈ X × X′| λ ∘ v bijective}

be the space of remote pairs and define V_{[λ]} , V′_{[v]} by Equations (2.2). As in the projective case it is seen that [λ] and [v] are remote iff W is the direct sum W = Im(v) ⊕ ker(λ), and thus M can be interpreted as the space of direct sum decompositions of W of the type L ⊕ H. We assume that the V_{[λ]} cover X and the U_{[v]} cover X′, i.e., every submodule of type L has a complement of type H and vice versa; then we say that (X, X′) is a Grassmannian geometry over 𝕂. If 𝕂 is a field, the assumptions are always fullfilled, and in the finite dimensional case we use also the notation Gras_{p}(𝕂^{p+q}) := Gras_{𝕂^{p}}(𝕂^{p+q}).

3.3. The sets V_{[λ]} and V′_{[v]} carry a natural structure of affine spaces over 𝕂; this is seen as in the preceding chapter (note that the space C_{L,W} from 1.5 is automatically included in X′ since all complements of a fixed submodule L are isomorphic as 𝕂modules). Therefore the multiplication maps μ_{r} and μ′_{r} can be defined in the same way as in 2.3.

3.4. The map μ_{r} and its dual μ′_{r} are given by the explicit formulas

μ_{r}([x], [a], [z]) = [(1 − r)x(ax)^{−1} + rz(az)^{−1}],  (3.1)
μ′_{r}([a], [x], [b]) = [(1 − r)(ax)^{−1}a + r(bx)^{−1}b].  (3.2)

In fact, the right hand side of (3.1) is well-defined, as is seen by replacing a by ga (resp. x by xg or z by zg). Affinizing with respect to a means to normalize x and z such that ax = az = id_{L}. Thus in the affine picture with respect to a we have μ_{r}(x, a, z) = (1 − r)x + rz as desired. (Multiplying (1 − r)x + rz with a from the left shows that this operator is indeed again injective.) The other equation is proved in the same way.

3.5. The group G := PGL(W) acts on X and on X′ by forward transport of sets; in our operator realization it acts on X by g.x = g ∘ x and on X′ by g′.a = a ∘ g^{−1} . Formula (3.1) now implies the analogue of (2.7), that is

gμ_{r}(x, g^{t}a, y) = μ_{r}(gx, a, gy)  (3.3)

with g^{t}a = a ∘ g. Next we write (3.1) and (3.2) in the form

\begin{aligned}μ_{r}([x], [a], [z]) & = [(1 − r)x(ax)^{−1}az + rz], \\ μ′_{r}([a], [x], [b]) & = [(1 − r)bx(ax)^{−1}a + rb]. \end{aligned}  (3.4)

Thus for s ∈ 𝕂, the map s_{[x],[a]} is induced by the linear map g = s id_{W} +(1 − s)x(ax)^{−1}a, and we can calculate its transposed:

\begin{aligned}g^{t}[b] = [b ∘ g] & = [sb + (1 − s)(bx)(ax)^{−1}a] \\ & = [s(bx)^{−1}b + (1 − s)(ax)^{−1}a] = μ_{s}([a], [x], [b]) = s_{[a],[x]}([b]), \end{aligned}

proving that

(s_{[x],[a]})^{t} = s_{[a],[x]}  (3.5)

which together with (3.3) implies the identity (PG1).

3.6. Next we are going to establish (PG2). It is not so easy to write the middle multiplication in operator form since in general the inverses appearing in (3.1) cannot be removed (in the projective case they can since then ax and ay commute). Therefore the proof from the preceding chapter does not carry over. First of all, as noticed in the preceding section, (PG2) is equivalent to saying that

M_{x,y}^{(r)} : X′ ⊃ V′_{M_{y,x}^{(r)}(o′)} → V_{o′} ⊂ X

is 𝕂-affine, where o′ is some fixed but arbitrary affinization (cf. Equation (2.8)). We will verify this explicitly by rewriting the formula (3.1) affinely. In order to simplify formulas, we assume without loss of generality that M_{y,x}(o′) = o where o ∈ X, o′ ∈ X′ are the canonical base points corresponding to the decomposition W = L ⊕ H. We normalize x : L → W, a : W → L etc. with respect to the decomposition W = L ⊕ H such that

x = ( \begin{matrix}id & L \\ X \end{matrix} ) , y = ( \begin{matrix}id & L \\ Y \end{matrix} ) , a = ( id_{L} A) .

Then our assumption that M_{[y],[x]}^{(r)}(o′) = o means that rX + (1 − r)Y = 0. Thus

r ax + (1 − r) ay = r(id_{L} +AX) + (1 − r)(id_{L} +AY) = id_{L} .

Using this, we get (for the second equality, multiplying with (ax)(ay) from the right in the bracket)

μ_{r}([x], [a], [y]) = [(1 − r)x(ax)^{−1} + ry(ay)^{−1}]

= [(1 − r)x(ay) + y(ay)^{−1}r(ax)(ay)]

= [(1 − r)x(ay) + y(ay)^{−1}(id_{L} −(1 − r)(ay))(ay)] = [(1 − r)x(ay) + ry(ax)]

= [(1 − r) ( \begin{matrix}id_{L} +AY \\ X(id_{L} +AY) \end{matrix} ) + r ( \begin{matrix}id_{L} +AX \\ Y (id_{L} +AX) \end{matrix} )]

= [( \begin{matrix}id & L \\ (1 − r)X + rY + (1 − r)XAY + rY AX \end{matrix} )]

= [( \begin{matrix}id & L \\ X + Y − (rXAX + (1 − r)Y AY) \end{matrix} )].

This is clearly affine in A, as we wanted to show. The preceding calculation has shown more than anounced, namely that the term in question is furthermore essentially quadratic in X; for r = \frac{1}{2} we have the simple expression

μ_{\frac{1}{2}} ([x], [a], [y]) = [( \begin{matrix}id & L \\ XAX \end{matrix} )].  (3.6)

Moreover, this formula shows that lines have no longer a geometric signification independent of the affinization: take r = \frac{1}{2} ; then the midpoint map is essentially given by A ↦ XAX; for generic X, XA runs through a set corresponding to GL(L), and the image of the midpoint map can contain a set that is much bigger than a line.

3.7. For the case of characteristic 2, the same remarks as in 2.8 apply. It is clear that the analoque of (T) still holds; however, the analogues of Formulas (2.13) are more complicated since the inverse appearing at various places cannot be removed.

3.8. The preceding discussion shows that an analogue of Theorem 2.9 holds in Grassmannian geometry.

3.9. (Extension of base ring.) If R is an extension of 𝕂, then the data (W ⊗_{𝕂} R, L ⊗_{𝕂} R, H ⊗_{𝕂} R) define a Grassmannian geometry (X_{R}, X′_{R}) over R which comes together with a map of (X, X′) into this space. For instance, real Grassmannians are naturally imbedded into complex ones. Of course, it does not make sense to say that scalar restriction turns the complex Grassmannians into real ones; but the collection of multiplication maps μ_{r}, r ∈ ℝ associated to the complex Grassmannians still satisfies all algebraic properties we have in the real case, and therefore it will make perfectly sense to consider the complex Grassmannians as a new “generalized projective geometry” over 𝕂 = ℝ. This remark also applies to quaternionic Grassmannians: the multiplication maps are defined as above; taking scalars only from Z(ℍ) = ℝ they have the properties from Theorem 2.9.

3.10. As for projective spaces, M carries the structure of a “symmetric space over 𝕂” (if 2 ∈ 𝕂^{∗}). The arguments from 2.11 show that M is homogeneous under G; it can be written as the homogeneous space M = PGL(W)/P(GL(L) × GL(H)) which is a symmetric space if 𝕂 = ℝ or ℂ.

4. Projective algebra + Polarity = Polar algebra

4.1. Let (X, X′) be a Grassmannian geometry over 𝕂 as in the preceding chapter, given by X = Gras_{L}(W), X′ = Gras_{H}(W) together with the multiplication maps μ_{r}, μ′_{r}. It is clear that (X′, X) is again a Grassmannian geometry, called the dual geometry. A corresponding polar geometry arises by a suitable identification of the partners X and X′: an antiautomorphism of (X, X′) is a pair of bijections (p : X → X′, p′ : X′ → X) compatible with the multiplication maps in the sense that the identities

pμ_{r}(x, a, y) = μ′_{r}(px, p′a, py), p′μ′_{r}(a, x, b) = μ_{r}(p′a, px, p′b)  (4.1)

hold (the first equation means that the restriction p : V_{a} → V′_{p′a} is 𝕂-affine, and dually for second equation). A correlation is an antiautomorphism “of order 2” which means that p′ = p^{−1} . A point p ∈ X is called isotropic if (x, p(x)) ∉ M and non-isotropic else. A polarity is a correlation having some non-isotropic points, and a null-system is a correlation for which all points are isotropic. Thus by definition, the set

M^{(p)} = {x ∈ X| (x, p(x)) ∈ M}  (4.2)

associated to a polarity p, is non-empty. Its complement, the associated quadric of a polarity p

Q^{(p)} = {x ∈ X| (x, p(x)) ∉ M},  (4.3)

may very well be empty; then the polarity is called elliptic.

4.2. For the general theory of polar geometries see Chapters 3 and 4 of Part II – there it is shown that the complement M^{(p)} of the quadric carries a natural structure of a symmetric space over 𝕂 (Part II, Theorem 4.2), given by the multiplication map (in the sense of O. Loos [Lo69]) μ(x, y) := μ_{−1}(x, p(x), y). If we are in the finite dimensional case over 𝕂 = ℂ or ℝ, then the general theory of symmetric spaces ([Lo69]) implies that the topological connected components are homogeneous symmetric spaces, that is, they are of the form G/H, G a Lie group and H an open subgroup of a non-trivial involution of G. For general base fields or rings this is no longer true.

4.3. In this chapter we will describe the most important examples of polar geometries and symmetric spaces associated to Grassmannians. In general, however, a Grassmannian geometry won’t admit any polarity, and therefore we will have to impose various restrictions in order to assure the existence of polarities.

4.4. Assume that W = L ⊕ L is the direct sum of two copies of L. Then, if H = L is the second factor, X = Gras_{L}(W) and X′ are really the same sets, and id : X → X′ clearly is an antiautomorphism; it is a null-system and not a polarity since every element of X is incident with itself. Note that this null-system is canonical in the sense that it commutes with the whole automorphism group. We can easily define polarities: for instance, we may take

g_{1} = ( \begin{matrix}0 & id_{L} \\ id_{L} & 0 \end{matrix} )  or  g_{2} = ( \begin{matrix}0 & id_{L} \\ − id_{L} & 0 \end{matrix} )

(matrix with respect to the decomposition W = L ⊕ L); then the square of g_{i} in PGL(W) is the identity, and g_{i} : X → X′, g_{i}^{−1} : X′ → X defines a polarity (the first copy of L is mapped onto the second one). In order to describe the quadric and its complement associated to g_{1} , we look at the polarity given by p = ({1_{n} \atop 0} {0 \atop −1_{n}} ) which is conjugate to g_{1} by the element R := ({1 \atop −1} {1 \atop 1}) (the Cayley transform) of PGL(W). Then the quadric is given by all subspaces F ⊂ W isomorphic to L and having non-trivial intersection with first and second factor of W = L ⊕ L, and thus M^{(p)} is the space of subspaces F ⊂ W isomorphic to L and having trivial intersection with first and second factor. This means that F can be written

F = {(x, α(x))| x ∈ L} =: Γ_{α}

for a unique α ∈ GL_{𝕂}(L). The map α ↦ Γ_{α} is a bijection of GL_{𝕂}(L) onto M^{(p)} ; it identifies a linear operator with its graph, and thus M^{(p)} ≅ GL_{𝕂}(L) as sets. They are also isomorphic as symmetric spaces: the subgroup of GL_{𝕂}(W) fixed under conjugation by p is GL_{𝕂}(L)×GL_{𝕂}(L), and M^{(p)} is homogeneous under the action of this group and can be written as the homogeneous space (GL_{𝕂}(L) × GL_{𝕂}(L)/GL_{𝕂}(L), the stabilizer being diagonally imbedded. In the finite dimensional case over 𝕂 = ℝ or ℂ, this is precisely the Lie group GL(n, 𝕂), seen as a symmetric space. We observe that also for general base rings the “symmetric space over 𝕂” GL_{𝕂}(L) is always homogeneous.

The matrix g_{1} leads to another realization of GL_{𝕂}(L) in the Grassmannian, related to the one just described via the Cayley transform R. The matrix g_{2} leads to a different symmetric space. In fact, one can apply arguments known from the real finite dimensional case showing that the associated symmmetric space is c-dual to the preceding one (see [Be01, Prop. X.1.3]). For 𝕂 = ℝ, the c-dual of GL(n, ℝ) is GL(n, ℂ)/GL(n, ℝ); in general, the c-dual of GL_{𝕂}(L) is isomorphic to GL_{𝕂}(L) itself if 𝕂 contains a square root i of −1, and to

GL_{𝕂(i)}(L ⊗_{𝕂} 𝕂(i)))/GL_{𝕂}(L)

else.

4.5. The preceding example was somewhat special: in ordinary projective geometry, it corresponds to the space 𝕂P^{1} = P(𝕂⊕𝕂). In general, one will look for polarities defined by non-degenerate bilinear forms. Let us assume that W is finite dimensional over a field 𝕂, so W ≅ 𝕂^{n}, L ≅ 𝕂^{p} is the first factor, (X, X′) = (Gras_{p}(𝕂^{n}), Gras_{q}(𝕂^{n})), and let b : W × W → 𝕂 be a non-degenerate bilinear form. Then, if F ∈ X = Gras_{p}(𝕂^{n}) is the image of x : L → W,

F^{⊥} := {v ∈ W | b(F, v) = 0} = ker(x^{∗} : W → L^{∗})

is an element of X′ = Gras_{q}(𝕂^{n}), and

p : X → X′, [x] ↦ [x^{∗}], p′ : X′ → X, [a] ↦ [a_{∗}]

(right adjoint in the first, left adjoint in the second term) is an antiautomorphism; in fact, (4.1) holds since

\begin{aligned}p(μ_{r}([x], [a], [y])) & = [((1 − r)a(xa)^{−1} + ra(ya)^{−1})^{∗}] \\ & = [((1 − r)a_{∗}(x^{∗}a_{∗})^{−1} + ra_{∗}(y^{∗}a_{∗})^{−1}] = μ_{r}(p[x], p′[a], p[y]). \end{aligned}

It is a correlation if the form b is symmetric or skew-symmetric. In order to distinguish between polarities and null-systems, we fix the base point (o, o′) ∈ M, o being the first and o′ being the second factor in the decomposition 𝕂^{n} = 𝕂^{p} ⊕ 𝕂^{q}. Any polarity can be normalized by an element of GL(W) such that p(o) = o′, that is, b is given by a block matrix

B = ( \begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix} )  (4.4)

with B_{1}, B_{2} non-degenerate. The following cases can arise:

(1) b is a quadratic form, and B_{1} , B_{2} may be assumed to be symmetric. The space M^{(p)} is the set of p-dimensional subspaces F such that W = F ⊕ F^{⊥}, that is, b non-degenerate on on the first and hence also on the second factor. The orthogonal group O(b, 𝕂) acts on M^{(p)} , but this action is in general not transitive. It is so if 𝕂 = ℂ; in this case M^{(p)} is the symmetric space O(n, ℂ)/(O(p, ℂ) × O(q, ℂ)). If 𝕂 = ℝ, then for B_{1} = 1_{p}, B_{2} = 1_{q} the polarity is elliptic, and then

M^{(p)} = X = O(n)/(O(p) × O(q))  (4.5)

is the Grassmannian, considered as a (compact) symmetric space. The c-dual case arises if B_{1} = −1_{p}, B_{2} = 1_{q}; then the topological connected component containing the base point is

M_{o}^{(p)} ≅ O(p, q)/(O(p) × O(q));  (4.6)

this is a real bounded symmetric domain in the sense of [Lo77], realized inside its compact dual via a real version of the Borel-imbedding. For the general choice B_{1} = I_{r_{1},s_{1}} , B_{2} = I_{r_{2},s_{2}} (with I_{k,l} = ({1_{k} \atop 0} {0 \atop −1_{l}} )) we get a symmetric space of the type

O(r, s)/(O(r_{1}, s_{1}) × O(r_{2}, s_{2}))  (4.7)

with r_{1} + r_{2} = r, s_{1} + s_{2} = s. If 𝕂 = ℚ, then for B_{1} = 1_{p}, B_{2} = 1_{q}, we still have an elliptic polarity, whence M^{(p)} = X, but the action of O(n, ℚ) is far from being transitive on X. These remarks show that the orbit structure in the general case is fairly complicated.

(2) b is a symplectic form; that is, we can assume that p = 2r, q = 2s, B_{1} = J_{r}, B_{2} = J_{s} with J_{l} = ({0 \atop −1_{l}} {1_{l} \atop 0} ). Since any two subspaces of equal dimension on which the symplectic form is non-degenerate are conjugate under the symplectic group, we can always write

M^{(p)} ≅ Sp(p + q, 𝕂)/(Sp(p, 𝕂) × Sp(q, 𝕂)).  (4.8)

Note that if b is symplectic but p and q are odd, then p is a null-system and not a polarity. Finally, under certain assumptions it is possible to generalize some of these examples to infinite dimension – if 𝕂 = ℝ we can, e.g., take Hilbert Grassmannians and then get in a similar way as in (4.5) and (4.6) the infinite dimensional real bounded symmetric domains introduced by W. Kaup (see [Kau97]).

4.6. Instead of bilinear forms we may also take sesquilinear forms associated to some antiautomorphism of 𝕂; they define polarities which are, however, defined only over the subfield fixed under the antiautomorphism (this construction applies also to quaternionic Grassmannians, as remarked in 3.9). Thus we get, similar as in (4.5), an elliptic polarity of complex Grassmannians, turning them into the symmetric space

X = U(n)/(U(p) × U(q)),

and for B_{1} = −1_{p}, B_{2} = 1_{q} we get as the zero-component the bounded symmetric domains U(p, q)/(U(p) × U(q)) (including, for p = 1, the complex hyperbolic spaces) and, in general, symmetric spaces of a form similar to (4.7). Similar formulas can be derived in the quaternionic case.

5. Projective algebra: the Lagrangian case

5.1. Lagrangian geometries are derived from the Grassmannian geometry (Y, Y′) = (Gras_{L}(W), Gras_{L}(W)), W = L ⊕ L, considered in 4.4. We assume that W is finite dimensional over a field 𝕂, W ≅ 𝕂^{n} ⊕ 𝕂^{n}, and that β is a non-degenerate symmetric or skew-symmetric form on W. Let

X := {F ∈ Gras_{L}(L ⊕ L)| F = F^{⊥}}

be the space of Lagrangians of β and let X′ := X. We assume that X is non-empty. Then (X, X′) has the same formal properties as (Y, Y′): every element of X′ defines an affinization of X and vice versa, the multiplication maps are defined and still satisfy the properties from Theorem 2.9. (We just need the fact that the set of all Lagrangians complementary to one given Lagrangian is, in a natural way, an affine space over 𝕂; this is seen as in 1.5. The multiplication maps of (X, X′) are just the restrictions of those of (Y, Y′) and therefore verify the same identities.) The space M ⊂ (X × X′) is now the space of complementary Lagrangian subspaces of W; from Witt’s theorem it follows that M is a homogeneous space,

M ≅ O(β)/GL_{𝕂}(L)  (5.1)

with GL_{𝕂}(L) realized as \{({g \atop 0} {0 \atop g^{−1}} )| g ∈ GL_{𝕂}(L)\}.

5.2. If β is a symplectic form, (X, X′) is called a symplectic Lagrangian geometry, and if β is a (neutral) quadratic form, (X, X′) is called an orthogonal Lagrangian geometry.

5.3. Polarities and null-systems of (X, X′) are defined in the same way as in the preceding chapter; as mentioned there, any polarity gives rise to a symmetric space over 𝕂. Note that the identification n : X ≅ X′ is a canonical null-system (it commutes with the whole automorphism group Aut(X, X′) which is the orthogonal group of the form). Composing with n, antiautomorphisms are identified with automorphisms; correlations thus correspond to automorphisms g of order 2. Combining choices of β and g we obtain several interesting polar geometries. Let us mention here some of them:

(1) Assume g = ({1_{n} \atop 0} {0 \atop −1_{n}} ). Then the arguments from 4.4 show that F ∩ g(F) = 0 iff F = Γ_{α} for a unique α ∈ GL(L). Next we fix β. Let A be a non-degenerate quadratic or symplectic form on L (which we may identify with a symmetric or skew-symmetric matrix if we write L ≅ 𝕂^{n}).

(1.1) β = ({0 \atop −A} {A \atop 0}). Then the defining relation of the adjoint operator with respect to

A,

0 = A(αv, w) − A(v, α^{∗}w) = β((v, αv), (w, α^{∗}w)),  (5.2)

implies that Γ_{α} is Lagrangian for β iff α is self-adjoint for A. Therefore M^{(p)} is the space of graphs of non-degenerate A-self-adjoint operators. Again we distinguish two cases:

- A is symplectic. (Thus β is symmetric and of signature (2r, 2r).) Then M^{(p)} is identified with the space of symplectic forms on L. Any two symplectic forms are conjugate under the general linear group, and thus

M^{(p)} ≅ GL(2r, 𝕂)/ Sp(r, 𝕂)  (5.3)

as a homogeneous space and also as a “symmetric space over 𝕂”.

- A is symmetric. (Thus β is symplectic.) Then M^{(p)} is identified with the space of non-degenerate quadratic forms on L. The action of the general linear group is in general far from being transitive on M^{(p)} . This is so if 𝕂 is algebraically closed; then

M^{(p)} ≅ GL(n, 𝕂)/O(n, 𝕂)

as a “symmetric space over 𝕂”. If 𝕂 = ℝ, then the topological connected components are symmetric spaces in the ordinary sense, isomorphic to the homogeneous spaces GL(n, ℝ)/O(n − l, l) parametrized by the signature of the form. .

(1.2) β = ({A \atop 0} {0 \atop −A}). Then the defining relation of the adjoint operator with respect to A implies in a similar way as above that Γ_{α} is Lagrangian for β iff α^{∗} = α^{−1} for the form A on L. Therefore M^{(p)} is the orthogonal group O(A, 𝕂) of A. This is a “symmetric space over 𝕂 of groupe type” and therefore always homogeneous under the group O(A, 𝕂) × O(A, 𝕂) acting from the left and from the right. For A = J_{r}, we get for M^{(p)} the symplectic groups Sp(r, 𝕂), and for A a diagonal matrix with entries a_{1}, …, a_{p} we get the orthogonal groups O(a_{1}, …, a_{n}; 𝕂). In particular, the complex and real cases lead to the orthogonal Lie groups O(p, ℂ) and O(r, s), r + s = p.

(2) Elliptic polarities exist for instance if 𝕂 = ℝ or 𝕂 = ℚ, given by the orthogonal complement with respect to a scalar product. Then, for 𝕂 = ℝ, if β is symplectic, X ≅ U(n)/O(n), and if β is symmetric neutral, X ≅ O(n) as a symmetric space.

5.3. As in the Grassmannian case, polarities may also be defined via sesquilinear forms, and also starting with skew-fields (see 4.6). The number of symmetric spaces thus obtained increases considerably – see [Be01, Sections I.6 and XII] for a classification in the real case, including among others the group cases U(r, s) and Sp(r, s) and spaces of non-degenerate Hermitian forms.

6. Affine metric algebra

6.1. It is known that a vector space V with a non-degenerate quadratic form can be “conformally” imbedded into a quadric – the most important case is the one-point-completion of the Euclidean vector space V = ℝ^{n} given by the sphere S^{n}, where the imbedding is defined by stereographic projection. More generally, the “conformal compactification” of the space ℝ^{n} with a form of signature (p, q) is the projective quadric defined by a form of signature (p + 1, q + 1), which leads to the action of the “conformal group” SO(p + 1, q + 1) on ℝ^{n}. This “conformal compactification” can be found, e.g., in the literature related to the classification of Cartan-domains (where it corresponds to the type IV-domains), but its construction usually is not intrinsic: one uses external direct sums in terms of which a new quadratic form is defined. In order to give an intrinsic version of this construction, we return in this chapter to the context of affine spaces (Chapter 1). The objects we are interested in are now affine spaces with a distribution q_{y} (y ∈ V ) of quadratic forms such that q_{y}(x) = q_{o}(x − y). We assume throughout that 𝕂 is a field of characteristic different from 2.

6.2. We define a quadratic affine space to be an affine space V over 𝕂 together with a binary map q : V × V → V , such that: (q1) For all x ∈ V , q_{x} := q(x, ·) : V → 𝕂 is a quadratic form on the vector space V

with zero vector x, (q2) The “field of quadratic forms” q_{x}, x ∈ V , is translation invariant in the sense that

q ∘ (τ × τ) = q for all translations τ; in other terms, having fixed a base point o, we have

q(x + v, y + v) = q(x, y).

The space is called non-degenerate if the symmetric bilinear form associated to one of the q_{p} ’s is non-degenerate. An isometry (resp. similarity) between quadratic affine spaces is an affine map α such that q′(αx, αy) = q(x, y) (resp. q(αx, αy) = λ_{α}q(x, y) for a scalar λ_{α} depending only on α), x, y ∈ V .

6.3. If (V, q) is a quadratic affine space, then q is actually symmetric in both arguments and hence biquadratic. In fact, let g_{p} be the symmetric bilinear form associated to q_{p}, i.e.

g_{p}(x, y) = q_{p}(x + y) − q_{p}(x) − q_{p}(y).

Then from the translation invariance we get, with respect to a base point o,

2q_{p}(x) = g_{p}(x, x) = g_{o}(x − p, x − p) = g_{o}(x, x) − 2g_{o}(x, p) + g_{o}(p, p),
q(p, x) = q_{p}(x) = q_{o}(x) − g_{o}(x, p) + q_{o}(p)  (6.1)

which implies the claim.

6.4. The set

Q := {q_{p}| p ∈ V }

is contained in the space of all quadratic functions on V . Let ⟨Q⟩ be the subspace generated by Q.

Theorem 6.5. Let (V, q) be a non-degenerate quadratic affine space over 𝕂. Then there is a unique symmetric bilinear form ⟨·, ·⟩ on ⟨Q⟩ such that for all x, y ∈ V we have

⟨q_{x}, q_{y}⟩ = q(x, y).

If a base point o ∈ V is fixed, then ⟨Q⟩ has a direct sum decomposition ⟨Q⟩ ≅ V ⊕𝕂⊕𝕂 into linear, constant and homogeneous quadratic terms, and the bilinear form ⟨·, ·⟩ is with respect to this decomposition given by the matrix

( \begin{matrix}−g_{o} \\ & 0 & 1 \\ & 1 & 0 \end{matrix} ) .

Moreover, the quadratic map

V → ⟨Q⟩, y ↦ q_{y}

is an imbedding of V into the set of isotropic vectors of the form ⟨·, ·⟩; its image is naturally identified with a paraboloid in a hyperplane of ⟨Q⟩.

Proof. Unicity of the form is clear since Q generates ⟨Q⟩. We write Equation (6.1) in the form

q_{p} = q_{o} − p^{∗} + q_{o}(p)

(where p^{∗} = g_{o}(·, p)); by applying translations, we see that the homogeneous parts q_{o}, p^{∗} and the constants belong again to ⟨Q⟩. Thus, having fixed the base point o, ⟨Q⟩ = V ⊕𝕂 1⊕𝕂 q_{o}. (By non-degeneracy, p ↦ p_{∗} is injective and we identify its image with V .) Now we define our form simply by the matrix given in the theorem and verify that the property ⟨q_{x}, q_{y}⟩ = q(x, y) holds:

⟨q_{x}, q_{y}⟩ = ⟨q_{o} − x^{∗} + q(x), q_{o} − y^{∗} + q(y)⟩
= ⟨q_{o}, q(y)⟩ + ⟨q(x), q_{o}⟩ + ⟨x^{∗}, y^{∗}⟩ = q_{o}(y) + q_{o}(x) + ⟨x^{∗}, y^{∗}⟩
= q_{o}(x) − g_{o}(x, y) + q_{o}(y) = q(x, y)

since by definition the bracket of the linear parts with the homogeneous quadratic and the constant terms is zero and ⟨1, 1⟩ = 0, ⟨1, q_{o}⟩ = 1. This proves existence of the form.

Having defined the form, it follows that ⟨q_{x}, q_{x}⟩ = q(x, x) = q_{x}(x) = 0, i.e. all q_{x} are isotropic vectors. It remains to be shown that the image of the imbedding is a paraboloid: according to (6.1), the homogeneous quadratic part of q_{p} is independent of p, i.e., Q is contained in the affine hyperplane E of ⟨Q⟩ having constant last coordinate, equal to q_{o}. We claim that Q is a paraboloid in E. In fact, identifying E with V ⊕𝕂1, Q is equal to the set

{(−p^{∗}, q_{o}(p))| p ∈ V }.

This set is the graph of the quadratic map V → 𝕂, p ↦ −q_{o}(p). But the graph of a quadratic form V → 𝕂 is (almost by definition) a paraboloid in V ⊕ 𝕂.

Proposition 6.6. The preceding construction is functorial: any isometry f : (V, q) → (V′, q′) (resp. similarity f with λ_{f} ∈ 𝕂^{∗}) induces an imbedding Q → Q′ wich extends to an isometry (resp. similarity) f̃: ⟨Q⟩ → ⟨Q′⟩. In particular, the group of bijective similarities of (V, q) acts by automorphisms of the quadric in P(⟨Q⟩).

Proof. One verifies that the extension given by f̃(q_{x}) = q_{fx} is well-defined and has the desired properties.

6.7. The lower right form in the matrix from the theorem is diagonalized by the “real Cayley transform”, yielding 1 and −1 on the diagonal. For instance, if b(x, x) = x^{t}I_{p,q}x is the form on ℝ^{n} with signature (p, q), then b̂ has signature (p + 1, q + 1). We will call Q̂ the “conformal completion” of the affine space (V, g). Note that every projective quadric can affinely be realized as a paraboloid, and thus every projective quadric arises in the way just described.

7. Projective algebra: the conformal case

7.1. Assume W is a vector space over a field 𝕂 of characteristic different from 2, equipped with a non-degenerate symmetric bilinear form b : W × W → 𝕂. We consider the associated projective quadric

X = {[x] ∈ P(W)| b(x, x) = 0}

together with its dual projective quadric which is, by definition, the space

X′ = {[x^{∗}]| [x] ∈ X} ⊂ P(W)′

of tangent hyperplanes of X (as usual, we write x^{∗} = b(x, ·) and identify the class of a linear form with its kernel). We say that elements [x] ∈ X, [y^{∗}] ∈ X′ are remote if they are remote in (P(W), P(W)′), i.e. if b(x, y) ≠ 0, and define M ⊂ (X × X′) to be the space of remote pairs. Note that [x] ∈ X and [x^{∗}] ∈ X′ are never remote (they are “as incident as possible”), and thus the canonical bijection

n : X → X′, [x] ↦ [x^{∗}]

turns out to be a null-system (in the sense of 4.1) and not a polarity; it is canonical in the sense that it commutes with the natural automorphism group O(b). For instance, if X is a sphere in ℝ(P^{n+2}), then any tangent space intersects X in only one point, and therefore M is essentially the complement of the diagonal in X × X.

7.2. We will show that every element [y^{∗}] ∈ X′ defines an affinization of V_{[y^{∗}]} = {[x] ∈ X| b(x, y) ∈ 𝕂^{∗}}: denote by E_{[y^{∗}]} := {[x] ∈ P(W)| b(x, y) ∈ 𝕂^{∗}} the affinization of P(W) having [y^{∗}] as “hyperplane at infinity”. Since [y] lies at infinity, it corresponds to a family of parallel lines in E_{[y^{∗}]} whose projective completions intersect all in [y]. Let ⟨y⟩ be the corresponding group of translations in direction of these lines and denote by

F_{[y^{∗}]} := E_{[y^{∗}]}/⟨y⟩

the corresponding quotient space; it is, in a natural way, an affine space over 𝕂. But F_{[y^{∗}]} clearly is in bijection with V_{[y^{∗}]} for the projective completion of any line l in direction ⟨y⟩ intersects the quadric X transversally at one point (namely at [y]) and therefore has to have exactly one other intersection point l_{X} ∈ X which is remote from [y^{∗}] since l is by definition remote from [y^{∗}]. Thus the map l ↦ l_{X} is a bijection F_{[y^{∗}]} → V_{[y^{∗}]} which we use to define on V_{[y^{∗}]} the structure of an affine space. The same construction can be done dually, with the rôles of X and X′ exchanged. Note that, in case X is a sphere, our construction is really nothing but ordinary stereographic projection, where one usually identifies F_{[y^{∗}]} with the tangent space at some “antipode” of [y] which, in fact, is not canonical from a projective point of view.

7.3. We can recover the structure of a quadratic affine space (in the sense of 6.2) on V_{[y^{∗}]} : it is given, for [x], [z] ∈ V_{[y^{∗}]} , by the formula

q([x], [z]) = b(x, z).

7.4. The multiplication maps μ_{r}, μ′_{r} can now be defined in the same way as in the preceding chapters. We claim that μ_{r} is given by the explicit formula

μ_{r}([x], [y^{∗}], [z]) = [(1 − r)b(z, y)x + rb(x, y)z − r(1 − r)b(x, z)y].  (7.1)

If [x] and [z] belong to V_{[y^{∗}]} ⊂ X and [f] := [f(x, y, z)] denotes the right hand side of (7.1), then an easy calculation shows that b(f, f) = 0. Moreover, since b(x, y), b(z, y) ∈ 𝕂^{∗}, one deduces that f ≠ 0, i.e. [f] ∈ X. Now we pass to the affine picture by taking [y^{∗}] as hyperplane at infinity: this means to normalize x and z such that b(x, y) = b(z, y) = 1. Then the first two terms in f(x, y, z) reduce to (1 − r)x + rz and the last term just adds a translation into direction of [y], and by definition of the affine chart this shows that [f(x, y, z)] is indeed the image of (1 − r)x + rz under our chart.

7.5. Clearly the construction from 7.2 is invariant under the natural automorphism group G = O(b), i.e. we have

gμ_{r}([x], g^{t}[y^{∗}], [z]) = μ_{r}(g[x], [y^{∗}], g[z])  (7.2)

for all g ∈ O(b), which can of course also be directly verified from Formula (7.1). If we identify X and X′, then μ_{r} is invariant under O(b) in the usual sense.

7.6. Let us write left and middle multiplications corresponding to μ_{r} in operator form: from (7.1) we get

r_{[x],[y^{∗}]} = [(1 − r)y^{∗} ⊗ x + rb(x, y) id +r(r − 1)x^{∗} ⊗ y],  (7.3)
M_{[x],[z]} = [(1 − r)z^{∗} ⊗ y + rx^{∗} ⊗ y − r(1 − r)b(x, z) id_{W}].  (7.4)

Via b, the transposed of, say, y^{∗} ⊗ x is identified with x^{∗} ⊗ y, and thus (7.3) and (7.4) imply

(r_{[x],[y^{∗}]})^{t} = r_{[y],[x^{∗}]},  (M_{[x],[z]})^{t} = M_{[z],[x]},  (7.5)

which combined with (7.2) yields once again the identities (PG1) and (PG2) from Chapter 2.

7.7. (Polarities and symmetric spaces.) Correlations, null systems and polarities of (X, X′) are defined as in 4.1. As already mentioned, the canonical identification X ≅ X′ is then a null-system. There exist also polarities: take any element p ∈ O(b) with p^{2} = id and admitting a point [x] ∈ X such that b_{p}(x, x) := b(x, px) ∈ 𝕂^{∗}; then p : X → X ≅ X′ is a polarity. As in the preceding cases, the general theory (Part II, Theorem 4.1) implies that M^{(p)} = {[x] ∈ X| b(x, px) ∈ 𝕂^{∗}} carries a natural structure of a “symmetric space over 𝕂”. For example, in the real case the usual symmetric space structure SO(n + 1)/ SO(n) on the sphere M arises for b of signature (n, 1) and b_{p} positive definite (elliptic polarity). Another choice of p leads to the real hyperbolic spaces M = SO(n, 1)/ SO(n). In the complex case, an elliptic polarity is obtained if b_{p} is a positive definite scalar product; then M = X is the compact Hermitian symmetric space SO(n + 2)/(SO(n) × SO(2)). For general 𝕂, the corresponding symmetric spaces are no longer homogeneous.

7.8. (Remarks on the incidence structure.) Our discussion of Grassmannian, Lagrangian and conformal geometry has featured the algebraic aspects and not the incidence geometry of the spaces in question. In fact, we have seen that the algebraic description of these geometries can be done in a rather uniform way (via affinizations plus the identities (PG1) and (PG2)). The incidence structure of these spaces is fairly complicated and differs significantly from case to case (see, e.g., [Ch49], [D63], [Hua45], [F96] for important results on the incidence structure in special cases); it even depends highly on the base field and partially breaks down over rings (where incidence is not the same as non-remoteness). For these reasons it is remarkable that in fact the algebraic structure entirely determines the incidence structure (work in progress).

References

[B94]  Berger, M., Geometry I (second edition), Springer-Verlag, Berlin 1994.

[Be00] Bertram, W., “Generalized projective geometries. II: General theory and equivalence with Jordan structures”, preprint, Nancy 2000 (referred to as “Part II”).

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

[Ch49] Chow, W.L., “On the geometry of algebraic homogeneous spaces”, Ann. Math. 50, no.1, 32 – 75 , 1949.

[D63] Dieudonné, J., La Géométrie des Groupes Classiques, Springer, New York, 1963.

[F96] Faulkner, J.R., “Projective remoteness planes”, Geom. Ded. 60, 237 – 275 (1996).

[Hel78] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York 1978.

[Hua45] Hua, L.-K., “Geometries of matrices I. Generalization of von Staudt’s Theorem”, Trans. AMS 57, 441 – 481, 1945.

[Kau97]  Kaup, W., “On real Cartan factors”, man. math. 92 (1997), 191 – 221.
[Lo69]  Loos, O., Symmetric Spaces I, Benjamin, New York 1969.

[Lo77] Loos, O., Bounded Symmetric Domains and Jordan Pairs, Lecture Notes, Irvine 1977.