Wolfgang Bertram
email: bertram@iecn.u-nancy.fr

Homotopes and Conformal Deformations of Symmetric spaces

Homotopy is an important feature of associative and Jordan algebraic structures: such structures always come in families whose members need not be isomorphic among each other, but still share many important properties. One may regard homotopy as a special kind of deformation of a given algebraic structure. In this work, we investigate the global counterpart of this phenomenon on the geometric level of the associated symmetric spaces -on this level, homotopy gives rise to conformal deformations of symmetric spaces. These results are valid in arbitrary dimension and over general base fields and -rings.

0. Introduction 0.1. Conformal deformations. Deformations of Lie and other algebras have been much studied since the fundamental work of Gerstenhaber [G64], and related constructions on the level of Lie groups ("contractions") have attracted the interest of physicists even earlier ( [START_REF] Inönu | On the contraction of groups and their representations[END_REF]; see also [START_REF] Dooley | On contractions of semisimple Lie groups[END_REF]). Following Gerstenhaber, one usually considers a given Lie algebra structure µ on g as a point of the variety L of all Lie algebra structures on g, and one studies the structure of L on a neighborhood of µ. In order to cover the case of base fields different from R, one considers formal deformations µ t = ∞ m=0 t m µ m , and then relates conditions on the formal derivatives of such one-parameter families of deformations to Lie algebra cohomology.

In this work we consider a special kind of deformations which we call conformal in order to distinguish them from the general and more formal approaches mentioned above. These deformations have the advantage to lift always to the space level and thus to define global deformations (or "contractions") of geometric spaces, given by explicit formulas which ensure a good control on the simultaneous deformation of geometric properties of the spaces in question, regardless of dimension and nature of the base field. For this reason we hope that our approach may turn out to be useful for harmonic analysis (in the spirit of [START_REF] Dooley | On contractions of semisimple Lie groups[END_REF]; cf. [START_REF] Faraut | Berezin kernels and Analysis on Makarevic spaces[END_REF], [START_REF] Pevzner | Analyse conforme sur les algèbres de Jordan[END_REF] for topics to which our methods may apply) and maybe for physics (the contraction of de Sitter or anti-de Sitter models of general relativity to Minkowski-space of special relativity is a special case of conformal deformation in our sense, but many other cases, both finite and infinite dimensional and potentially interesting for physics, are also included in our setting).

The possibility of conformal deformations of a geometric space (symmetric space or Lie group) relies on the existence of an additional structure which we call generalized conformal structure. Algebraically, this means that we consider symmetric spaces or Lie groups which lie in the the image of the Jordan-Lie functor (see Section 1.6 for its definition): there is an "overlying" Jordan algebraic structure from which one can recover the Lie algebraic structure, and this overlying Jordan structure corresponds geometrically to a kind of conformal structure. Now, on the Jordan algebraic level there exists a notion of homotopy -Jordan algebraic structures J always come in natural families (J i ) i∈I , where the index set I typically is a vector space or a certain algebraic variety; the Jordan structures J i are then called homotopes of J , and if in turn J can be obtained as a homotope of J i , then J i is called an isotope of J . In our previous work [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF], the relation between geometric structures belonging to isotopic Jordan structures could be fully understood, but it was not clear how to understand "deformations" or "contractions" to more singular values, where J i is only homotopic, but no longer isotopic to J . In the present work we close this gap.

To our knowledge, there is no interesting notion of homotopy in a purely Lie-theoretic setting, and therefore it seems quite hard to recover the present results without using Jordantheory (if one wishes to avoid case-by-case calculations). On the other hand, once the geometric framework is clarified, some results given in this paper could be used to derive in a geometric way many of the identities and properties of Jordan pairs from [START_REF] Loos | [END_REF]. We illustrate this remark by giving independent proofs of some of our results, one proof based on the algebraic results from [START_REF] Loos | [END_REF], and another based on the geometric framework of generalized projective geometries ( [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF]; cf. Appendix B). 0.2. Example: associative algebras and general linear groups. For readers who are not experts in Jordan theory, let us give a very simple example that is well-suited to illustrate what we mean by "conformal deformation", and by "homotopy": if G is any group with unit element 1, then we can define new group laws on G by choosing an arbitrary element a ∈ G and defining x • a y := xay.

The unit element of (G, • a ) is then a -1 . Of course, all these group laws are isomorphic among each other: left translation l a : G → G is an isomorphism from (G, • a ) onto (G, •) (one may also use right translation), and hence this construction seems not to be very interesting. However, things would start to become more interesting, if one could take a "limit" for a converging to some element in the "boundary of G"; then (G, • a ) should converge to some deformed group which is no longer isomorphic to G (a "homotope" of G).

In order to make this idea more precise, we need some additional structure. For instance, assume that G = A × is the unit group of an associative unital algebra A over a commutative ring K . As explained above, G := A × is a group with product • a , for any a ∈ A × . We want to allow a to "converge" to some non-invertible element of A -the problem being that, although the homotope algebra (A, • a ) with product x • a y = xay is an associative algebra for all a ∈ A, it will not have a unit element as soon as a is non-invertible. In order to circumvent this problem, we translate, for invertible a, the unit element a -1 by the translation τ = τ -a -1 to the origin 0 ∈ A, that is, for invertible a we consider the group law x ⋄ a y = τ (τ -1 (x) • a τ -1 (y)) = (x + a -1 )a(y + a -1 ) -a -1 = xay + y + x (0.1) which is defined on the set

G a = {x ∈ A| a -1 + x ∈ A × } = {x ∈ A| 1 + ax ∈ A × }. (0.2)
Now it is easy to take the "limit as a becomes non-invertible": one checks that the last expressions of (0.1) and (0.2) define a group (G a , ⋄ a ) for any element a ∈ A. (The inverse of x ∈ G a is y = -x(1+ax) -1 , as is seen by direct calculation.) For a ∈ A × these groups are, by construction, all isomorphic to G, but for non-invertible a ∈ A they are not: they are "homotopes" , but not isotopes of G. In particular, for a = 0 , we get the translation group (A, +) as a deformation of A × . Also, it is clear that, if a, b ∈ A belong to the same A × -orbit (under left, or under right action), then G a and G b are isomorphic. Therefore, if we can classify the orbits of the action of A × × A × on A, then we can also classify the various groups G a obtained by the preceding construction. For instance, this is possible in the case where A = M (n, n; K) is the algebra of square matrices over a field K , and then a complete description of the groups G a is not difficult (see Section 5.1). Finally, one remarks that, on the level of Lie algebras, the corresponding situation is very simple: whenever the notion of a "Lie algebra of G a " makes sense, then this is simply A with the a-deformed Lie bracket UJA Unital (linear) Jordan algebras (J, •) (Section 1.7), with the a-homotopes • a , a ∈ J . JP (Linear) Jordan pairs (V + , V -) (Section 1.4), with the a-homotopes V + a , a ∈ V -. JTS (Linear) Jordan triple systems (V, T ), with the α -homotopes (V, T α ), where α is a structural transformation (Lemma 1.13).

In Chapter 2 we describe the Jordan theoretic analog of the associative construction given above (Prop. 2.2, related to the a-homotope • a , and Th. 2.3, related to V + a ). In Chapter 3 we give an independent and geometric version of these results (Th. 3.1 and Th. 3.3). Finally, in Chapter 4 we construct the geometric version of deformations corresponding to the α -homotope T α (Th. 4.3). In some sense, this is the most general (but also most abstract) result, since one may choose α = Q(a), the quadratic operator corresponding to a ∈ V -, in order to recover the a-homotope of T . In view of applications in harmonic analysis or physics (see above), we describe in Chapter 5 the behavior of geometric structures (for instance, affine connections or duals of metrics) under conformal deformations -for brevety, whenever calculations are essentially the same as in [START_REF] Bertram | The Geometry of Jordan and Lie Structures[END_REF] and [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF], we refer to these works for further details. In Chapter 6 we make our abstract results more explicit for some cases, namely for a-deformations belonging to idempotents or tripotents (in the semisimple case of finite dimension over a field this makes a classification possible), and for the case of Grassmannians. In the appendices, we collect some basic facts on reflection spaces and symmetric spaces (A), on generalized projective geometries (B) and on structural transformations (C); the simple Lemma C.3 on globalizations of such transformations is a main ingredient to be used in the proof of Theorem 4.3.

Acknowledgement. I thank Ottmar Loos for helpful comments.

Notation. Throughout, we work over a commutative base ring K in which 2 and 3 are invertible and hence use the concepts linear Jordan pairs, algebras and triple systems.

Homotopes of Lie and Jordan algebraic structures

1.1. Scalar homotopes and c-duality. If V is an algebra over K (Lie or other), with product m(x, y), then for any scalar λ ∈ K , we may define a new product m λ (x, y) := λm(x, y), called a scalar homotope of(V, m). Then ϕ := λ id V is a homomorphism from (V, m λ ) to (V, m). In particular, the algebras (V, m λ ) for invertible λ are all isomorphic among each other. Thus, if K is a field, this construction gives us nothing interesting. Now let (V, R) be a triple system (i.e., R : V 3 → V is a trilinear map). Then R λ (X, Y, Z) := λR(X, Y, Z) defines a new triple product on V , again called a scalar homotope. But now λ id V is a homomorphism from (V, R λ 2 ) to (V, R), and hence we can only conclude that all homotopes (V, R µ ), with µ a square of an invertible element in K , are isomorphic to (V, R). For instance, if K = R, we get a new triple system (V, R -1 ) which is called the c-dual of R . For a general field, we say that the (V, R λ ) with invertible λ are scalar isotopes of (V, R); they need not be isomorphic to (V, R).

The aim of this chapter is to recall some basic results showing that for Jordan algebraic structures there are interesting generalizations of the notion of scalar homotopy.

1.2. Graded Lie algebras. Let (Γ, +) be an abelian group. A Γ-graded Lie algebra is a Lie algebra g over K with a decomposition g = n∈Γ g n such that [g m , g n ] ⊂ g n+m for all n, m ∈ Γ. The Z/(2)-graded Lie algebras correspond bijectively to symmetric Lie algebras, i.e., to Lie algebras together with an automorphism of order 2 . A 3-graded Lie algebra is a Z-graded Lie algebra such that g n = 0 if n = -1, 0, 1 . Note that then g = g 0 ⊕ (g 1 ⊕ g -1 ) gives rise to a Z/(2)-grading, so g is then also a symmetric Lie algebra (of a rather special type).

1.3. Lie triple systems. A Lie triple system (LTS) is a K -module q together with a trilinear map q × q × q → q, (X,

Y, Z) → [X, Y, Z]
such that (where we use also the notation R(X, Y, Z) := R(X, Y )Z := -[X, Y, Z], alluding to the interpretation of this expression as a curvature tensor in differential geometry)

(LT1) R(X, X) = 0 (LT2) R(X, Y )Z + R(Y, Z)X + R(Z, Z)Y = 0 (the Jacobi identity) (LT3) For D := R(X, Y ) we have DR(U, V, W ) = R(DU, V, W ) + R(U, DV, W ) + R(U, V, DW ).
If g is a Z/(2)-graded Lie algebra, then g -1 with [X, Y, Z] := [[X, Y ], Z] becomes an LTS, and every LTS arises in this way since one may reconstruct a Lie algebra from an LTS via the standard imbedding (due to Lister, cf. [START_REF] Loos | Symmetric Spaces I, Benjamin[END_REF]).

1.4. Linear Jordan pairs. If g = g 1 ⊕ g 0 ⊕ g -1 is a 3 -graded Lie algebra, then we let V ± := g ±1 and define trilinear maps by

T ± : V ± × V ∓ × V ± → V ± , (x, y, z) → [[x, y], z].
The maps T ± satisfy the following identies:

(LJP1) T ± (x, y, z) = T ± (z, y, x) (LJP2) T ± (u, v, T ± (x, y, z)) = T ± (T ± (u, v, x), y, z) -T ± (x, T ∓ (v, u, y), z) + T ± (x, y, T ± (u, v, z))
In general, a pair of K -modules (V + , V -) with trilinear maps

T ± : V ± × V ∓ × V ± → V ± is called a linear Jordan pair if (LJP1
) and (LJP2) hold. Every linear Jordan pair arises by the construction just described.

1.5. Jordan triple systems. An involution of a 3 -graded Lie algebra is an automorphism θ of order 2 such that θ(g ± ) = g ∓ and θ(g 0 ) = g 0 . Then we let V := g + and define

T : V × V × V → V, (X, Y, Z) → [[X, θY ], Z].
Then T satisfies the identities (LJT1) and (LJT2) obtained from (LJP1) and (LJP2) by omitting the indices ± . A K -module together with a trilinear map satisfying these identities is called a (linear) Jordan triple system (JTS). Every linear Jordan triple system arises by the construction just described. One may also define the notion of an involution of a Jordan pair; then Jordan triple systems are the same as Jordan pairs with involution (see [START_REF] Loos | [END_REF]). For any JTS (V, T ), the Jordan pair (V + , V -) = (V, V ) with T ± = T is called the underlying Jordan pair.

1.6. The Jordan-Lie functor.

If T is a JTS on V , then [X, Y, Z] = T (X, Y, Z) -T (Y, X, Z)
defines a LTS on V . This defines a functor from the category of JTS to the category of LTS over K , which we call the Jordan-Lie functor (cf. [START_REF] Bertram | The Geometry of Jordan and Lie Structures[END_REF]). The LTS (V, R) belongs the θ -fixed subalgebra g θ of g whose involution is induced by the canonical involution of the 3 -graded Lie algebra g.

Jordan algebras.

A Jordan algebra is a K -module J with a bilinear and commutative product x • y such that the identity [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF]; the reader not familiar with Jordan theory may throughout think of the example J = A τ , the set of fixed points of an anti-automorphism τ of an associative algebra A, with bilinear product x • y = xy+yx 2 .) Jordan algebras always come in families: for any Jordan algebra J and element a ∈ J , the homotope J a is defined as follows. Writing L(x) for the left multiplication operator L(x)y = x • y , we can define a new product on J by

(J2) x • (x 2 • y) = x 2 • (x • y) holds (cf.
x • a y := (L(xa) + [L(x), L(a)])y = (x • a) • y + x • (a • y) -a • (x • y).
One proves that J a = (J, • a ) is a again a Jordan algebra (see [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF]), called the a-homotope of (V, •), and that T (x, a, y) := x • a y defines a Jordan triple product on J (see [START_REF] Bertram | The Geometry of Jordan and Lie Structures[END_REF]Ch. II] for a geometric proof of this fact). Thus (J, J) is a Jordan pair, but not all Jordan pairs are of this form.

1.8. Jordan pairs with invertible elements. For any Jordan pair (V + , V -) and every a ∈ V -, V + with the bilinear product

x • a y := 1 2 T + (x, a, y)
is a Jordan algebra, again called the a-homotope algebra. This algebra is unital if, and only if, the element a is invertible in (V + , V -), which means that the operator

Q(a) := Q -(a) := 1 2 T -(a, •, a) : V + → V - is invertible; then a -1 := (Q(a)) -1 (a)
is the unit element of J a . Every unital Jordan algebra arises in this way from a Jordan pair having invertible elements, after a choice of such an element.

1.9. Invertibility. In a Jordan algebra (J, •), the quadratic operator U x is defined by

U x y = (2L 2 x -L x 2 )y = 2x • (x • y) -(x • x) • y. An element x ∈ J is called invertible if U x is invertible; let J × = {x ∈ J| U x ∈ GL(J)},
be the set of invertible elements. For such x, we define the inverse x -1 = U -1 x x and call the map j : J × → J , x → x -1 the Jordan inverse. Then the following identities hold (cf. [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF]p. 200 and 201]):

(SB1) U Uxy = U x U y U x (the "fundamental formula") (SB2) U x -1 = (U x ) -1 (SB3) x -1 = U -1
x x 1.10. Quasi-invertibility. In general, a Jordan pair (V + , V -) does not have invertible elements. The important notion is the one of quasi-inverse: let

B(x, y) : = id V -T (x, y) + Q(x)Q(y) τ y (x) : = x y := B(x, y) -1 (x -Q(x)y)
the Bergman operator, resp. the quasi-inverse (the latter provided that (x, y) is quasi-invertible, which means that B(x, y) is invertible). If x is invertible and (x, y) quasi-invertible, then we have τ y (x) = (x -1 + y) -1 . 1.11. Structural transformations of Jordan triple systems and associated homotopes. For Jordan triple systems, there is a much richer supply of homotopes than for Lie triple systems: let (V, T ) be a Jordan triple system. A structural transformation of (V, T ) is an endomorphism α of V with the property ∀x, y, z ∈ V : T (αx, y, αz) = αT (x, αy, z).

(1.1) Lemma 1.12. If α a structural transformation, then the formula

T (α) (x, y, z) := T (x, αy, z) (1.2)
defines a Jordan triple product T α on V , called the α -homotope of T . The quadratic map and the Bergmann operator of T α are given by

Q α (x) = Q(x) • α, B α (x, y) = B(x, αy).
Proof. It is clear that T α satisfies (LJT1). For the proof of (LJT2) take the identity (LJT2) for T and replace the middle elements v and y by αv and αy and get T (u, αv, T (x, αy, z)) = T (T (u, αv, x), αy, z) -T ± (x, T (αv, u, αy), z) + T ± (x, αy, T ± (u, αv, z))

Then apply the defining relation of α to the second term on the right hand side, and one gets the identity (LJT2) for T α . The quadratic operator for this JTS clearly is

Q α (x) = Q(x)
• α , and it follows that

B α (x, y) = id -T α (x, y) + Q α (x)Q α (y) = id -T (x, αy) + Q(x) • αQ(y) • α = id -T (x, αy) + Q(x)Q(αy) = B(x, αy).
The set of structural transformations of (V, T ),

Svar(T ) := {α ∈ End(V )| ∀x, y, z ∈ V : T (αx, y, αz) = αT (x, αy, z)}
is called the structure variety of the Jordan triple system T . Note that, by the fundamental formula

Q(Q(x)z) = Q(x)Q(z)Q(x)
, all quadratic maps Q(x) belong to Svar(T ), and so do the Bergmann operators of the form B(x, x) for all x ∈ V (by Bergmann structurality Q(B(x, y)z) = B(x, y)Q(z)B(y, x), cf. [Lo75, JP26]). Also, Svar(T ) contains all involutive automorphisms of T , together with their negatives. More generally, Svar(T ) is a cone, i.e., invariant under taking scalar multiples.

Deformations of quadratic prehomogeneous symmetric spaces

2.1. The symmetric space of invertible elements in a Jordan algebra. Assume J is a Jordan algebra over a ring K with Jordan product x • y and unit element 1 . The set J × of invertible elements of J is stable under the maps j(x) = x -1 (the Jordan inverse) and U x for every x ∈ J × . Let Q(x) := U x . Then the identities (SB1) -(SB3) from Section 1.9 are exactly the identities (SB1) -(SB3) from Appendix A.1, and therefore Prop. A.2 implies that J × carries a natural structure of a reflection space given by

µ(x, y) = σ x (y) = U x (y -1 ) = U x U -1 y y.
The differential at 1 of the symmetry σ 1 = j can be calculated algebraically by using scalar extension by dual numbers

K[ε] = K⊕εK , ε 2 = 0 . It is easily checked that, in J ⊗ K K[ε] = J ⊕εJ , we have (1 + εv) -1 = 1 -εv ,
and it follows that the tangent map of j et the identity is -id J . If a ∈ J × is another base point, the maps Q and j are replaced by the quadratic map, resp. Jordan inverse in the homotope algebra (J, • a -1 ) (which has a as unit element), and hence we have also T a σ a = -id J . Thus also property (S4) defining a symmetric space is satisfied (in an algebraic sense), and hence J × is a symmetric space. We call such spaces quadratic pre-homogenoues symmetric spaces (cf. [Be00, Chapter II]). Now we describe what happens when we translate the space J × such that a base point a is translated into the origin 0 ∈ J . Recall from Section 1.10 the definition of the Bergmann-operator and the quasi-inverse.

Proposition 2.2. We equip J × with its symmetric space structure µ(x, y) = σ x (y) = U x y -1 , let a ∈ J × and τ = τ -a -1 : J → J be translation by -a -1 . Then the translated symmetric space τ (J × ) is given by τ

(J × ) = {x ∈ J| B(x, -a) ∈ GL(J)} with product map τ µ(τ -1 (x), τ -1 (y) = 2x + U x a + B(x, -a) τ a (-y) .
Proof.

First of all, x ∈ τ (J × ) if and only if a -1 + x is invertible, if and only if U a -1 +x is invertible; but since U a is assumed to be invertible, we have the identity

B(x, a) = U x-a -1 U a (2.1)
([Lo75, I.2.12]) and hence x belongs to τ (J × ) if and only if B(x, -a) is invertible, i.e., iff (x, -a) is a quasi-invertible pair. Now let us calculate the translated product map, or, which amounts to the same, via Prop. A.2, the new quadratic map and inversion with respect to a base point. With respect to the old base point a -1 ∈ J × , the quadratic map was given by Q(x)y = U x U a y , and the symmetry was the Jordan inverse j a in the Jordan algebra J a with unit element a -1 . We have to show that transformation by τ leads to the new quadratic map and new inverse

τ Q τ -1 (x) τ -1 (y) = 2x + U x a + B(x, -a)y, τ (σ a (τ -1 (x)) = τ a (-x).
(2.2)

The first equality in (2.2) follows from (2.1):

U x+a -1 U a (y + a -1 ) -a -1 = B(x, -a)(y + a -1 ) = B(x, -a)y + a -1 + T (x, a, a -1 ) + U x U a a -1 = B(x, -a)y + a -1 + 2x + U x a
For the proof of the second equality from (2.2), note that the inverse in J a is j a (x) = U -1 a x -1 , and hence

τ • j a • τ -1 (x) = -a -1 + U -1 a (x + a -1 ) -1 = -a -1 + U -1 a (a -x ) = (-x) a
by the symmetry principle for the quasi-inverse, see [START_REF] Loos | [END_REF]3.3].

Next we will show that the formulae of the preceding proposition define a symmetric space for any a ∈ J , invertible or not. Note that for a = 0 this is trivial: in this case we get the flat symmetric space structure on J . The proof becomes more transparent in the more general framework of Jordan pairs -in this context, the following equivalent version of the formulae from Prop. 2.2 holds:

Theorem 2.3. Assume (V + , V -) is a linear Jordan pair over K and fix an element a ∈ V -. Let U a := {x ∈ V + | B(x, -a) ∈ GL(V + )} be the set of elements x ∈ V + such that (x, -a) is quasi-invertible.
Then, for all x ∈ U a , the set U a is stable under the transformations

Q(x) := τ 2x+Q(x)a • B(x, -a), σ 0 := -τ -a = τ a • (-id),
and (U a , Q, σ 0 ) is a reflection space with base point 0 . Moreover, the algebraic differential of the symmetry σ x at x, defined by σ x (x + εv) = σ x (x) + εDσ x (x)v , equals -id V + , and thus U a is a symmetric space.

Proof.

We use the homotope J = V + a with product x • a y = 1 2 T (x, a, y) and adjoin a unit element J = K1 ⊕ J (if J happens to be unital, we forget the unit element of J ). For x, y ∈ J , we calculate in J ,

U x y = (2L 2 x -L x 2 )y = 2x • a (x • a y) -(x • a x) • a y = Q + (x)Q -(a)y U 1⊕x (y) = y + 2x • a y + U x y = (id +T (x, a) + Q(x)Q(a))y = B(x, -a)y, U 1⊕x (1 ⊕ y) = U 1⊕x (1) + U 1⊕x (y) = (1 + x) 2 + U 1⊕x y = 1 ⊕ 2x + x 2 + U 1⊕x y = 1 ⊕ 2x + Q(x)a + B(x, -a)y . (2.3)
The second equality shows that B(x, -a) is the restriction of U 1⊕x to J , and hence (x, -a) is quasi-invertible in (V + , V -) if and only if 1 + x is invertible in J , i.e.,

1 + U a = J × ∩ (1 + J).
This set is a symmetric subspace of V × . In fact, the third equation of (2.3) shows that it is stable under the quadratic map Q (with respect to the origin 1 ), and it is also stable under

inversion: if (x, -a) is quasi-invertible, then the inverse of 1 + x in J is given by (see [Lo75, I.3]) (1 + x) -1 = 1 -τ -a (x).
Transferring the symmetric space structure from 1 + U a to U a via the map 1 ⊕ x → x, we obtain the claimed formula. Algebraically, the differential Dσ 0 (0) of σ 0 at the origin may be defined by scalar extension with dual numbers K[ε] = K ⊕ εK , ε 2 = 0 :

σ 0 (0 + εv) = -B(εx, -a) -1 (εx + Q(εx)a) = -(id -εT (x, a))εx = -εx, whence Dσ 0 (0) = -id V + . Since σ x = Q(x) • σ 0 , it follows that Dσ x (x) = -id V + .
Remark. By the preceding calculations, the fundamental formula

U 1-x U 1-y U 1-x = U U1-x(1-y) in 1 -U a translates to the identity B(x, a)B(y, a)B(x, a) = B 2x -Q(x)a + B(x, a)y, a ,
which in turn can be interpreted by saying that, for every a ∈ V -, the map x → B(x, a) is a homomorphism of the reflection space U a into the automorphism group of (V + , V -), seen as a reflection space.

2.4. The topological case. Assume K is a topological ring with dense unit group. A topological Jordan pair is a Jordan pair together with topologies on V + and V -such that V ± are Hausdorff topological K -modules V ± and the trilinear maps T ± are continuous. Following [START_REF] Bertram | Projective completions of Jordan pairs. II[END_REF], we say that V ± satisfies the condition (C2) if the following holds: (C2) for all a ∈ V ∓ , the sets U a are open and the maps U a × V ± → V ± , (x, z) → B(x, -a) -1 z are continuous.

Theorem 2.5. Assume (V + , V -) is a topological Jordan pair satisfying the condition (C2).

(1) The symmetric space (U a , µ) from Theorem 2.3 is smooth, i.e., it is a symmetric space in the category of manifolds over K .

(2) With respect to the base point 0 , the Lie triple system of the symmetric space (U a , Q, σ 0 ) is given by antisymmetrizing the Jordan triple product S(u, v, w) := T + (u, Q(a)v, w) in the first two variables: [x, y, z] = S(x, y, z) -S(y, x, z).

Proof.

(1) Smoothness of µ follows from [BeNe05, Prop. 5.2], and the Property (S4) of a symmetric space follows from the calculation of Dσ 0 (0) given above (the differential in the sense of differential calculus is the same as the algebraic differential, cf. [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF]).

(2) Once again, the easiest proof is by reduction to unital Jordan algebras: for a unital Jordan algebra (J, e), the Lie triple system of the pointed symmetric space (J × , e) is given by

[x, y, z] = T (x, y, z) -T (y, x, z) = [L x , L y ]z = x • (y • z) -y • (x • z) (2.4)
(see [BeNe05, Theorem 3.4] and its proof which is algebraic in nature). For invertible a, the Lie triple system of (J × , a) is then given by the same formula, but replacing • by • a and T by T a (x, y, z) = T (x, Q(a)y, z) = S(x, y, z).

Starting now with a Jordan pair (V + , V -) and arbitrary a ∈ V -, we apply the preceding remarks to the unital Jordan algebra J = K1 ⊕ J with J = V + a , as in the proof of Theorem 2.3. Then the Lie triple system of the pointed symmetric space (1 + U a , 1) = J × ∩ (1 + J) is given by restricting the triple product (2.4) (with

• = • a ) to J = V +
a . The preceding result applies to all finite-dimensional and to all Banach Jordan pairs over R or C, but also to many other not so classical cases. Note that the (C2)-condition assures in particular that the sets U a are not reduced to {0} (if V + = {0} ). As to Part (2) of the theorem, it is a general fact that antisymmetrizing a Jordan triple system in the first two variables gives a Lie triple system (the "Jordan-Lie functor", see Section 1.6). The preceding construction works also in other contexts where some sort of differential calculus can be implemented (cf. Remark 10.1 in [START_REF] Bertram | Projective completions of Jordan pairs. II[END_REF]), and it can also be applied if we start with any non-unital Jordan algebra instead of J a ; then U a corresponds to the set of quasi-invertible elements in this algebra.

Deformations of generalized projective geometries

The geometric or "integrated" version of a Jordan pair (V + , V -) is a generalized projective geometry (X + , X -), see Appendix B. We give an analog of Theorem 2.3 in this context; it generalizes [Be03, Section 6], where only the case of Jordan algebras has been treated (cf. last part of the claim).

Theorem 3.1.

Let (X + , X -) be the generalized projective geometry associated to a Jordan pair

(V + , V -), fix a, b ∈ X -such that U a,b := v a ∩ v b is non-empty, and let p : U a,b → X -, x → Π 1 2 (a,
x, b) be the midpoint map assigning to x the midpoint of a and b in the affine space v x . Then the set U a,b becomes a symmetric space when equipped with the multiplication map

µ(x, y) = (-1) x,p(x) (y) = Π -1 x, Π 1 2 (a, x, b), y .
In case (V + , V -) = (J, J) comes from a unital Jordan algebra J , then p extends to a bijection denoted by p : X + → X -; in this case the symmetric space U a,b is the symmetric space associated to a polar geometry (see Section 4.1), and it is isomorphic to the quadratic prehomogeneous symmetric space J × .

Proof.

Let us show first that the set U a,b is indeed stable under the symmetries σ x = (-1) x,p(x) for x ∈ U a,b . To this end, note that, since (-1) = (-1) -1 , the identity (PG1) implies that, for x ∈ U c,b , the pair

(g + , g -) := (σ + x , σ - x ) = (-1) x,p(x) , (-1) p(x),x
acts as an automorphism on (X + , X -). By definition, The defining identies (S1) -(S4) of a symmetric space are now all easily checked: x is fixed by σ x since (-1) x,c (x) = x for all c⊤x; σ 2

σ x = σ + x is the first component of this automorphism. Then, using that Π r (u, c, v) = Π 1-r (v, c, u) and that r -1 v,c = (r -1 ) v,c , we see that σ - x (c) = Π -1 Π 1 2 (a, x, b), x, a) = Π 2 (a, x, Π 1 2 (a, x, b)) = b, and similarly σ - x (b) = a. Therefore σ + x (v b ) = v σ - x (b) = v a
x = id since (-1) 2

x,c = 1 x,c = id; the tangent map of σ x at x is -id TxM since in the chart v p(x) , centered at x, this map is just the negative of the identity; finally, in order to prove (S3) note first that, because of the fundamental identity (PG2), the pair (p, p) is self-adjoint, and thus for all scalars r ,

p • r x,p(x) (y) = p Π r (x, p(x), y) = Π r (p(x), x, p(y)) = r p(x),x • p(y)
In particular, for r = -1 , we have p

• (-1) x,p(x) = (-1) p(x),x • p, i.e., p • σ x = σ ′ x • p.
Using this, we obtain σ x µ(y, z) = σ x Π -1 (y, p(y), z) = Π -1 (σ x y, σ - x p(y), σ x z) = Π -1 (σ x y, p(σ x y), σ x z) = µ(σ x y, σ x z), which is precisely the property (S3).

Finally, assume that (V + , V -) comes from a unital Jordan algebra. In this case, there is a canonical identification of X + and X -, denoted by n : X + → X -(see [START_REF] Bertram | Projective completions of Jordan pairs. I[END_REF], [START_REF] Bertram | The geometry of null systems, Jordan algebras and von Staudt's Theorem[END_REF]), and then the property that v a ∩ v b be non-empty is equivalent to saying that the pair (a, n(b)) be quasi-invertible. If this is the case, the map p is, via n, identified with (-1) a,n(b) ([Be03, 2.1]), and hence extends canonically to a bijection X -→ X + (called an inner polarity). In [Be03, Section 6] it is shown that the symmetric space associated to an inner polarity is, in a suitable chart, the same as the quadratic prehomogeneous space J × .

The quasi-inverse revisited.

In order to show that the symmetric spaces of Theorem 3.1 and of Theorem 2.3 coincide, we need the global interpretation of the quasi-inverse via translation groups: in any module m over a ring K define for r ∈ K the proper dilations r x (y) = (1 -r)x + ry ; then if 2 is invertible in K , vector addition can be recovered from the dilations via x + y = 2 x+y 2 = 2 0 2 -1 x (y), i.e., translations are given by τ x = 2 0 2 -1 x . In a generalized projective geometry, and with respect to a fixed base point (o, o ′ ), the identity (PG1) then implies that the pair

(τ x , τ x ) = (2 o,o ′ 2 -1 x,o ′ , 2 -1 o ′ ,o 2 o ′ ,x )
is an automorphism. Dually, ( τ a , τ a ) for a ∈ V -is then defined. One can show that, in the affine chart (V + , V -) corresponding to the base point (o, o ′ ), τ a is indeed the quasi-inverse from Jordan theory. Similarly, we have

(τ -x , τ -x ) = (2 x,o ′ 2 -1 o,o ′ , 2 -1 o ′ ,x 2 o ′ ,o ).
Theorem 3.3.

Let notation be as in Theorem 3.1 and fix o ′ := b as base point in X -and choose an arbitrary base point o ∈ v a ∩ v b . We write (V + , V -) for the Jordan pair associated to the base point (o, o ′ ). Then U a,b = U a,o ′ is naturally identified with U a , and in the global chart V + of U a,b , the symmetric space structure from Theorem 3.1 is given by the explicit formula from Theorem 2.3.

Proof. We calculate in the Jordan pair (V

+ , V -) = (v o ′ , v ′ o )
. For a ∈ V -and x ∈ V + , the following are equivalent:

x⊤a ⇐⇒ x ∈ v a = τ a (v o ′ ) = τ a (V + ) ⇐⇒ τ -a (x) ∈ V + ⇐⇒ (x, -a) quasi-invertible. Hence v a ∩ v o ′ = τ a (V + ) ∩ V + = U a .
Now let us show that the symmetric space structure from Theorem 3.1 is given by the explicit formulae

σ o = -τ -a σ x σ o = Q(x) = τ 2x+Q(x)a • B(x, -a). (3.1)
The proof is in several steps. Let

d := 1 2 a, i.e., a = 2d = 2 o,o ′ d. Then σ x (y) = Π -1 (x, Π 1 2 (o ′ , x, 2d), y) = Π -1 (x, 2 -1 o ′ ,x 2 o ′ ,o d, y) = Π -1 (x, τ -x d, y) = τ -x Π -1 (2x, d, y + x) = τ -x • (-1) 2x,d • τ x (y), (3.2)
and for x = o this gives

σ o = (-1) o,d = (-1) τ d .o,τ d .o ′ = τ d (-1) o,o ′ τ -d = -τ -2d = -τ -a ,
proving the first relation from (3.1). Now we are going to prove the second relation from (3.1). Recall from [Be02, Cor. 5.8] or [Be00, X.3.2] that one can express the dilations r z,b for a scalar r ∈ K and a quasi-invertible pair (z, b) via translations, quasi-inverses and the dilation r id V + = r o,o ′ . For the scalar r = -1 , the formula reads:

(-1) z,b = τ z • τ 2 τ-z(b) • τ z • (-1) = (-1) • τ -z • τ -2 τ-z(b) • τ -z (3.3)
where (-1) = (-1) o,o ′ = -id. Using this, we get from (3.2) along with σ o = (-1) o,d ,

σ x σ o = τ -x • (-1) 2x,d • τ x • (-1) o,d = τ -x • (-1) • τ -2x • τ -2 τ-2x(d) • τ -2x • τ x • (-1) • τ -2d = τ x • τ 2 τ-2x(d) • τ x • τ -2d (3.4)
From the proof of Theorem 3.1, we know already that σ x σ o acts as an affine transformation on U a = V + ∩ v a . Thus, in order to prove the second relation from (3.1), it suffices to determine its linear part and its value at o. The value at o of this transformation is

σ x σ o (o) = x + τ 2 τ-2x(d) (x) = x + 1 2 τ τ-2x(d) (2x) = x + 1 2 (2x + Q(2x)d) = 2x + 2Q(x)d = 2x + Q(x)a
where we passed to the last line by using the covariance property of the quasi-inverse map [Lo75, Th. I.3.7] which implies, in the notation of [START_REF] Loos | [END_REF],

τ τ-zb (z) = z (b -z ) = B(-z, b) (z -z) b -(-z) b = -B(-z, b)B(-z, b) -1 (-z -Q(-z)b) = z + Q(z)b.
Let us determine the linear part of σ x σ o . For a general automorphism g such that g(o) ∈ V + , the linear part in the triple decomposition g ∈ τ

V + Aut(V + , V -) τ V -is the inverse of the denominator d g (o) (see [BeNe04,
Theorem 2.10 and its proof]). Now, the denominator

d g (o) of the transformation g = τ w • τ v is d g (o) = B(v, w) (see [BeNe04]
), and hence the linear part of

σ x σ o is (d σxσo (o)) -1 = (d τ 2 τ -2x (d) •τx (o)) -1 = B x, 2 τ -2x (d) -1 = B 2x, τ -2x (d) -1 = B(2x, -d) = B(x, -2d) = B(x, -a)
where we used JP35 in passing to the last line. Summing up, σ x σ 0 = τ 2x+Q(x)a • B(x, -a), and (3.1) is proved.

Note that combining Theorem 3.1 and Theorem 3.3 gives another (in fact, independent) proof of Theorem 2.3. Such a proof does not involve the technique (which is artificial from a geometric point of view) of adjoining a unit element to the Jordan algebras V + a .

Conformal deformations of polar geometries

4.1. The symmetric space of a generalized polar geometry. Recall from Appendix B.3 that Jordan triple systems (V, T ) correspond to generalized polar geometries (X + , X -; p), i.e., to generalized projective geometries with polarity p : X + → X -. We denote by

M (p) := {x ∈ X + | x⊤p(x)} (4.1)
the set of non-isotropic points of the polarity p. This set becomes a symmetric space when equipped with the product µ(x, y) := σ x (y) := (-1) x,p(x) (y) = Π -1 x, p(x), y

(see [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF]th. 4.1]; the proof is similar to the one of Theorem 3.1). Fixing a point o ∈ M (p) and choosing the transversal pair (o, o ′ ) = (o, p(o)) as base point in (X + , X -), we can describe the affine image of M (p) as follows: when we identify X + and X -and V + and V -via p, then (x, p(x)) is a transversal pair iff (x, -x) is a quasi-invertible pair (see Chapter 3), i.e., iff the Bergman operator B(x, -x) is invertible, and hence 1

M (p) ∩ V = {x ∈ V | B(x, -x) is invertible }. (4.3)
1 One should note that the sign in the correspondence between spaces and triple systems is a matter of convention. Geometers look at Hermitian symmetric spaces of non-compact type as having "negative" curvature, and on the other hand, one is used to associate them to "positive" Hermitian JTS, see [START_REF] Loos | Bounded symmetric domains and Jordan pairs[END_REF]. In order to check signs, one may use that for the JTS R with T (x, y, z) = 2xyz , the positive Bergmann polynomial B(x, -x) = (1 + x 2 ) 2 must belong to the compact space RP 1 and the "opposite" Bergmann polynomial B(x, x) = (1 -x 2 ) 2 to its non-compact dual.

4.2. The symmetric space associated to the α -homotope. Now consider the Jordan triple systems V α = (V, T α ), the α -homotopes of V with respect to structural transformations α : V → V (cf. Lemma 1.12). Since V α is a JTS in its own right, the preceding construction associates to V α a generalized polar geometry (X + α , X - α , p α ) with symmetric space M α . The explicit formula for the affine image (analog of the "bounded realization") is of the same type as (4.3), where the underlying K -module is the same, but T is replaced by T α and B by B α (u, v) = B(u, αv). However, a priori, the global spaces X + α for various α , are different spaces. In order to compare the geometries X α and the symmetric spaces M α with each other and to realize them as deformations of M (p) , we need a common realization, which is given by the following result:

Theorem 4.3.
Let (X + , X -; p) be the generalized polar geometry of a Jordan triple system (V, T ), and let α : V → V be a structural transformation, i.e., α ∈ Svar(T ). Let V α = (V, T α ) be the α -homotope of (V, T ) and let (X + α , X - α ; p α ) be the generalized polar geometry associated to (V, T α ), with symmetric space denoted by M α ⊂ X + α . Then there exists a canonical embedding Φ α : X + α → X + having the following properties: (1) The affine image of M α under this imbedding coincides with the description of M α in the chart V α , i.e.

Φ α (M α ) ∩ V = {x ∈ V | B(x, -αx) ∈ GL(V )}
(2) The tangent map of Φ α at the origin can be algebraically defined, and it is the identity map V α → V . (3) If α is invertible, then the image of this imbedding is precisely the symmetric space M (pα) associated to the polarity p • α : X + → X -.

Proof.

First we describe the construction of Φ α . Saying that α : V → V is a structural transformation of the JTS (V, T ) is equivalent to saying that the map of Jordan pairs (α, α) : (V, V ) → (V, V ) is a structural transformation of the Jordan pair (V, V ) (see Appendix C). We apply Lemma C.3, with (W + , W -) = (V + , V -) and (f, g) = (α, α); then the Jordan pair (V + , W -) from Lemma C.3 is (V α , V α ), with the JTS V α defined by Lemma 1.12. The second point of Lemma C.3 says that ϕ = (id, α)

: (V α , V α ) → (V, V )
is a homomorphism of Jordan pairs. As in Section C.4, the homomorphism (id, α) induces a homomorphism (Φ + , Φ -) :

(X + (V α ), X -(V α )) → (X + (V ), X -(V )) with injective first component Φ + : X + (V α ) → X + (V ).
We define Φ α := Φ + and show that this imbedding has the desired properties. Since (Φ + , Φ -) is a homomorphism of generalized projective geometries, its restriction to affine parts with respect to the base points is given by the pair of linear maps which induces it, namely by (id, α), and hence the algebraic tangent map can be defined and is equal to the pair (id, α) (cf. [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF]). Thus the tangent map of Φ + at the base point o is the identity map V α → V . This proves (2).

We prove (1). Let x ∈ V + = V . Recall that the element p(x) ∈ X -is identified with x ∈ V -= V . By definition, x belongs to the symmetric space M (p) if and only if x and p(x) are transversal, if and only if the pair (x, -p(x)) = (x, -x) is transversal (see proof of 3.1), if and only if B(x, -x) is invertible. The same argument, applied to the JTS T α , shows that x belongs to M α if and only if B α (x, -x) is invertible. But B α (x, y) = B(x, αy) (Lemma 1.12), and hence x belongs to M α if and only if B(x, -αx) is invertible.

We prove (3). Assume α is T -selfadjoint and invertible. Then (id, α) : (V α , V α ) → (V, V ) is a Jordan pair isomorphism, and the induced map (Φ + , Φ -) is an isomorphism of geometries. Under this isomorphism, the polarity of (X + (V α ), X -(V α )) defining the symmetric space M α (the exchange automorphism) corresponds to the map p • α which is therefore the polarity defining the symmetric space Φ α (M α ). This had to be shown.

, Γ α = {(v, αv)| v ∈ V } , is a sub-JTS of the polarized JTS (V ⊕ V, T ) with T (x, a), (y, b), (z, c) = (T (x, b, z), T (a, y, c)),
and that Γ α is isomorphic to V α as JTS. By the preceding construction, this situation globalizes as follows: the polarized JTS (V ⊕V, T ) corresponds to the polar geometry (X + ×X -, X -×X + ) with the exchange automorphism (see [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF]Prop. 3.6]) and with associated symmetric space

M = {(x, α) ∈ X + × X -| x⊤α} .
Then M α is the intersection of M with the graph of the map

Φ + (X + ) → X -Φ + (x) → Φ -(x).
Remark 4.5. (Alternative proof in the real finite-dimensional case.) In the finite-dimensional real or complex case, the main statements of Theorem 4.3 can be proved in a different way: if g = g 1 ⊕ g 0 ⊕ g -1 with involution θ is the involutive 3 -graded Lie algebra associated to the JTS T , then

q α := {v + θ(αv)| v ∈ g 1 }
is a sub-Lie triple system of g (a "defining plane" in the terminology of [START_REF] Rivillis | Homogeneous locally symmetric regions in homogeneous spaces associated with semi-simple Jordan algebras[END_REF]). Let h α = [q α , q α ] and define the subalgebra g α := h α ⊕ q α of g. Let G α and H α be the analytic subgroups of G = Aut(g) with Lie algebra g α , resp. h α . Then the orbit M α := G α .o of the base point o in X + is open, and it is a homogeneous symmetric space isomorphic to G α /H α coinciding (possibly up to topological connected components) with the space M α from Theorem 4.3. This construction can already been found in the work [START_REF] Rivillis | Homogeneous locally symmetric regions in homogeneous spaces associated with semi-simple Jordan algebras[END_REF] by A.A. Rivillis. Since it uses the exponential map as essential ingredient, it gives rather results of local nature, and it does not work for general base fields.

Theorem 4.6. Assume that K is a topological ring with dense unit group, and that (V, T ) is a topological Jordan triple symstem over K such that

(C1) The set (V × V ) × = {(x, y) ∈ V × V | B(x, y) ∈ GL(V )} of quasi-invertible pairs is open in V × V ,

and the map

(V × V ) × × V → V, (x, y, z) → B(x, y) -1 z is continuous.
Assume moreover that α is a continuous structural transformation map. Then M α is a smooth symmetric space over K , and it is an open submanifold of X = X + . The Lie triple system of the pointed symmetric space (M α , o) is V equipped with the Lie triple product

[X, Y, Z] α := T α (X, Y, Z) -T α (Y, X, Z) = T (X, αY, Z) -T (Y, αX, Z).
Proof. This follows directly from [BeNe05, Theorem 6.3] since the JTS T α again satisfies the property (C1).

In particular, the LTS [•, •, •] α and [•, •, •] -α are negatives of each other, which means that the symmetric spaces M α and M -α are c-duals of each other.

Behavior of geometric structures under deformation

For applications in geometry and harmonic analysis one not only needs to know how the affine image M α ∩ V of the symmetric space M α as a set, but also how its geometric structure depends on the deformation parameter α . We give some answers to this question. As a rule, the dependence on the parameter α can be made explicit if we simply mind the rules: in the following formulae, replace M by M α , T by T α , B(x, y) by B(x, αy) and Q(x) by Q(x) • α .

5.1. The product map. The product map of the symmetric space M is given by Formula (3.3), by letting x = z , b = p(x):

σ x (y) = (-1) x,p(x) (y) = τ x τ 2 τ-x(x) τ x (-1) o,o ′ (y) = τ 2 τx(x) B(2 τ x (x), -x) -1 τ 2 τx(x) (-y) (5.1)
In particular, the "square"

x 2 := σ x (o) in the symmetric space M is σ x (o) = 2 τ x (x) = 2(id V -Q(x)) -1 x (5.2)
(the last equality is proved as in [START_REF] Bertram | The Geometry of Jordan and Lie Structures[END_REF]p. 195]).

5.2. Second derivatives and second order tangent bundles. The tangent bundle T M of M , which is again a symmetric space, is simply obtained by scalar extension by dual numbers, as already used in Section 2.1 (cf. [BeNe05, Th. 5.5]). Hence the geometrically important second tangent bundle T T M is given by scalar extension by the second order tangent ring

T T K = K[ε 1 , ε 2 ].
The same statement holds for the geometry (X + , X -) itself. For instance, by a straightforward calculation using that ε 2 1 = 0 = ε 2 2 , we get from the definition of the quasi-inverse and from (5.1)

τ a (ε 1 v 1 + ε 2 v 2 ) = B(ε 1 v 1 + ε 2 v 2 , a) -1 (ε 1 v 1 + ε 2 v 2 -Q(ε 1 v 1 + ε 2 v 2 )a) = ε 1 v 1 + ε 2 v 2 + ε 1 ε 2 T (v 1 , a, v 2 ), σ x (x + ε 1 v 1 + ε 2 v 2 ) = x -ε 1 v 1 -ε 2 v 2 + ε 1 ε 2 T (v 1 , τ -x (x), v 2 ).
On the other hand, ordinary second differentials and second order tangent maps of twice continuously differentiable maps f are related via [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF] Eqn. (7.18)]), so that from the preceding calculation we get the second differential of τ a at the origin and the one of σ x at x.

f (x + ε 1 v 1 + ε 2 v 2 ) = f (x) + ε 1 df (x)v 1 + ε 2 df (x)v 2 + ε 1 ε 2 d 2 f (x)(v 1 , v 2 ) (see

The canonical affine connection.

It is possible to define the canonical affine connection of the symmetric space in a purely algebraic way. In the situation of Theorem 4.6 it coincides with any of the various differential geometric definitions of a connection (cf. [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF]Chapter 26]). More precisely, in [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF]Chapter 26] it is shown that the canonical connection ∇ of a symmetric space M can be defined in terms of the second order tangent bundle T T M , and we have the following chart description: let x ∈ M , V a chart around x inducing the flat connection ∇ 0 on some neighborhood of x; then the difference C := ∇ -∇ 0 (i.e., the Christoffel tensor of the canonical connection with respect to the chart) at the point x is simply the second derivative of the symmetry σ x at x: C x = d 2 σ x (x). Since we have calculated this second differential above, we get

C x (u, v) = T (v 1 , τ -x (x), v 2 ) = T (v 1 , (id V -Q(x)) -1 x, v 2 ) (5.3)
(for the last equality, see (5.2)). In the finite-dimensional real case this result is already given in [BeHi01, Theorem 2.3]; however, the proof given there does not carry over to the general situation considered here. As explained in [START_REF] Bertram | Characterization of the Kantor-Koecher-Tits algebra by a generalized Ahlfors operator[END_REF], (5.3) implies that the canonical connections of the symmetric spaces M α are conformally equivalent among each other. Note also that, for α = 0 , we get C x = 0 for all x, and hence ∇ = ∇ 0 , as expected.

5.4. Trace form, pseudo-metrics and their duals. It is clear that semi-simplicity is not preserved under deformations. Let us assume that K is a field, the JTS V is finite-dimensional over K , and that the trace form g 0 (u, v) := tr(T (u, v, •)) is non-degenerate. If this is the case, then g 0 extends to a pseudo-Riemannian metric tensor (non-degenerate symmetric two fold covariant tensor field) with affine picture given by

g x (u, v) := g o (u, B(x, -x) -1 v), (u, v ∈ V, x ∈ M ) (5.4)
(see [Be00, X.6] for the case K = R). On the other hand, the dual two fold contraviant tensor field with affine picture γ x (ϕ, ψ) = g o (ϕ, B(x, -x)ψ) can always be defined and exists also for the deformed spaces.

5.5. Invariant measures: group case. Let K = R and A a finite dimensional associative unital R-algebra. In general, the Lie group G = A × and its deformations G a will not be unimodular (take, e. g., the algebra of upper triangular square matrices). However, if G = A × happens to be unimodular (e.g, the case of the general linear groups), then also its deformations G a remain unimodular: the modular function is g → det(Ad(g)); if this function is trivial for all groups G a , a ∈ A × , then by density of A × in A it remains trivial for all a ∈ A.

5.6. Invariant measures on symmetric spaces. Let K = R and V a finite-dimensional JTS over R. Then in general there is no measure on M which is invariant under the transvection group G(M ). Such a measure exists if (V, T α ) has non-degenerate trace form: in this case, the corresponding invariant integral is given by the explicit formula

I α (f ) = Mα∩V f (x)| det B(x, -αx)| -1 2 dx, (5.5)
where dx is Lebesgue measure on V (see [Be00, X.6.2]). The term | det B(x, -αx)| -1 2 dx translates the existence of an invariant half-density on the symmetric space M α (cf. [START_REF] Bertram | Geometric Bergman and Hardy spaces[END_REF]). But one may define a half-density on M α ∩ V by the same formula for any α in the structure variety. Since the invariance condition is an equality of two functions depending continuously on α , these equality will still hold for all α in the closure of the subset S of the structure variety for which the trace form of T α is non-degenerate, and hence for α ∈ S there still exists an invariant measure on M α . (It is not clear wether this property holds for all α in the structure variety. There may be topological connected compenents of the structure variety containing only "degenerate" α 's, see end of Section 6.5.) In particular, for α → 0 , the invariant measure tends to the Lebesgue measure of V . 5.7. Further results and topics. Further results on geometric structures on the spaces M α in the real, non-degenerate case can be found in the paper [START_REF] Makarevic | Jordan algebras and orbits in symmetric R -spaces[END_REF] by B.O. Makarevic, and it would certainly be interesting to generalize some of these results. Finally, one would like to study the dependence of harmonic analysis and representation theory on the symmetric spaces M α on the deformation parameter α . Although the situation here is different from the one considered, e.g., in [START_REF] Dooley | On contractions of semisimple Lie groups[END_REF], it seems possible to transfer ideas developed in the context of contractions of semisimple Lie groups to the set-up just described. Since all relevant geometric data of M α depend algebraically or at least analytically on α , one may expect that something similar is true on the level of data of harmonic analysis. See, for instance [START_REF] Faraut | Berezin kernels and Analysis on Makarevic spaces[END_REF] and [START_REF] Pevzner | Analyse conforme sur les algèbres de Jordan[END_REF] for examples of harmonic analysis on symmetric spaces of the type M α . The parameter α is fixed in these works, so that the dependence on α is not visible. When varying α , not only the aspect of deformations of α to singular values is of interest, but also the one of analytic continuation between isotopic, but non isomorphic values of α . For instance, the duality between compactly causal and non-compactly causal symmetric spaces is reflected by the isotopy between α and -α (cf. [Be00, XI.3]).

Some examples

6.1. Quadratic matrices and associative algebras. Let us return to the setting of Section 0.2 and consider the associative algebra A = M (n, n; K) of square matrices over a field K . Then the A × × A × -orbits are represented by the idempotent matrices e := e r := 1 r 0 0 0 , r = 0, . . . , n.

The group

G e = {x ∈ A| 1 + ex ∈ A × } is explicitly given by G e = x = α β γ δ | det 1 r + α β 0 1 n-r = 0 = 1 r + α β γ δ | det(1 r + α) = 0
(where β, γ, δ are arbitrary), with product x ⋄ e y = xey + x + y , i.e.

α β γ δ ⋄ e α ′ β ′ γ ′ δ ′ = αα ′ αβ ′ γα ′ γβ ′ + α β γ δ + α ′ β ′ γ ′ δ ′ .
This group is isomorphic to a semidirect product of GL(r, K) and a Heisenberg type group: for β, γ, δ zero, we get a subgroup isomorphic to GL(r, K), and for α = 0 , we obtain a subgroup with product

0 β γ δ ⋄ e 0 β ′ γ ′ δ ′ = 0 0 0 γβ ′ + 0 β γ δ + 0 β ′ γ ′ β ′ .
It is easily checked that G e is the semidirect product of these two subgroups. The Lie algebra of G e is given by the bracket [x, y] e = xey -yex, which yields explicitly

α β γ δ , α ′ β ′ γ ′ δ ′ e = αα ′ -α ′ α αβ ′ -α ′ β γα ′ -γ ′ α γβ ′ -γ ′ β
so that for α = 0 = α ′ we get the 2-step nilpotent algebra

0 β γ δ , 0 β ′ γ ′ δ ′ e = 0 0 0 γβ ′ -γ ′ β .
If A is a general associative algebra and e an idempotent in A, then the preceding calculations carry over, where matrices have to be understood with respect to the Peirce-decomposition of A, where c := 1 -e :

A = cA ⊕ eA = cAc ⊕ cAe ⊕ eAc ⊕ eAe.
Then G e is isomorphic to a semidirect product of (eAe) × with a Heisenberg type group cAe ⊕ eAc ⊕ cAc. However, in general not every element of A is similar to an idempotent, and hence other types of deformations can appear.

6.2. Rectangular matrices. We consider the Jordan pair (V + , V -) = (M (p, q; K), M (q, p; K)) of rectangular matrices over a field K , with trilinear maps T ± (x, y, z) = xyz + zyx. The group GL(p, K) × GL(q, K) acts by automorphisms of this Jordan pair(in the usual way from the right and from the left), and every matrix a ∈ V -is conjugate under this action to a matrix of the form e = e r with e ii = 1 for i = 1, . . . , r and 0 else. Thus we may assume a = e , and then the symmetric space structure on U e from Theorem 2.3 is calculated in a similar way as above.

Recall that a pair (x, y) ∈ (M (p, q; K), M (q, p; K)) is quasi-invertible if and only if the matrix 1 p -xy is invertible in M (p, p; K); therefore U e is the set of matrices x ∈ M (p, q; K) such that 1 p + xe is invertible, i.e. Again one finds that U e has a bundle structure: the basis is given be letting β, γ, δ equal to zero; it is the space of r × r -Matrices, with the symmetric space structure

U e = 1 r + α β γ δ | det(1 r + α) = 0,
Q(α)α ′ = 2α + α ′ + α 2 + αα ′ + α ′ α + αα ′ α
which is isomorphic to the group GL(r, K), seen as a symmetric space (group case, cf. Appendix A.3). The fiber is given by letting α = 0 , and we get explicitly

Q 0 β γ δ 0 β ′ γ ′ δ ′ = 0 2β + β ′ 2γ + γ ′ 2δ + δ ′ + 0 0 0 γβ + γβ ′ + γ ′ β .
Knowing that the quasi-inverse in the Jordan pair of rectangular matrices is given by τ a (x) = (1 -xa) -1 x = x(1 -ax) -1 , we find that the symmetry with respect to the origin in the fiber is

σ 0 0 β γ δ = 0 -β -γ -δ + γβ ,
so that the product map of the fiber is finally

µ 0 β γ δ , 0 β ′ γ ′ δ ′ = 0 2β -β ′ 2γ -γ ′ 2δ -δ ′ + 0 0 0 (γ -γ ′ )(β -β ′ )
.

In contrast to the preceding section, where the fiber was a two-step nilpotent group, for the symmetric space structure the fiber is now simply a flat a symmetric space. This is not surprising since the Lie triple bracket on a two-step nilpotent group is zero. Explicitly, one can check by direct calculation that the map

F : 0 β γ δ → 0 β γ δ + γβ
is an isomorphism from the fiber onto the same set, but equipped with the flat symmetric space structure 2x -y . (The situation is completely similar to the one considered in [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF], where the The preceding proof was purely Jordan-theoretic. It would be interesting to state and prove Theorem 6.4 in a geometric way, i.e., in the context of Theorem 3.1. The starting point should be a suitable geometric interpretation of von Neumann regularity. This is indeed possible and will be taken up elsewhere. 6.5. Grassmannians. Next we give some examples illustrating Theorem 4.3. Assume W = K p+q is a vector space over a field K of characteristic different from 2 and let (X + , X -) = (Gras p (W ), Gras q (W )) the Grassmannian geometry of p-and q -dimensional subspaces of W , respectively. Then (X + , X -) is a generalized projective geometry; with respect to the base point (o + , o -) = (K p ⊕0, 0⊕K q ), it corresponds to the Jordan pair (M (q, p; K), M (p, q; K)). We define a family of symmetric space structures associated to (X + , X -): assume β : W × W → K is a symmetric or skew-symmetric bilinear form on W such that β is non-degenerate on the first summand o + = K p . If β is non-degenerate on all of W , then p : X + → X -, E → E β , where E β is the orthogonal complement of E with respect to β , is a globally defined polarity, but if β is degenerate, then p is only defined on the {E ∈ X + | dim(E β ) = q} . In any case, the space of non-degenerate linear subspaces of W ,

M β := {E ⊂ W | W = E ⊕ E β } is a symmetric space with product map µ(E, F ) = σ E (F ) = (-1) E,E β (F ),
where, for any scalar r ∈ K and decomposition W = U ⊕ V , we let r U,V be the linear map which is 1 on U and r on V . Let us study how the symmetric space structure and the Jordan triple system of M β depends on β . In order to do this, we have to assume that p(o + ) = o -, i.e., that β(o + , o -) = 0 . In other words, β is of the form β = B 1 0 0 B 2 , i.e. β((x, y), (x ′ , y ′ )) = β 1 (x, x ′ ) + β 2 (y, y ′ ) = x t B 1 x ′ + y t B 2 y ′ with x, x ′ ∈ K p , y, y ′ ∈ K q and matrices B 1 (size p × p) and B 2 (size q × q ) which are both either symmetric or skew-symmetric. The property that p(o + ) = o -assures that the affine picture of p is a linear map from M (q, p; K) to M (p, q; K). We calculate this map explicitly: let X : K p → K q be linear and consider its graph

Γ X = (x, Xx)| x ∈ K p ∈ X + .
Using that B 1 is invertible, we find that the orthogonal complement of Γ X is given by

(Γ X ) β = {(u, v) ∈ K p × K q | ∀x ∈ K p : β((u, v), (x, Xx)) = 0} = {(u, v) ∈ K p × K q | ∀x ∈ K p : u t B 1 x + v t B 2 Xx = 0} = {(u, v) ∈ K p × K q | ∀x ∈ K p : u t x + v t B 2 XB -1 1 x = 0} = {(u, v) ∈ K p × K q | ∀x ∈ K p : u = -B -1 1 X t B 2 v} = Γ -B -1 1 X t B2
. The linear picture of the map p is therefore

M (q, p; K) → M (p, q; K), X → -B -1 1 X t B 2 = -(B 2 XB -1 1 ) t = -(αX) t
with α(X) = B 2 XB -1 1 , and the JTS of the symmetric space M β is

T α (X, Y, Z) = X(αY ) t Z + Z(αY ) t X = X(B 2 Y B -1 1 ) t Z + Z(B 2 Y B -1 1 ) t X, which is the α -homotope of T (X, Y, Z) = XY t Z + ZY t X .
Let us consider some special choices of B 1 and B 2 . Recall that B 1 has to be invertible, and B 1 and B 2 are either both symmetric or both skew-symmetric. In case K = R and K = C, topological connected components of the symmetric spaces we obtain are homogeneous and of the type

M β ∼ = O(B 1 ⊕ B 2 )/(O(B 1 ) × O(B 2 ))
. The isomorphism classes of symmetric spaces obtained for invertible matrices B 2 are listed in [START_REF] Bertram | The Geometry of Jordan and Lie Structures[END_REF]Tables XII.4]:

• Symmetric case: by choosing for B 1 and B 2 invertible diagonal matrices having coefficients b ii ∈ {±1} , we get all symmetric spaces of the type O(l + j, k + i)/(O(l, k) × O(i, j)). • Skew-symmetric case: by choosing the usual matrices of symplectic forms (the dimensions have to be even), we get all symmetric spaces of the type Sp(l + j, K)/(Sp(l, K) × Sp(j, K)). Standard examples of deformations to degenerate cases are obtained by taking some of the diagonal coefficients of B 2 equal to zero. Since orthogonal groups of degenerate forms can be described as semidirect products of vector groups, general linear groups and orthogonal groups of non-degenerate forms, it is possible to give explicit descriptions of such symmetric spaces. Note that in case p even and q odd a new series of symmetric spaces arises which does not arise as deformation of a semisimple series: choose B 1 to be a symplectic form and B 2 skew-symmetric and non-zero.

6.6. Other examples, and some final comments. Similar examples of deformations of semi-simple symmetric spaces arise also in all other classical series of symmetric spaces (see the classification in [Be00, Chapter XII]) and also for about half of the simple exceptional symmetric spaces (cf. comments in [Be00, Section 0.6]). In order to keep this work in reasonable bounds, we cannot go here into further details. Let us just mention that all classical groups arise as group cases by our construction (the general linear groups are related to associative algebras, see Section 0.1, and all orthogonal, symplectic and unitary groups are related to certain Jordan triple systems); it is then an interesting problem to determine the deformations of these symmetric spaces that are again of groupe type. In other words, we raise the problem: which deformations of group type symmetric spaces come from deformations of the overlying group structure (as was the case for the general linear group in Section 0.1)? This question should not be confused with another, also very natural problem: if (M α ) α∈Svar(T ) is a deformation of the symmetric space M = M id , in which sense are then the transvection groups G α := G(M α ) deformations of the transvection group G = G(M )? They will in general not be "contractions of G" in the sense of [START_REF] Dooley | On contractions of semisimple Lie groups[END_REF] since the dimension of G(M α ) will not be constant as a function of α . Replacing transvection groups by automorphism groups leads to a variant of this problem. It seems that contractions of Aut(M ) in the sense of [START_REF] Dooley | On contractions of semisimple Lie groups[END_REF] can only be obtained by using some additional parameter which in some sense memorizes the structure of the stabilizer group of a point.

Appendix A. Symmetric spaces

A.1. Reflection spaces with base point: a group-like concept. A reflection space is a set M together with a family (σ x ) x∈M of symmetries σ

x : M → M such that (S1) σ x (x) = x (S2) σ x • σ x = id M (S3) for all x, y ∈ M , σ x σ y σ x = σ σx(y) . The map µ : M × M → M, (x, y) → µ(x, y) := σ x (y)
is called the multiplication map. The concept of a reflection space is "affine", in the sense that it does not refer to a distinguished base point. On the other hand, for some purposes it is necessary to choose a base point o ∈ M . Following Loos [START_REF] Loos | Symmetric Spaces I, Benjamin[END_REF], we then define the quadratic map and the square, for x ∈ M ,

Q(x) := Q o (x) := σ x • σ o : M → M, x 2 := Q(x)o = σ x (o).
The symmetry σ o is called inversion and is sometimes denoted by x -1 := σ o (x). Note that µ(x, y) can be recovered from these data via

µ(x, y) = σ x (y) = σ x σ o σ o (y) = Q(x)j(y) = Q(x)Q(y) -1 y.
The following relations are easily checked (cf. [START_REF] Loos | Symmetric Spaces I, Benjamin[END_REF] or [Be00, Chapter I]): for all x, y ∈ M ,

(SB1) Q(Q(x)y) = Q(x)Q(y)Q(x) (SB2) Q(x -1 ) = Q(x) -1 (SB3) x -1 = Q(x) -1 x Proposition A.2.
There is an isomorphism of categories between the category of reflection spaces with base point and the category of pointed sets (M, o) equipped with two maps Q :

M × M → M , (x, y) → Q(x)y and σ o : M → M , x → x -1 such that (SB1) -(SB3) hold. Proof. Given (M, o; Q, σ o ), we define µ(x, y) := σ x (y) := Q(x)y -1 = Q(x)Q(y) -1 y,
and (S1) -(S3) are easily proved from (SB1) -(SB3). (For (S3), start by proving that σ o Q(x)σ o = Q(x) -1 .) This construction clearly is inverse to the one described above, and it is also clear that we get the same morphisms.

The various spaces (M, a, Q a , σ a ) for a ∈ M are in general not isomorphic among each other (the map Q(a) is an isomorphism from (M, o) onto (M, a 2 ) and not onto (M, a)). In order to change base points, we have the "isotopy formula":

Q a (x) = σ x σ a = σ x σ o (σ a σ o ) -1 = Q(x)Q(a) -1 .
A.3. Symmetric spaces. Assume K is a topological ring with dense unit group. A smooth symmetric space is a smooth manifold over K together with a reflection space structure such that the multiplication map µ is smooth and (S4) the tangent map of the symmetry σ x at the fixed point x is equal to the negative of the identity of the tangent space T x M .

(Cf. [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF] for the notion of smpooth manifold over K ; if K = R or K = C, it coincides with the usual notion.) We will use the term symmetric space, somewhat informally, also in other, purely algebraic contexts, whenever the notion of tangent maps of the symmetries σ x makes sense and (S4) is satisfied. For instance, this will be the case if there is some natural chart around x such that σ x is, in this chart, represented by the negative of the identity . If (M, o) is a smmoth symmetric space with base point, one may define its Lie triple system in a similar way as the Lie algebra of a Lie group; this is done most naturally by using the quadratic maps Q(x), cf. [START_REF] Loos | Symmetric Spaces I, Benjamin[END_REF] for the real finite-dimensional and [Be06, Chapters 5 and 27] for the general case.

Important examples of symmetric spaces arise from the group cases: a group G with the new product µ(g, h) = σ g (h) = gh -1 g satisfies (S1), (S2), (S3), and if G is, e.g., a Lie group, then also (S4) holds. Subspaces of (G, µ) are given by spaces of symmetric elements: if τ is an involution of a group G (anti-automorphism of order 2), then M := {g ∈ G|τ (g) = g -1 } is stable under µ, i.e., it is a subspace. 1. Duality. All axioms of an affine pair geometry appear together with their dual version: thus (X -, X + ; ⊤; Π -, Π + , Σ -, Σ + ) is again an affine pair geometry, called the dual geometry of (X + , X -; ⊤; Π + , Π -, Σ + , Σ -). 2. Homomorphisms. These are pairs of maps (g + : X + → Y + , g -: X -→ Y -) which are compatible with the transversality relations and with the structure maps, in the obvious sense. 3. Antiautomorphisms. These are isomorphisms (g + , g -) : (X + , X -) → (X -, X + ) onto the dual geometry. If g -= (g + ) -1 , then (g + , g -) is a correlation, and if, moreover, there exists a non-isotropic point (i.e., there is x ∈ X + with x⊤g + (x)), then (g + , g -) is called a polarity. 4. Adjoint (or: structural) pairs of morphisms. These are pairs of maps (f : X + → Y + , h : Y -→ X -) which are compatible with transversality and with structure maps in the sense that x⊤h(a) iff f (x)⊤a and f.Π s (u, h.v, w) = Π s (f.u, v, f.w), f.Σ(u, h.v, w, h.z) = Σ(f.u, v, h.w, z).

It is necessary to allow here also maps that are not everywhere defined (see Appendix C). A map f : X + → X -is called selfadjoint if (f, f ) is an adjoint pair of morphisms into the dual geometry (compare with Section 1.12). 5. Stability. A pair geometry (X + , X -, ⊤) is called stable if any two points x, y ∈ X ± are on a common affine chart, i.e., there exists a ∈ X ∓ such that x, y ∈ v a .

B.3. The affine pair geometry associated to a Jordan pair. Assume that V = (V + , V -) is a linear Jordan pair over K , which we obtain in the form (V + , V -) = (g 1 , g -1 ) for a 3graded Lie algebra g = g 1 ⊕ g 0 ⊕ g -1 with T ± (x, y, z) = [[x, y], z] (see Section 1.4). Let G := PE(V + , V -) = U + , U + be the projective elementary group associated to this 3 -graded Lie algebra, i.e., the group generated by the vector groups U ± = exp(g ±1 ), let H be the intersection of G with the automorphism group of the 3 -grading (called the (inner) structure group) and let P ± := U ± H . Since U ± are vector groups, it is easily seen that (X + , X -) := (X + (V ), X -(V )) := (G/P -, G/P + ) is an affine pair geometry. The base point (eP -, eP + ) is often denoted by (o + , o -), and the affine parts (v o -, v o + ) corresponding to this base point are identified with (V + , V -) = (exp(g + ).o + , exp(g -).o -). (In [START_REF] Bertram | Projective completions of Jordan pairs. I[END_REF] a realization of this geometry and of its transversality relation in terms of Lie algebra filtrations is given.)

The construction is functorial: a Jordan pair homomorphism induces a homomorphism of affine pair geometries ([Be02, Th. 10.1]). In particular, an involution of a Jordan pair induces a polarity of (X + , X -). Since Jordan triple systems are nothing but Jordan pairs with involution, we thus associate a polar geometry to any JTS.

The geometry associated to a Jordan pair (V + , V -) is stable if and only if the maps V ∓ × V ± → X ± , (a, x) → exp(a) exp(x).o ± are surjective; then X ± can be seen as V ± × V ∓ modulo the equivalence relation given by fibers of this map (called projective equivalence; cf. [START_REF] Loos | Decomposition of projective spaces defined by unit-regular Jordan pairs[END_REF]). Another equivalent version of stability is that the projective group G admits the "Harish-Chandra decomposition" G = exp(V + ) exp(V -)H exp(V + ) and G = exp(V -) exp(V + )H exp(V -), where H is the structure group.

B.4. Generalized projective geometries are affine pair geometries (X + , X -) such that the structure maps satisfy some algebraic identities (PG1) and (PG2) which should be seen as global analogs of certain Jordan algebraic identies. Recall from B.1 that r x,a (y) = Π r (x, a, y), r a,x (b) = Π - r (a, x, b) ("left multiplication operators"), and denote by M (r)

x,y (a) := Π r (x, a, y), M (r) a,b (x) := Π r (a, x, b) the "middle multiplication operators". Then the geometry (X + , X -) associated to a Jordan pair (V + , V -) satisfies the following properties ([Be02, Th. 10.1]):

(PG1) For any invertible scalar r ∈ K and any transversal pair (x, a), the pair of maps (g, g ′ ) =

(r x,a , r -1 a,x ) is an automorphism of the affine pair geometry (X + , X -).

b,a ) is an adjoint pair of (locally defined) morphisms from (X + , X -) to (X -, X + ), and dually.

Applied to elements of X + , resp. X -, (PG1) and (PG2) are indeed algebraic identities for the structure maps. One should think of (PG1) as a global analog of the defining identity (LJP2) (Section 1.4) of a Jordan pair, and of (PG2) as the global analog of the fundamental formula.

  [x, y] a = xay -yax. (0.3) 0.3. Plan of this work. In the first chapter we recall the definition of the three basic Jordan theoretic categories and the notions of homotopy we are interested in:

  and σ x (v a ) = v b and hence σ x preserves U a,b . Moreover, it follows that the group G(U a,b ) generated by all products of two symmetries preserves both a and b and hence acts affinely both on v a and on v b .

  Remark 4.4. (Realisation as a graph.) Assume that α is a T -symmetric map. Then it is easily checked that the graph of α

  β, γ, δ arbitrary . Using that Q(x)e = xex and B(x, -e)y = (1 + xe)y(1 + ex), we get the quadratic map of the symmetric space U e Q(x)y = 2x + Q(x)e + B(x, -e)y = 2x + y + xex + xey + yex + xeyex.

  (PG2) For any r ∈ K and a, b ∈ X -with v a ∩ v b = Ø , the pair of middle multiplications (f, h) = (M (r) a,b , M

cf.[START_REF] Loos | Decomposition of projective spaces defined by unit-regular Jordan pairs[END_REF] 1.3 (3)] for the case of a stable geometry; the proof of the corresponding fact for the general non-stable case is similar but rather lengthy and is therefore omitted here.

Appendix B: Generalized projective geometries Appendix C: Globalization of structural maps AMS-classification: 17C37, 32G99, Secondary: 17C27, 14D99 Key words: Homotope, isotope, Jordan algebras (-triple systems, -pairs), Lie triple system, symmetric space, generalized projective (and polar) geometries. symmetric space structure on the second order tangent bundle T T M of a symmetric space M is calculated, see loc. cit., Chapters 26 and BA.) 6.3. Symmetric spaces associated to idempotents. The preceding calculations and remarks can be generalized for an arbitrary Jordan pair (V + , V -), as follows: we assume that e := e -∈ V -is an element which can be completed to an idempotent, i.e., there existes e + ∈ V -such that Q + (e + )e -= e + and Q -(e -)e + = e + . (In other words, e is von Neumann regular, cf. [START_REF] Loos | [END_REF].) Associated to an idempotent (e + , e -), there is a Peirce-decomposition

which is the eigenspace decomposition of the operator T (e + , e -). The space V + 1 ⊕ V + 0 depends only on e -since V + 1 ⊕ V + 0 = ker Q(e -), and V + 2 depends only on e + since V 2 = Q(e + )V -. Moreover, V + 2 is a unital Jordan algebra with unit element e + (see [START_REF] Loos | [END_REF] for all this). Theorem 6.4.

(Bundle structure of U e .) Assume e = e -∈ V -as a von Neumann regular element and choose e + ∈ V + such that (e + , e -) is an idempotent.

(1)

2 is a symmetric subspace of U e (called the base), isomorphic to the quadratic pre-homogeneous symmetric space (V + 2 ) × .

(3) The space V + 1 ⊕ V + 0 is a symmetric subspace of U e (called the fiber), and its symmetric space structure is isomorphic to the flat symmetric structure 2x -y on this set.

Proof. (1) We use the identity JP24 of [START_REF] Loos | [END_REF] which gives

, Q(e)x = 0 . Hence B(x, e)B(x, -e) = B(x, 0) = id . In the same way, B(x, -e)B(x, e) = id, and so B(x, -e) is always invertible, with inverse B(x, e). This shows that V + 0 ⊕ V + 1 ⊂ U e . Moreover, then τ x (e) = e since B(e, x) -1 (e -Q(e)x) = B(e, -x)(e -Q(e)x) = B(e, -x)e = e . Assume x ∈ V + 2 . Then B(x, -e) is invertible in End(V + ) if and only if its restriction to V + 2 is invertible, and hence, by the results on the Jordan algebra case (Prop. 2.2), we see that this is equivalent to saying that e + + x is invertible in V + 2 . Now let z ∈ V + and decompose z = x + y with y ∈ V + 0 ⊕ V + 1 and y ∈ V + 2 . Then (y, e) is quasi-invertible with τ y (e) = e , as we have seen above, and hence we may apply the identity JP34 of [START_REF] Loos | [END_REF] in order to obtain B(x + y, e) = B(x, τ y (e))B(y, e) = B(x, e)B(y, e). This operator is invertible in End(V + ) if and only if x and y satisfy the previously described conditions, and this proves the claim.

(2) This follows directly from (1) since the formula for the product µ(x, y) with x, y ∈ (V × 2 + e + ) uses only data which depend on the Jordan algebra V + 2 .

(3) We first calculate σ 0 (y) for an element y ∈ ker Q(e): using e -y = e , we get from the symmetry formula ([Lo75, 3.3])

Next we calculate the map Q(x)y = 2x+Q(x)e+B(x, -e)(y). Since B(x, -e)y = y +T (x, e, y)+ Q(x)Q(e)y = y + T (x, e, y), we get

y).

As in the matrix case (Section 6.2), it is checked by a direct calculation that the map

e is an isomorphism from the symmetric space V + 0 ⊕V + 1 with the usual flat structure (x, y) → 2x-y onto the symmetric space V + 0 ⊕ V + 1 equipped with the symmetric space structure that we have just described.

Appendix B: Generalized projective geometries

The geometric counterpart of a Jordan pair is a generalized projective geometry, and the one of a JTS a generalized polar geometry. In this appendix we recall the basic definitions and refer to [START_REF] Bertram | Generalized projective geometries: General theory and equivalence with Jordan structures[END_REF] and [START_REF] Bertram | Projective completions of Jordan pairs. I[END_REF] for further details.

B.1. Affine pair geometries.

A pair geometry is given by a pair of sets, denoted by (X + , X -), and a binary transversality relation (X + × X -) ⊤ ⊂ X + × X -; we write x⊤α or α⊤x if (x, α) belongs to (X + × X -) ⊤ , and we require that for all α ∈ X ∓ there exists x ∈ X ± such that x⊤α . In other words, X ± is covered by the sets

as α runs through X ∓ . An affine pair geometry over a commutative ring K is a pair geometry such that, for every α ∈ X ∓ , the set v α is equipped with the structure of an affine space over K ; then the sets v α will be called affine parts or affine charts of X ± . Given an affine pair geometry, we let, for (x, a), (y, a) ∈ (X + × X -) ⊤ and r ∈ K , r x,a (y) := ry be the product r • y in the K -module v a with zero vector x, and we define the structure maps by

where (X + × X -× X + ) ⊤ := {(x, a, y) ∈ X + × X -× X + | x⊤a, y⊤a}, (B.3) and dually the maps Π - r are defined. Similarly, we get structure maps defined by vector addition:

where the sum is taken in the K -module (v a , x) and the set (X + × X -× X + × X + ) ⊤ is defined by a similar condition as in (B.3). Dually, Σ -is defined.

B.2. Categorical notions. Assume (X + , X -) and (Y + , Y -) are affine pair geometries over K .

Appendix C: Globalization of structural transformations C.1. Homomorphisms and structural transformations. There are two ways to turn Jordan pairs over K into a category: one defines homomorphisms in the usual way as pairs of linear maps (h + , h -) : (V + , V -) → (W + , W -) which are compatible with the maps T ± , and one defines a structural transformation to be a pair of linear maps f : [START_REF] Loos | Decomposition of projective spaces defined by unit-regular Jordan pairs[END_REF]). In both cases we get a category; these categories are different, but have essentially the same isomorphisms.

C.2. Globalization: example of projective spaces. A homomorphism of Jordan pairs (ϕ + , ϕ -) : (V + , V -) → (W + , W -) always lifts to a homomorphism of associated geometries ([Be02, Th. 10.1]), whereas for structural transformations the situation is more difficult. In order to show that structural transformations do not always extend globally, consider the example of ordinary projective geometry X + = KP n over a field. Its Jordan pair is (W, W * ), with W = K n , and

This Jordan pair is simple, and hence homomorphisms (W, W * ) → (V + , V -) are either injective or trivial. Clearly, every injective linear map K n-1 → K m-1 of affine parts lifts to a map KP n → KP m of associated projective spaces, which shows that homomorphisms globalize, as stated above. On the other hand, every linear map f : V → W together with its dual map defines a structural transformation (f, g) = (f : V → W, f * : W * → V * ). If f is not injective, then f does not lift to a globally defined map of projective spaces. The best we can do is to extend f to a not everywhere defined map f from KP n to KP m . The "natural domain of definition" of f is

The set D( f ) is strictly bigger than the affine part V (except for the trivial case f = 0 ). The purpose of this appendix is to define such a natural domain of definition for any structural transformation of Jordan pairs.

) is a structural transformation between Jordan pairs, then the pair (V + , W -) with w) is a linear Jordan pair, and the maps ϕ := (ϕ + , ϕ -) := (f, id) : (V + , W -) → (W + , W -), (x, y) → (f (x), y) ψ := (ψ + , ψ -) := (id, g) : (V + , W -) → (V + , V -), (x, y) → (x, g(y)) are homomorphisms of Jordan pairs.

Proof. The first claim can be checked by a direct computation which is completely analogous to the proof of Lemma 1.12. We show that ϕ is a homomorphism:

Similarly, we see that ψ is a homomorphism. C.4. The globalization of a structural transformation. Assume (f, g) is a structural transformation as above, and let ϕ and ψ the homomorphisms defined in the lemma. By functoriality, we have induced homomorphisms on the level of geometries:

In general, if the first (resp. second) component of a Jordan pair homomorphism is injective, then so is the first (resp. second) component of the induced map of geometries 1 , and hence the first component Ψ + and the second component Φ -are injective. The following maps together with their domains of definition are called the globalization of (f, g): The following result will not be needed in this work and its proof is therefore omitted:

Theorem C.5. The pair ( f , g) is an adjoint pair of morphisms between X (V ) and X (W ) whose domain of definition (D( f ), D( g)) is complete in the following sense: whenever x ∈ D( f ), then f (x) ⊤ ⊂ D( g), and whenever a ∈ D( g), then g(a) ⊤ ⊂ D( f ). Moreover, (D( f ), D( g)) is a generalized projective geometry, isomorphic to X (V + , W -), with structure maps given by Π + r (x, a, y) = Π r (x, g(a), y), Π - r (a, y, b) = Π - r (a, f (y), b),


Marc Chaumont

Louis Tribouillard

Gérard Subsol

Florian Courtade

Jérôme Pasquet

Mustapha Derras

Automatic localization of tombs in aerial imagery: application to the digital archiving of cemetery heritage

come

I. INTRODUCTION

In France and more generally in Western Europe, cemeteries are composed of built tombs which are generally aligned. Most cemeteries were located around churches until the late 18th century. After this period, due to lack of space, cemeteries were established away from heavily populated areas, at the periphery of towns and villages.

Cemeteries constitute a very valuable database of the history of a local population. Each tomb gives information on the history (birth and death dates), the culture (style of the monument, name typology), and the social status (size of the monument) of a person or a group of persons.

Nevertheless, very little advantage has been taken of this historical heritage, except in a few rare cases such as Père Lachaise cemetery in Paris. This cemetery attracts hundreds of thousands of visitors annually to the graves of famous people and a virtual tour of the most famous graves is even available on the Web 1 . In 2010, the European Community launched the European Cemeteries Route 2 , which is an effort to recover, maintain and provide access to the most significant cemeteries of the European continent to the public. 1 www.pere-lachaise.com/ 2 www.cemeteriesroute.eu/en/ To go further, there is a crucial need to digitally archive cemetery data for heritage purposes. In a lot of cemeteries, especially the older ones located in small villages, this work has not been done and very few cemeteries are described, even partially, in an exploitable digital database.

The first step is to localize and map all the tombs in a cemetery, which may be a very complex procedure (there are about 69,000 tombs in Père Lachaise cemetery). A practical way would be to to use aerial images but manually pointing and delineating several tens or hundreds of tombs in such an image is very tedious.

We propose to automate this process by using image processing algorithms. This is a very challenging problem as tombs vary substantially in appearance, size and disposition on aerial images. Moreover, vegetation (there are more than 5,300 trees in Père Lachaise cemetery), shadows created by the numerous buildings, walking people or utility vehicles may create occlusions and other distortions on the images.

We have compared three automated localization methods: the first one is based on the well-known watershed image processing algorithm which gives very limited results; the two others first require a supervised learning of a database of tomb images which takes the high variability in appearance into account. In section II, we introduce the problem and the segmentation methodology. In Section III, we detail the approach which is based on a recent algorithm. Lastly, in Section IV, we give some experimental results on aerial images of cemeteries of French villages and discuss other approaches related to digital archiving of cemetery heritage.

II. LOCALIZING TOMBS IN AERIAL IMAGES: A COMPLEX

PROBLEM

To the authors' knowledge, no papers have dealt with tomb localization from aerial images. A first idea would be to use standard image processing algorithms to segment multiple compact objects. For example, we can extract the strong gradient areas in the image, use them as seeds for the watershed algorithm and keep only the largest basins. Figure 1 shows the result of such a procedure on an aerial image (3360 × 4200 pixels) of a village cemetery in France (Chanoy in the Haute-Marne department). We can see a lot of mistakes, which are both undetected tombs and regions wrongly assigned as tombs. Indeed, in Figure 1, tombs are highly variable in size, shape and color, they are not evenly aligned, and there are many shadows associated with buildings. All of these conditions are difficult to manage using standard image processing algorithms.

A more satisfactory approach could be to introduce a model of the tomb object which may take its specificities and variability into account. Among these approaches, the most successful are those based on learning. They do not require formal modelling but just a sufficient number of examples of the object in order to learn to recognize the object.

The Viola-Jones framework is a very classical learningbased method [START_REF] Viola | Robust real-time face detection[END_REF]. This method, presented for the first time in 2001, was primarily used to detect faces and has been applied to other objects such as hands, persons or vehicles.

Figure 2 shows the result of a segmentation approach based on the Viola-Jones framework on an aerial image of another French village cemetery (Saint-Gatien in the Basse-Normandie department). We use a learning database based on 21 aerial images of cemeteries, all located in the same department (some details of this database are given in Section IV). From these cemeteries, 1,348 images of tombs, and 2,000 images with the same sized background were extracted manually to set-up the learning database.

We performed a quantitative assessment of the two methods on the Saint-Gatien cemetery [START_REF] Courtade | Segmentation automatique d'images numériques : application la détection et à la géo-localisation des tombes dans un cimetière[END_REF] which contains 636 tombs. With the Viola-Jones framework, the object recall is 49%, the object precision is 72%, and the object F-score is 53%. These indicators reveal a significant improvement compared to the watershed-based approach, where these parameters are respectively 24%, 23%, and 24%. This confirms that learningbased approaches give better results than approaches without any modelling. However, although the results obtained with the Viola-Jones framework are better, the localization results are insufficient for practical applications.

New algorithms for object localization have been proposed over the last five years [START_REF] Harzallah | Combining efficient object localization and image classification[END_REF], [START_REF] Aldavert | Fast and Robust Object Segmentation with the Integral Linear Classifier[END_REF], [START_REF] Fulkerson | Class segmentation and object localization with superpixel neighborhoods[END_REF]. In particular, the Viola-Jones framework is hampered by a long learning time, the empirical adjustment of false positive and false negative rates, and from the use of cascades of classifiers which reduces the classification performance [START_REF] Harzallah | Combining efficient object localization and image classification[END_REF]. Moreover, the features which characterize the object are too simple. We investigated several of these new methods and selected one which we describe in the next section.

III. TESTING A RECENT METHOD

In 2010, Aldavert et al. [START_REF] Aldavert | Fast and Robust Object Segmentation with the Integral Linear Classifier[END_REF] proposed to use more descriptive features than in the Viola-Jones framework, to integrate the concept of bag of visual features, and described a low complexity solution that overcomes the drawbacks associated with the cascade of classifiers. In their paper, they obtained state-of-the art results of much lower complexity, on generic image databases.

The Aldavert approach initially requires a learning phase. During this phase, one or several hundreds of images with their ground truths (i.e the label "tomb" or "not tomb" associated with each pixel) are presented as algorithm input. The learning algorithm, in its C++ implementation 3 , which slightly differs from the article, contains four major steps:

1) Definition of a vector of 32 scalar features computed on small image areas which will be used as a descriptor of the center pixel of the area. At this step, the vector is based on the HOG descriptor [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] which characterizes the distributions of gradient orientations.

2) Determination of a dictionary composed of visual words.

Here, each visual word is a vector of 32 real values representative of all the vectors computed on the image. Said in another way, a visual word is a representative element of clusters of vectors which have some close features. At this stage, the dictionary is built with an Extremely Random Forest (ERF) approach of low complexity. 3) Determination, on a small area, of an histogram of visual words. As each pixel is described only by a very limited number of visual words, it is very easy to obtain an histogram representing the proportion of each visual word for this area. 4) The learning phase, using a linear classifier that takes a subset of the histograms, that were computed in the previous step, as input. Note also that the method may be extended to a multi-resolution version where one classifier is used per resolution.

During the test phase, steps (1) and ( 3) are successively performed and step ( 4) is replaced by a prediction step (also called classification step). In this last step, the classifier decides to label the pixel as "tomb" or "not a tomb". Note that an additional step may also be included to refine (and denoise) the results. In the original paper, this additional step is achieved by using a mean-shift segmentation approach applied on the "probability" map which gives the degree of membership to the "tomb" or "not a tomb" label.

IV. EXPERIMENTS AND DISCUSSION

We work with 21 cemeteries, all located in the Haute-Marne department. These cemeteries contain between 33 and 533 tombs but most of them contain less than a hundred tombs. The tombs are not always uniformly spaced, nor perfectly aligned. Some of them are covered with vegetation, and most of them are made of marble and covered with few flower pots. Almost all the cemeteries are located around a church.

The sizes of the 21 aerial images are 3761 × 4092 pixels on average, and tombs have a height between 100 -120 pixels, and between 50 -60 pixels wide. The space between tombs may vary a lot, from 5 to 20 pixels. Most of the aerial views contain shadows due to surrounding buildings and trees.

19 cemeteries were used to build the learning image database. Each of the 19 aerial images was split in multiple non-overlapping smaller images of 640 × 480 pixels in size. From all those "small" images, the more representative were selected in order to constitute the learning database consisting of 150 images, including 90 images containing tombs.

The Lecey cemetery will be used for the tests. Of course it does not belong to the learning database. Lecey is a small village of 7.85 km 2 with 223 habitants (2012 census), located in the Haute-Marne department. Its cemetery contains 93 tombs distributed all around the church. Figures 3 and4 show an aerial view 4 of Lecey cemetery (taken in November 2011), The Aldavert algorithm first learns to recognize the tombs using the learning database. In this phase, the HOG descriptor is computed on small 32 × 32 pixel areas. The dictionary of visual words is composed of 655, 360 vectors and the histogram of visual words is obtained on small 40 × 40 pixel areas. Learning of the linear classifier is based on histograms computed at 300, 000 pixels which are pseudorandomly selected. The parameters of the linear classifier are tuned by evaluating the results on a subset of 50 images of the learning database.

Figure 5.a shows a part of the aerial image of the Lecey cemetery, close to the entrance. Figure 5.b shows the probability map obtained for each pixel. Red pixels (resp. blue) correspond to areas which have a high (resp. low) probability of being part of a tomb. We can see that most of the pixels are well classified and these preliminary results are very promising. Figure 6 shows the result of a very simple approach used to localize the tombs, by taking the probability map into account. The method consists of keeping only the highly probable regions of sufficiently large sizes. If the obtained bounding boxes are too large, they are split, by taking the image gradients inside these boxes into account.

The main drawback of the Aldavert approach is the difficulty of extending the pixel classification of Figure 5 to an object classification as presented in Figure 6. This may be improved either by adding a low level segmentation approach based on the probability map as we did or by concatenating the pixel classification results in order to get an area classification.

A quantitative comparison between Viola-Jones and Aldavert approaches was performed on an aerial view of Signy-le-Petit cemetery in the Ardennes department. Both algorithms give comparable performances with an object recall and precision of respectively 58% and 72% for Viola-Jones and 53% and 76% for Aldavert. In fact, the Viola-Jones approach is hampered by the limits of cascading classifiers which reduce performance whereas the Aldavert approach is very sensitive to any pixel errors. Nevertheless, the computational gain during the learning phase is huge when using the Aldavert approach which is 42 times faster than for the Viola-Jones approach.

V. CONCLUSION In this paper, we strongly stress the necessity of digital archiving of cemeteries. Except for some cemeteries such as Père Lachaise, most of them are not described, even partially, in an exploitable digital database. The first step in archiving is to precisely locate the tombs and a convenient way is to use aerial images. We propose to automate this process by using image processing algorithms. We especially focus on learningbased approaches, and show their superiority compared to low-level approaches such as watershed. After recalling the general framework of recent learning-based approaches, we test one of them, on a database of aerial images of cemeteries in the Haute-Marne French department. The first results are promising and should lead to an automated method.

We could extend our work by integrating additional information on the localized tombs. For example, some researchers have proposed to apply Optical Character Recognition to headstones [START_REF] Christiansen | Capstone: Gather valuable information found on headstones as an automated process[END_REF]. A user walking in a cemetery alley takes photographs of headstones, and image processing algorithms automatically extract the writings and could recognize names or dates. All of these data (including photographs) could then be automatically integrated in the tomb database in order to achieve an exhaustive digital archiving of the cemetery heritage.

Fig. 1 .

 1 Fig. 1. Results of a watershed-based procedure on an aerial view of a part of a French village cemetery (Chanoy in the Haute-Marne department). Red circles and green rectangles respectively indicate centers and bounding boxes of the detected regions.

Fig. 2 .

 2 Fig. 2. Results of the Viola-Jones framework on an aerial view of a part of a French village cemetery (Saint-Gatien in the Basse-Normandie department). Green rectangles represent the bounding boxes of the detected tombs.

Fig. 3 .

 3 Fig. 3. An aerial view of Lecey cemetery taken in November 2011.

Fig. 4 .

 4 Fig. 4. A Google Street View image of Lecey cemetery taken in October 2010.

Fig. 5 .

 5 Fig. 5. Part of Lecey cemetery (a) and its probability map (b). Red (resp. blue) pixels correspond to areas with a high (resp. low) probability of being part of a tomb.

Fig. 6 .

 6 Fig. 6. Results obtained on an aerial view of a part of Lecey cemetery. Green rectangles represent the bounding boxes of the detected tombs.

www.cvc.uab.cat/˜aldavert/plor/software.html

http://www.leuropevueduciel.com/

ACKNOWLEDGEMENT

The authors would like to thank the Berger-Levrault group for supporting this research. Berger-Levrault is a French public regulation expert that addresses healthcare and local public administrations. With almost 60 thousands customers and more than 1,000 employees, the Berger-Levrault group is a key enabler for the development of e.Government in France. Bringing people/citizens/patients and their public administrations closer is a key focus of the Berger-Levrault group.


Haritza Camblong

Octavian Curea

Aitor Etxeberria

Alvaro Llaria

Research experimental platforms to study microgrids issues

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