Complexifications of Symmetric Spaces and Jordan theory. I.

Wolfgang Bertram

Abstract. Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use work of B.O. Makarevič [Ma73] in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan- and Lie triple systems.

Key words: symmetric space, complexification, Jordan– and Lie triple system

AMS-classification: 17 C 36, 22 E 15, 53 B 35, 53 C 15

0. Introduction

0.1. Straight and twisted complexifications. We distinguish two types of complexifications of a symmetric space M = G/H: the first type is of the form M_{ℂ} = G_{ℂ}/H_{ℂ}, where G_{ℂ} and H_{ℂ} are suitable complexifications of the Lie groups G and H. Although one can in general not imbed M globally into M_{ℂ}, this is locally always possible. We call the corresponding local complexification of M straight.

We now give some examples of the second type of complexifications which is called twisted: the unit disc N = SU(1, 1)/ SO(2) is a twisted complexification of its real form M = ]−1, 1[ ≅ ℝ; the complex projective space N = U(n + 1)/(U(n) × U(1)) is a twisted complexification of the real projective space M = O(n + 1)/(O(n) × O(1)); the variety N = Gl(2n, ℝ)/ Gl(n, ℂ) of complex structures on ℝ^{2n} is a twisted complexification of the symmetric space of group type M = Gl(n, ℝ). Twisted complexifications are characterized by the property that the invariant almost complex structure J and the curvature tensor R of the symmetric space N satisfy the relation

R(J X, Y)Z = −R(X, J Y)Z  (T)

for all vector fields X, Y, Z; then we say that N is a twisted complex symmetric space. This is a generalization of the class of pseudo-Hermitian symmetric spaces. In general, it is not known how many twisted complexifications a symmetric space admits. For irreducible spaces we give the answer by classification, see below.

0.2. The Jordan theoretic viewpoint. In the late sixties M. Koecher discovered a bijection between Hermitian symmetric spaces and an algebraic object called positive Hermitian Jordan triple systems, cf. [Lo77] and [Sa80]. We generalize this correspondence in a first step to twisted complex symmetric spaces: they are equivalent to Hermitian Jordan triple systems (Th. 1.3.6), and in a second step to their real forms: they are equivalent to general real Jordan triple systems (Th. 2.2.6). The idea is simple: we introduce the structure tensor of a twisted complex symmetric space with curvature tensor R and almost complex structure J by the formula

T (X, Y, Z) := \frac{1}{2}(R(X, Y)Z + J R(X, J Y)Z)

and prove that it is a Hermitian Jordan triple system (Prop. 1.3.2). It can be restricted to real forms (Lemma 2.2.1) and it contains all information in order to reconstruct (locally) the twisted complex symmetric space from its real form (Th. 2.2.2). This is due to the remarkable relation

T (X, Y, Z) − T (Y, X, Z) = R(X, Y)Z;

we say that T is a Jordan-extension of the curvature R. We are thus lead to the category of symmetric spaces with a Jordan extension, called the category of symmetric spaces with twist (Def. 2.2.5). The advantage of our method compared with the more traditional one used in [Sa80] is not so much that it is more general but that it is simpler and more natural. In fact, we obtain not only bijections of objects, but also equivalences of categories even in the nonsemisimple case. The equivalences of categories constructed in [Sa80] are not as natural and do not generalize to the non-semisimple case.

0.3. Twisted para-complexifications. Our method can also be applied to paracomplexifications: these are defined in the same way as complexifications, but using a structure with J^{2} = id instead of J^{2} = − id. They lead to a class of symmetric spaces generalizing the para-Hermitian symmetric spaces investigated by Kozai and Kaneyuki (cf. [KanKo85]). It turns out that a symmetric space admits a twisted para-complexification if and only if it admits a twisted complexification (Th. 2.2.6). For instance, the one-sheeted hyperboloid is the twisted para-complexification of ℝ corresponding to the twisted complexification given by the unit disc.

0.4. Classification. Given a symmetric space, there is no general method to decide whether it admits a twisted complexification, and if so, how many there are and how to describe them as a quotient space. In Chapter 4 we present the classification of simple twisted complexifications and para-complexifications of symmetric spaces of classical type. The result is rather surprising:

Theorem A. (Existence) Every (modulo possibly a central extension) irreducible symmetric space of classical type admits a twisted complexification.

The central extension is needed e.g. in the group case M = Sl(n, ℝ) which does not admit a twisted complexification; but its central extension Gl(n, ℝ) does (see above).

Theorem B. (Unicity) The number of inequivalent twisted complexifications in the situation of Th. A is either 1,2 or 3, and “generically” it is equal to 1.

On on algebraic level, Th. A and Th. B are equivalent to the statement that the algebraic Jordan-Lie functor assigning to a Jordan triple system T the Lie triple system

R_{T} (X, Y)Z := T (X, Y, Z) − T (Y, X, Z)

is (for the classical types) “generically bijective”. This fact has already been remarked by E. Neher ([N85]) by using both the classification of Lie triple systems ([B57]) and the classification of Jordan triple systems ([N80], [N81]), where also the exceptional spaces are considered. A classification-free approach to the Jordan-Lie functor in the compact case is due to O. Loos ([Lo85]). Our method of classification is different from the one used by E. Neher; it follows closely the classification of open symmetric orbits in symmetric R-spaces by B.O. Makarevič ([Ma73]).

0.5. Integrability. It is well-known that invariant almost complex structures on symmetric spaces are integrable (cf. [Ko69, Prop. X.6.5]); thus the twisted complex symmetric spaces are actually complex manifolds and the real forms considered here are their real forms as manifolds. However, the theory becomes clearer if in a first step one does not use this fact. We will use it in a second step ([Be98]) where we introduce the conformal group of a symmetric space with twist and characterize it by a theorem generalizing at the same time the fundamental theorem of projective geometry and the Liouville theorem for Jordan algebras ([Be96a, Th. 2.3.1]). Finally, we have in mind applications of the structure theory developed here and in [Be96c], [Be98] to harmonic analysis on symmetric spaces with twist.

In [Be96c] the present work was announced as “Algebraic Structures of Makarevič spaces. II”, but we have chosen a more definite titel in order to make clear that the present work is self-contained and independent of our previous work. In particular, we do not assume the reader to be familiar with Jordan theory.

Acknowledgements. I thank Joachim Hilgert for many helpful discussions.

1. The structure tensor of a twisted complex symmetric space

1.1. The Lie functor for symmetric spaces. A symmetric space (with base point) is a homogeneous space M = G/H where G is a Lie group and H an open subgroup of the group of fixed points of some non-trivial involution σ of G. The base point eH ∈ M is denoted by o.

Morphisms of symmetric spaces (with base point) are smooth maps φ : M → M′ of symmetric spaces which are base-point preserving and compatible with the canonical connections ∇ and ∇′ associated to M and M′ (cf. [Lo69]).

The infinitesimal object associated to a symmetric space M = G/H is the −1-eigenspace 𝔮 in the eigenspace-decomposition 𝔤 = 𝔥 ⊕ 𝔮 of the Lie algebra 𝔤 of G under the differential σ̇ of σ at the origin. If X, Y, Z ∈ 𝔮, then [X, Y, Z] := R(X, Y)Z := [[X, Y], Z] again belongs to 𝔮, and we have the rules

(LT1): R(X, Y) = −R(Y, X),

(LT2): [X, Y, Z] + [Y, Z, X] + [Z, X, Y] = 0,
(LT3): R(U, V)[X, Y, Z] = [R(U, V)X, Y, Z] + [X, R(U, V)Y, Z] + [X, Y, R(U, V)Z].

Condition (LT3) means that R(U, V) is a derivation of the triple bracket. If we identify 𝔮 with the tangent space T_{o}M via the bijective evaluation map ev_{o} : 𝔮 → T_{o}M, X ↦ X_{o}, the triple product [X, Y, Z] of 𝔮 is the curvature tensor R of the canonical connection of M, evalutated at o ([Lo69, II.2.5]; note that our curvature tensor differs by a sign from the more usual one).

Definition 1.1.1. A vector space together with a trilinear map 𝔮 × 𝔮 × 𝔮 → 𝔮, (X, Y, Z) ↦ [X, Y, Z] = R(X, Y)Z satisfying (LT1), (LT2) and (LT3) is called a Lie triple system (abbreviated LTS). Morphisms of Lie triple systems are linear maps which are compatible with the respective triple products.

It is a matter of straightforward verification (cf. [Lo69, Prop.II.2.3]) that for any LTS 𝔮, the direct sum 𝔤 of 𝔮 and the algebra Der(𝔮) of derivations of 𝔮 is a Lie algebra with the Lie bracket defined in the obvious way, such that the abstract LTS 𝔮 is imbedded as the sub-triple system 𝔮 of 𝔤 with its ordinary triple Lie bracket [X, Y, Z] = [[X, Y], Z]. The subalgebra [𝔮, 𝔮] ⊕ 𝔮 ⊂ 𝔤 is called the standard-imbedding of the LTS 𝔮.

Theorem 1.1.2. The functor of evaluating the curvature tensor at the base point defines an equivalence of the categories of connected simply connected symmetric spaces with base point and the category of finite-dimensional real Lie triple systems.

Proof. [Lo69,Th.II.4.12]

We will only need a weaker version of Theorem 1.1.2. Two symmetric spaces M, M′ with base points o, o′ are called locally isomorphic if there exist neighbourhoods U, U′ of the base points and an affine diffeomorphism φ : U → U′ with φ(o) = o′. This is an equivalence relation, and we may speak of local equivalence classes or germs of symmetric spaces and of their morphisms.

Theorem 1.1.3. The category of germs of symmetric spaces is equivalent to the category of finite-dimensional real Lie triple systems.

Proof. We define the functor from Lie triple systems to germs of symmetric spaces by taking the local equivalence class of the connected simply connected space given by Theorem 1.1.2. Then the claim follows from this theorem.

If G and H are complex Lie groups and σ is a holomorphic involution of G, then M is a complex symmetric space, 𝔮 is a complex vector space and the associated Lie triple-product is a ℂ-trilinear map. A complex version of Theorem 1.1.3. holds.

The main point in the proof of Th. 1.1.2 is to show that every morphism of Lie triple systems extends to a morphism of the corresponding symmetric spaces. It is important to use here the correct notion of morphisms of symmetric spaces; the notion of (weakly) equivariant maps of symmetric spaces (cf. [Sa80, p.47]) is in general not the correct one, which is the cause of some problems in the presentation given in [Sa80]. The relation between morphisms of symmetric spaces and weakly equivariant maps (on the algebraic level) is dicussed in [Jac51].

1.2. Straight and twisted almost complex structures. An almost complex structure on a manifold M is a tensor field J of type (1, 1) (i.e., J = (J_{p})_{p∈M} where J_{p} is an endomorphism of the tangent space T_{p}M) such that J_{p}^{2} = − id_{T_{p}M} for all p ∈ M.

If M = G/H is a symmetric space, we say that J is invariant if it is invariant under the action of the group G. Then J := J_{o} is H -invariant in the sense that J commutes with all tangent maps T_{o}h, h ∈ H, and any J having this property can be extended uniquely to a G-invariant tensor field. Thus G-invariant almost complex structures correspond to H -invariant complex structures on T_{o}M. If we identify 𝔮 with T_{o}M as above, the first isotropy representation of the Lie algebra 𝔥 of H is just the usual action of 𝔥 on 𝔮 by the adjoint representation. If J is H -invariant on 𝔮 = T_{o}M, then it is 𝔥-invariant:

(I)  ∀D ∈ 𝔥 : D ∘ J = J ∘ D.

Since [𝔮, 𝔮] ⊂ 𝔥, (I) implies the weaker condition

(I′)  ∀X, Y, Z ∈ 𝔮 : [X, Y, JZ] = J[X, Y, Z].

If (I′) holds, we say that J is an invariant complex structure on the LTS 𝔮. Lie triple systems with invariant complex structure form a category whose morphisms are ℂ-linear LTShomomorphisms.

Proposition 1.2.1. The category of LTS with invariant complex structure is equivalent to the category of germs of symmetric spaces with invariant almost complex structure, morphisms being morphisms of symmetric spaces which are almost-holomorphic, i.e. compatible with the respective almost complex structures.

Proof. It is clear that the objects of both categories correspond to each other under the bijection from Th. 1.2.3. The differential at the base point of an almost-holomorphic morphism of symmetric spaces is clearly ℂ-linear and is thus a morphism of twisted complex LTS. Let us show that conversely a ℂ-linear LTS-homomorphism φ̇ induces an almost holomorphic morphism φ of germs of symmetric spaces. In fact, the differential of φ at the point o is ℂ-linear by assumption, and from the condition φ ∘ exp(X) = exp(φ̇X) ∘ φ for X ∈ 𝔮 ([Lo69, Th.II.4.6]) it follows that the differential of φ at the point p = exp(X) · o is ℂ-linear.

Definition 1.2.2. An invariant complex structure J on a LTS 𝔮 is called straight if

(S)  ∀X, Y, Z ∈ 𝔮 : [JX, Y, Z] = [X, JY, Z]

holds, and twisted if

(T)  ∀X, Y, Z ∈ 𝔮 : [JX, Y, Z] = −[X, JY, Z]

holds. A straight complex symmetric space M = G/H is a symmetric space with an invariant almost complex structure J such that R(J X, Y)Z = R(X, J Y)Z, and a twisted complex symmetric space is a symmetric space with an invariant almost complex structure such that R(J X, Y)Z = −R(X, J Y)Z, where R is the curvature tensor and X, Y, Z are vector fields on M.

Proposition 1.2.3. An invariant complex structure J on a LTS 𝔮 is straight if and only if the triple product is J -linear in all three variables.

Proof. We only have to show that a straight invariant complex structure is J -trilinear. By invariance, it is J -linear in the last variable. Using the the Jacobi-identity, (S) and (I′) yield

0 = [JX, Y, Z] + [JZ, X, Y] + J[Y, Z, X].

We now add up the three equations obtained by cyclic permutation of X, Y, Z, one of it multiplied by −1, and use again the Jacobi-identity to obtain

0 = 2[JX, Y, Z] − 2J[X, Y, Z],

whence J[X, Y, Z] = [JX, Y, Z] = [X, JY, Z], implying that the triple product is J -linear in the first two variables.

Proposition 1.2.4. If J is an invariant complex structure on a LTS 𝔮, then the following are equivalent:

(i) J is twisted. (ii) J is an automorphism of 𝔮, i.e.

(T′)  ∀X, Y, Z ∈ 𝔮 : [JX, JY, JZ] = J[X, Y, Z].

(iii) J is a derivation of 𝔮, i.e.

(T′′)  ∀X, Y, Z ∈ 𝔮 : J[X, Y, Z] = [JX, Y, Z] + [X, JY, Z] + [X, Y, JZ].

Proof. In view of the invariance, the equivalence of (i) and (iii) is clear. Using in addition that J^{2} = − id_{𝔮}, the equivalence of (i) and (ii) follows immediately.

Remark 1.2.5. Since the exponential of a derivation is an automorphism and J = e^{\frac{π}{2} J}, one gets another proof of (iii) ⇒ (ii).

1.3. Structure tensor and Hermitian Jordan triple products. Let M = G/H be a symmetric space with curvature tensor R and invariant almost complex structure J . For any p ∈ M, the space End(T_{p}M) of endomorphisms of the tangent space T_{p}M splits into a direct sum of complex-linear and complex-antilinear (w.r.t. J_{p}) elements. The endomorphism R(X, Y) of 𝔛(M) is by assumption complex-linear, but the endomorphism R(X, ·)Z in general decomposes into two terms. We are thus lead to define the structure tensor T of J by

T (X, Y)Z = \frac{1}{2}(R(X, Y)Z − J R(X, J^{−1} Y)Z) = \frac{1}{2}(R(X, Y)Z + J R(X, J Y)Z).

Sometimes we will write T (X, Y, Z) instead of T (X, Y)Z. Since R and J are G-invariant, T is a G-invariant tensor field of the same type as the curvature tensor. We write J for J_{o}. Then

T_{o}(X, Y)Z = \frac{1}{2}([X, Y, Z] − J[X, J^{−1}Y, Z]) = \frac{1}{2}([X, Y, Z] + J[X, JY, Z])

is an H -equivariant map ⊗^{3}𝔮 → 𝔮, called the structure tensor of J. Since T_{o} and R_{o} are, by invariance, equivalent to T and R, we will suppress the index o when there is no risk of confusion.

Proposition 1.3.1.

(i) The structure tensor vanishes if and only if J is straight. (ii) The Lie algebra 𝔤 acts as a Lie algebra of derivations of T, and 𝔥 acts as a Lie algebra of

derivations of T_{o}. (iii) For all vector fields X, Y, Z on M,

T (X, Y, Z) = T (Z, Y, X).

(iv) For all vector fields X, Y, Z on M,

T (X, Y, JZ) = −T (X, JY, Z) = T (JX, Y, Z) = J T (X, Y, Z).

(v) J is twisted if and only if for all vector fields X, Y, Z on M,

T (X, Y)Z − T (Y, X)Z = R(X, Y)Z,

or, equivalently, T_{o}(X, Y)Z − T_{o}(Y, X)Z = [X, Y, Z], X, Y, Z ∈ 𝔮. Proof. (i) is clear from 1.2.3, and (ii) is the infinitesimal version of the invariance under G, resp. H.

(iii): Using the Jacobi-identity, we get

\begin{aligned}2(T (X, Y, Z) − T (Z, Y, X)) & = [X, Y, Z] − [Z, Y, X] + J([X, JY, Z] − [Z, JY, X]) \\ & = −[Z, X, Y] − J[Z, X, JY] = 0. \end{aligned}

(iv): T (X, Y)JZ = J T (X, Y)Z since J is invariant;

T (X, JY)Z = \frac{1}{2}([X, JY, Z] − J[X, Y, Z]) = −J T (X, Y)Z;

T (JX, Y)Z = T (Z, Y)JZ = J T (Z, Y)X = J T (X, Y)Z,

using twice part (iii).

(v): The tensor

T (X, Y)Z − T (Y, X)Z − [X, Y, Z] = \frac{1}{2}J([X, JY, Z] − [Y, JX, Z]) = \frac{1}{2}J([X, JY, Z] + [JX, Y, Z])

clearly vanishes if and only if J is twisted.

Proposition 1.3.2. The structure tensor T of an invariant twisted almost complex structure J on a symmetric space satisfies the following identities: (JT1) T (X, Y, Z) = T (Z, Y, X) (JT2)

T (X, Y) · T (U, V, W) = T (T (X, Y)U, V, W ) − T (U, T (Y, X)V, W ) + T (U, V, T (X, Y)W ).

Proof. The identity (JT1) has been proved above (Prop. 1.3.1 (iii)). We now establish the property (JT2). Since [𝔮, 𝔮] ⊂ 𝔥, Prop. 1.3.1 (ii) implies that R_{o}(X, Y) is a derivation of T_{o}. If D is a derivation of T_{o}, then JD is a skew-derivation of T_{o} in the following sense: using 1.3.1 (iv) we get

\begin{aligned}JD T_{o}(X, Y, Z) & = J(T_{o}(DX, Y, Z) + T_{o}(X, DY, Z) + T_{o}(X, Y, DZ)) \\ & = T_{o}(JD X, Y, Z) − T_{o}(X, JD Y, Z) + T_{o}(X, Y, JD Z). \end{aligned}

Thus 2T_{o}(X, Y) ∈ End(𝔮) is the sum of the derivation R_{o}(X, Y) ∈ 𝔥 and the skew-derivation J R_{o}(X, JY) ∈ End(𝔮). If J is twisted, we obtain for the difference of these two elements

\frac{1}{2}(R_{o}(X, Y) − JR_{o}(X, JY)) = −T_{o}(Y, X).

Using this, we can write (dropping the indices o)

T (X, Y) · T (U, V, W) = \frac{1}{2}(R(X, Y) · T (U, V, W) + J R(X, JY) · T (U, V, W))

= \frac{1}{2}(T ((R(X, Y) + J R(X, JY))U, V, W )+

T (U, (R(X, Y) − J R(X, JY))V, W )+
T (U, V, (R(X, Y) + J R(X, JY))W ))

= T (T (X, Y)U, V, W ) − T (U, T (Y, X)V, W ) + T (U, V, T (X, Y)W ).

This is the identity (JT2). Since both sides are H -invariant, this identity can be read as an identity for the whole tensor field T and arbitrary vector fields.

Definition 1.3.3. A Jordan triple system (abbreviated JTS) is a vector space V together with a trilinear map T : V × V × V → V , (X, Y, Z) ↦ T (X, Y, Z) = T (X, Y)Z satisfying the identities (JT1) and (JT2). A Hermitian JTS is a JTS T together with a complex structure J on V such that

T (JX, Y, Z) = −T (X, JY, Z) = T (X, Y, JY) = JT (X, Y, Z)

holds for all X, Y, Z ∈ V . Morphisms of (Hermitian) JTS are (complex) linear maps compatible with the respective triple products.

The preceding proposition together with 1.3.1 (iv) shows that the structure tensor of an invariant twisted almost complex structure is a field of Hermitian Jordan triple products.

Lemma 1.3.4. Let T be a Jordan triple product defined on a vector space 𝔮. Then

R_{T} (X, Y)Z := T (X, Y)Z − T (Y, X)Z = T (X, Y)Z − T (Z, X)Y

defines a Lie triple product on 𝔮. Proof. Clearly R_{T} is antisymmetric in the first two variables and satisfies the Jacobi-identity (LT2). The identity (JT2) implies that R_{T} (X, Y) is a derivation of T and therefore also of R_{T} , whence (LT3).

We call R_{T} the LTS associated to the JTS T, and if R = R_{T} , then we say that T is a Jordan-extension of the LTS R. It is clear that JTS-homomorphisms induce homomorphisms of associated LTS’s. Therefore we call T ↦ R_{T} the (algebraic) Jordan-Lie functor. The Jordan-Lie functor is not bijective, but it does induce bijections of some important sub-categories.

Proposition 1.3.5. The Jordan-Lie functor induces a bijection between the category of Hermitian JTS’s and the category of twisted complex LTS’s. Its inverse is given by the formula

T (X, Y)Z := \frac{1}{2}(R(X, Y)Z − JR(X, J^{−1}Y)Z).

Proof. If T is a Hermitian JTS, then J is a twisted invariant complex structure for R_{T} :

\begin{aligned}R_{T} (X, Y)JZ & = T (X, Y, JZ) − T (Y, X, JZ) = JR_{T} (X, Y)Z; \\ R_{T} (JX, Y) & = T (JX, Y) − T (Y, JX) = −(T (X, JY) − T (JY, X) = −R_{T} (X, JY). \end{aligned}

Conversely, if J is a twisted invariant complex structure for R, then, as seen above, the given formula defines a Hermitian JTS T. Thus both functors are well-defined. It is easily checked that they are inverse to each other: if J is twisted invariant for R, then R(X, JY) is symmetric in X and Y , and antisymmetrizing T in the first two variables, we get back R. If (T, J) is Hermitian, then

\begin{aligned}R_{T} (X, Y)Z + JR_{T} (X, JY)Z & = T (X, Y, Z) − T (Y, X, Z) + J(T (X, JY, Z) − T (JY, X, Z)) \\ & = 2T (X, Y, Z). \end{aligned}

Finally, homomorphisms in both categories are the same since compatibility with R and J is equivalent to compatibility with T and J.

Theorem 1.3.6. The following three categories are equivalent:

(1) tcLTS: twisted complex Lie triple systems,

(2) hJTS: Hermitian Jordan triple systems,

(3) tcSS: germs of twisted complex symmetric spaces. The functor tcSS → tcLTS is given by evaluating curvature tensor and almost complex structure at the base point, and the functor tcSS → hJTS by evaluating structure tensor and almost complex structure at the base point. Proof. The equivalence of (1) and (2) has been established in the previous proposition. The equivalence of (1) and (3) follows immediately from Prop. 1.2.1.

Example 1.3.7. (The space of complex structures) Let

N := {I ∈ Gl(2n, ℝ)| I^{2} = −1_{2n}}

be the space of complex structures on V := ℝ^{2n} and choose a base point I_{o} = ({0 \atop −1_{n}} {1_{n} \atop 0} ) ∈ N. By linear algebra, the action of Gl(2n, ℝ) on N by g · I = gIg^{−1} is transitive, and

N = Gl(2n, ℝ)/ Gl(n, ℂ)

is a symmetric space. Let us show that it carries a natural twisted complex structure: the tangent space T_{I_{o}}N is naturally identified with

𝔮 := {X ∈ M(2n, ℝ)|I_{o}X = −XI_{o}}

equipped with the Lie triple product

[X, Y, Z] = [[X, Y], Z] = XY Z + ZY X − ZXY − Y XZ

induced from the ambient Lie algebra 𝔤𝔩(2n, ℝ). By a simple calculation one verifies that

J : 𝔮 → 𝔮, X ↦ I_{o}X

is an invariant twisted complex structure on 𝔮. The corresponding structure tensor is given by

T (X, Y, Z) = \frac{1}{2}([X, Y, Z] + I_{o}[X, I_{o}Y, Z]) = XY Z + ZY X.

1.4. Invariant polarizations and Jordan pairs. Most of what has been said so far remains valid for tensor fields having the property J^{2} = id instead of J^{2} = − id.

Definition 1.4.1. An invariant polarization on a LTS (𝔮, R) is an endomorphism J such that J^{2} = id_{𝔮} and [X, Y, JZ] = J[X, Y, Z]. We call J straight if [JX, Y, Z] = [X, JY, Z] and twisted if [JX, Y, Z] = −[X, JY, Z] (X, Y, Z ∈ 𝔮). A polarized LTS is an LTS together with a twisted polarization. If the ±1-eigenspaces of J have the same dimension, then J is also called a paracomplex structure.

As in the proof of Prop. 1.2.3 it is seen that straight polarized LTS are precisely the J - trilinear LTS, and these are nothing but direct products of two LTS. Analogously to Prop. 1.2.4, an invariant polarization is seen to be twisted if and only if it is a derivation. The structure tensor of J is defined by

T (X, Y)Z = \frac{1}{2}([X, Y, Z] − J[X, J^{−1}Y, Z]) = \frac{1}{2}([X, Y, Z] − J[X, JY, Z]).

Then Propositions 1.3.1 and 1.3.2 and their proofs carry over to the polarized case without any changes, except that we have to replace “Hermitian Jordan triple systems” by “polarized Jordan triple systems” which are defined as Hermitian ones, but using a polarization instead of a complex structure. From a Jordan theoretic point of view, the important new feature is that polarized JTS are equivalent to Jordan pairs; cf. [Lo75] for definition.

Theorem 1.4.2. The following four categories are equivalent:

(1) pLTS: polarized Lie triple systems,

(2) pJTS: polarized Jordan triple systems,

(3) JP: Jordan pairs,

(4) tpSS: (germs of) symmetric spaces with invariant twisted polarization. The functor tpSS → pLTS is given by evaluating curvature tensor and involutive structure at the base point, and the functor tpSS → pJTS by evaluating structure tensor and involutive structure at the base point.

Proof. The equivalence of the categories (1), (2) and (4) is proved as in Th.1.3.6, and the equivalence of (2) and (3) is proved in [Lo75, p.10].

Example 1.4.3. (Space of polarizations) Let I_{p,q} := ({1_{p} \atop 0} {0 \atop −1_{q}} ) be the base point in the space

Y := {I ∈ Gl(p + q, ℝ)| I^{2} = 1_{p+q}, sgn(I) = (p, q)}

of polarizations on V = ℝ^{p+q} with signature (p, q). By calculations similar to those given in Example 1.3.5, we see that

Y = Gl(p + q, ℝ)/(Gl(p, ℝ) × Gl(q, ℝ))

is a symmetric space with invariant twisted para-complex structure. This structure is given by the map J : X ↦ I_{p,q}X on 𝔮 = \{({0 \atop B^{t}} {A \atop 0})| A, B ∈ M(p, q; ℝ)\}, and both eigenspaces are of equal dimension, naturally isomorphic to M(p, q; ℝ). All formulas remain valid if one replaces ℝ by ℂ. The space Y can be interpreted as the open dense subset of elements (E, F) in the direct product Gr_{p,p+q} × Gr_{q,p+q} of Grassmannians such that E ∩ F = 0.

2. Complexifications and symmetric spaces with twist

2.1. Real forms and complexifications.

Definition 2.1.1.

(1) A global complexification of a symmetric space M is a symmetric space N together with an invariant almost complex structure J and a conjugation τ (i.e. an almost antiholomorphic automorphism with τ^{2} = id_{N}) such that τ(o) = o and the symmetric space M is isomorphic to the τ -fixed space N^{τ} (or to a union of connected components of N^{τ}). In this situation we say that M (or τ) is a real form of N.

(2) A complexification of a germ M^{o} of a symmetric space is defined as a germ N^{o} of a symmetric space with invariant almost complex structure and germ τ of a conjugation such that M^{o} ≅ (N^{o})^{τ}, and again M^{o} is called a real form of N^{o}. The complexification is called straight, resp. twisted if N^{o} is straight complex, resp. twisted complex.

(3) A morphism of complexifications is a morphism of germs of symmetric spaces with an extension to an almost-complex morphism of the complexifications.

(4) Replacing complex structures by para-complex structures, we get the corresponding notions of para-complexifications.

Definition 2.1.2.

(1) A complexification of a LTS (𝔮, R) is a LTS 𝔭 with an invariant complex structure J and a conjugation τ (i.e. a complex antilinear automorphism with τ^{2} = id_{𝔭}) such that the LTS (𝔮, R) is isomorphic to the τ -fixed LTS 𝔭^{τ}. The complexification is called straight, resp. twisted if 𝔭 is straight complex, resp. twisted complex.

(2) A morphism of complexifications of LTS is a morphism of LTS together with a ℂ-linear extension to a morphism of the complexified LTS.

(3) Replacing complex structures by para-complex structures, we get the corresponding notions of para-complexifications.

Proposition 2.1.3. The category of local (para-) complexifications of germs of symmetric spaces is equivalent to the category of (para-) complexifications of Lie triple systems, and the subcategories of straight resp. twisted (para-) complexifications correspond to each other under this equivalence.

Proof. This is proved in the same way as Prop. 1.2.1.

Proposition 2.1.4. Any Lie triple product R : 𝔮 × 𝔮 × 𝔮 → 𝔮 has a unique straigt complexification, given by the ℂ-trilinear extension

R_{ℂ} : 𝔮_{ℂ} × 𝔮_{ℂ} × 𝔮_{ℂ} → 𝔮_{ℂ}.

Proof. Because of Prop. 1.2.3, a straight complexification is necessarily given by ℂ-trilinear extension. In order to verify that this indeed defines a LTS, we express the identities (LT1), (LT2) and (LT3) by commutative diagrams of linear maps of some tensor products and apply to these the ordinary complexification functor for linear maps.

Note that, if 𝔮 is the −1-eigenspace of σ̇ in 𝔤, then (𝔮_{ℂ}, R_{ℂ}) is the −1-eigenspace of the ℂ-linear extension σ̇_{ℂ} of σ onto the complexified Lie algebra 𝔤_{ℂ}. Thus, if M^{o} is the germ of the symmetric space G/H, then the germ M_{ℂ}^{o} belonging to (𝔮_{ℂ}, R_{ℂ}) is the germ of M_{ℂ} := G_{ℂ}/H_{ℂ}, where G_{ℂ} is the connected simply connected group with Lie algebra 𝔤_{ℂ} and H_{ℂ} = G_{ℂ}^{σ} the group fixed under the unique holomorphic involution whose differential at the origin is σ̇_{ℂ}. It is clear, however, that M_{ℂ} is in general not a global complexification of M = G/H. Note that straight para-complexifications always exist globally: they are just direct products of a space with itself.

2.2. Twisted complexifications and Jordan extensions.

Lemma 2.2.1. Let τ be a (para-) real form of a LTS (𝔮, R) with (para-) complex structure J. Then τ is an automorphism of the structure tensor T (X, Y)Z = \frac{1}{2}([X, Y, Z] − J[X, J^{−1}Y, Z]). In particular, the restriction T |_{𝔮^{τ}} of T to 𝔮^{τ} is well-defined. Proof.

τT (X, Y, Z) = \frac{1}{2}(τ[X, Y, Z] − τ(J [X, J^{−1}Y, Z]))

= \frac{1}{2}([τX, τY, τZ] − J [τX, J^{−1}τY, τZ])

= T (τX, τY, τZ),

using twice that J anticommutes with τ.

Theorem 2.2.2. Let (𝔮, R) be a Lie triple system. The following objects are in one-to-one correspondence:

(1) twisted complexifications of R,

(2) twisted para-complexifications of R,

(3) Jordan-extensions T of R (i.e. Jordan triple systems T such that T (X, Y) − T (Y, X) = R(X, Y)).

Proof. The correspondences (1) → (3) and (2) → (3) are given by restricting the structure tensor of the complexified space to the real form; the preceding lemma shows that this is welldefined. Moreover, the structure tensor of a twisted (para-) complex space is, according to Prop. 1.3.1 (v), a Jordan-extension of the curvature tensor, and this relation carries over to real forms. In order to give the inverse constructions we have to establish first some basic results on Jordan triple systems.

Lemma 2.2.3. Let T be a Jordan triple product on a vector space V and α an endomorphism of V with the property

(∗)  ∀x, y, z ∈ V : T (αx, y, αz) = αT (x, αy, z).

Then the formula

T^{α}(x, y, z) := T (x, αy, z)

defines a Jordan triple product T^{α} on V . Proof. The identity (JT2) for T with v and y replaced by αv and αy, respectively, yields

T (T (u, αv)x, αy, z) − T (x, T (αv, u)αy, z) + T (x, αy, T (u, αv)z) = T (u, αv)T (x, αy, z).

We apply (∗) to the middle triple product in the middle term and obtain

T (T (u, αv)x, αy, z) − T (x, αT (v, αu)y, z) + T (x, αy, T (u, αv)z) = T (u, αv)T (x, αy, z).

This is precisely the identity (JT2) for T^{α}. Since T^{α} clearly satisfies (JT1), it is a Jordan triple product.

Note that any involutive automorphism of T satisfies (∗). An important application of the previous lemma is the existence of a canonical (para-) Hermitian complexification in the category of JTS’s:

Theorem 2.2.5. Let (V, T) be a Jordan triple system. We imbed V into Ṽ := V ⊕ V as first factor and equip Ṽ with its canonical (para-) complex structure J(x, y) = (−y, x) (resp. I(x, y) = (y, x)). Then there exist four Jordan triple systems T_{ℂ}, T_{2} , T_{hℂ} and T_{phℂ} on Ṽ extending T and uniquely determined by the following properties:

(i) T_{ℂ} is J -linear in all three variables. (ii) T_{2} is I -linear in all three variables. (iii) T_{hℂ} is Hermitian with respect to J. (iv) T_{phℂ} is polarized with respect to I.

Proof. Uniqueness is immediate from the definitions. We now prove existence: (i) is proved in the same way as Prop. 2.1.4. In order to prove (iii), one verifies that complex conjugation τ(x, y) = (x, −y) is an automorphism of T_{ℂ}. Since it is an involution, it satisfies condition (∗) of the lemma, and therefore

T_{hℂ} := (T_{ℂ})^{τ}

again is a JTS. It clearly is Hermitian and extends T. We now prove (ii): let W^{±} be the eigenspaces of I; then W^{+} is the diagonal and W^{−} the antidiagonal, and V → W^{±}, v ↦ \frac{1}{2}(v, ±v) are vectorspace isomorphisms which we use as identifications in order to define the JTS T_{2} := T ⊕ T on W^{+} ⊕ W^{−}. It extends T and is I -trilinear. This proves (ii). The para-conjugation τ(x, y) = (x, −y) is an involution of T_{2} = T ⊕ T; in fact, in the coordinates W^{+} ⊕ W^{−} it is just exchange of the two factors. The lemma implies that T_{phℂ} := (T_{2})^{τ} is a JTS. It clearly is polarized by I and extends T, thus (iv) is proved.

Now we finish the proof of Th. 2.2.2.by constructing the correspondence (3) → (1): let

R̃(X, Y) := R_{T_{hℂ}}(X, Y) = T_{hℂ}(X, Y) − T_{hℂ}(Y, X).

Then R̃ is twisted since T_{hℂ} is Hermitian (Prop. 1.3.4). Further, restriction of R̃ to 𝔮 yields R since the restriction of T_{hℂ} to 𝔮 is T, and R_{T} = R by assumption. Therefore R̃ is a twisted complexification of R.

The correspondence (3) → (2) is constructed in the same way, using this time the para-Hermitian complexification T_{phℂ} of T. Finally, it is clear that these constructions are inverse to the ones given previously.

The theorem implies that the image of the Jordan-Lie functor T ↦ R_{T} is given by the LTS admitting a twisted (para-) complexification.

Definition 2.2.5. A (germ of a) symmetric space with twist is a (germ of a) symmetric space together with a Jordan-extension T of the curvature tensor R (i.e. an invariant field of Jordan triple products such that T (X, Y) − T (Y, X) = R(X, Y)). Morphisms of (germs of) symmetric spaces with twist are morphisms of symmetric spaces which are compatible with the respective Jordan-extensions.

Theorem 2.2.6. The following categories are equivalent:

(1) (real finite dimensional) Jordan triple systems.

(2) germs of symmetric spaces with twist.

(3) germs of symmetric spaces with local twisted complexification.

(4) germs of symmetric spaces with local twisted para-complexification.

Proof. The bijection between the objects of the respective categories is given by Th. 2.2.2 in combination with Prop. 2.1.3. It remains to be shown that all correspondences are functorial.

We show first that the constructions from Th. 2.2.4 are functorial. If φ : V → V′ is a homomorphism of JTS T and T′, then we complexify the commutative diagram

V × V × V →^{T} V
×^{3}φ ↓  ↓ φ
V′ × V′ × V′ →^{T′} V′,

and see that φ_{ℂ} is a homomorphism from T_{ℂ} to T′_{ℂ}. From this it follows that

φ_{ℂ}T_{hℂ}(u, v, w) = φ_{ℂ}T_{ℂ}(u, \overline{v}, w) = T′_{ℂ}(φ_{ℂ}u, φ_{ℂ}\overline{v}, φ_{ℂ}w) = T_{hℂ}(φ_{ℂ}u, φ_{ℂ}v, φ_{ℂ}w)

since φ\overline{v} = \overline{φv}. Thus φ_{ℂ} is a homomorphism from T_{hℂ} to T′_{hℂ}. Similarly for the paracomplexifications.

Next we use the fact that the Jordan-Lie functor T ↦ R_{T} is functorial, i.e. we obtain homomorphisms of the twisted (para-) complex LTS equivalent to T_{hℂ} resp. T_{phℂ}.

Applying now Prop. 2.1.5, we can lift these homomorphisms on a local level, i.e. we have shown that (1) → (3) and (1) → (4) are functorial.

It is clear that restriction to a real form is functorial; thus (3) → (2) and (4) → (2) are functorial. It is also clear that evaluating at the base point is a functorial construction; thus (2) → (1) is functorial, and the theorem is proved.

Definition 2.2.7. The geometric Jordan-Lie functor is the forgetful-functor from the category of (germs of) symmetric spaces with twist to the category of (germs of) symmetric spaces.

The algebraic Jordan-Lie functor T ↦ R_{T} is the infinitesimal version of the geometric Jordan-Lie functor:

JTS ⟷ germs of symmetric spaces with twist
↓  ↓
LTS ⟷  germs of symmetric spaces.

Here the top line is the equivalence of categories from Th. 2.2.6; the bottom line is the Lie functor for symmetric spaces (Th. 1.1.3); the right column is the geometric Jordan-Lie functor and the left column is the algebraic Jordan-Lie functor.

Remark 2.2.8. The Makarevič spaces introduced in [Be96b,c] are examples of symmetric spaces with twist: in fact, according to [Be96b, 2.3.3], the Makarevič spaces X^{(α)} defined there admit a global twisted or “Hermitian” complexification. The fact that they also admit globally a twisted para-complexification has been remarked in [Be96c, 2.1, Ex.4]. The classification [Ma73] of irreducible Makarevič spaces essentially coincides with the classification of irreducible symmetric spaces with twist (Chapter III); however, the close relation with Jordan triple systems and twisted complexifications was not remarked in [Ma73].

4.3. Complexification functors and complexification diagrams. We write ℂ, 2, hℂ and phℂ for the functors JTS → JTS defined in Th. 2.2.4; we call them the functors of complexification, doubling, Hermitian complexification and twisted para-complexification in the category of Jordan triple systems.

Theorem 2.3.1. The category of germs of symmetric spaces with twist is stable under the functors ℂ, 2, hℂ and phℂ. Any two of the four functors commute, i.e. the opposite composition yields functors isomorphic to those obtained by the usual composition. Further,

(i) ℂ ∘ ℂ = 2 ∘ ℂ, hℂ ∘ hℂ = 2 ∘ hℂ, phℂ ∘ phℂ = 2 ∘ phℂ, (ii) ℂ ∘ hℂ ≅ ℂ ∘ phℂ ≅ hℂ ∘ phℂ, (iii) phℂ ∘ hℂ ∘ ℂ ≅ 2 ∘ hℂ ∘ ℂ.

Proof. Since the functors ℂ, 2, hℂ and phℂ are functors in the category of Jordan triple systems, they are by Th. 2.2.6 also functors in the category of germs of symmetric spaces with twist.

The “commutation relations” (i)–(iii) are verified by elementary calculations, using the “multiplication table” for the four kinds of structures: the product of two commuting complex structures is para-complex; the product of two commuting twisted structures is straigt, and so on.

Note that exactly one interesting new functor arises by composition: it is a sort of double complexification-functor ℂ ∘ hℂ ≅ ℂ ∘ phℂ ≅ hℂ ∘ phℂ in the category of symmetric spaces with twist. It assigns to a space another one having four times its real dimension and carrying two commuting almost complex structures. We thus have four non-trivial complexification-functors ℂ, hℂ, phℂ and ℂ ∘ hℂ which are related by (i) – (iii). The situation may be visualized by the complexification diagram of a symmetric space with twist T:

↗ M_{ℂ} ↘
M → M_{hℂ} → (M_{hℂ})_{ℂ}
↘ M_{phℂ} ↗

Example 2.3.2. Using the notation of Example 1.3.7, let s(z) = \overline{z} be the standard complex conjugation of ℝ^{2n} = ℂ^{n}. Then τ(I) := −sIs defines a real form of N fixing the base point I_{o}. The group commuting with s = ({1 \atop 0} {0 \atop −1}) is isomorphic to Gl(n, ℝ) × Gl(n, ℝ), and

N^{τ} = (Gl(n, ℝ) × Gl(n, ℝ))/ Gl(n, ℝ)

is the general linear group, viewed as a symmetric space. Thus, with the Jordan-extension T (X, Y, Z) = XY Z + ZY X of 𝔤𝔩(n, ℝ), the complexification diagram of M = Gl(n, ℝ) reads

↗  Gl(n, ℂ)  ↘
Gl(n, ℝ) →  Gl(2n, ℝ)/ Gl(n, ℂ)  → Gl(2n, ℂ)/(Gl(n, ℂ) × Gl(n, ℂ)).
↘ Gl(2n, ℝ)/(Gl(n, ℝ) × Gl(n, ℝ)) ↗

3. Non-degenerate spaces

3.1. Pseudo-Hermitian and para-Hermitian symmetric spaces. We content ourselves with explaining the relation of these kinds of spaces with the concepts introduced previously and refer to [B57], [Koh65], [KanKo85], [FG96] and the references given there for a more detailed study.

Definition 3.1.1. A pseudo-Hermitian symmetric space (M, J , g) is a symmetric space M = G/H with an invariant pseudo-metric tensor field g and an invariant almost complex structure J such that g(J X, Y) = −g(X, J Y) for all vector fields X, Y on M. A para-Hermitian symmetric space is defined similarly, with J^{2} = − id replaced by J^{2} = id.

Proposition 3.1.2.

(1) A pseudo-Hermitian symmetric space is twisted complex, and a para-Hermitian symmetric space is twisted para-complex.

(2) A semisimple twisted complex symmetric space is pseudo-Hermitian symmetric, and a semisimple twisted polarized symmetric space is para-Hermitian.

Proof. (1) (Cf. [Koh65, Th.5]) According to [Hel62,p.68], the relation g(R(X, Y)V, W) = g(R(V, W)X, Y) holds for an invariant pseudo-metric tensor field g and the curvature R. Using that J is antisymmetric for g and invariant for R, we deduce that

g(R(J X, Y)V, W) = g(R(V, W) J X, Y) = −g(R(V, W)X, J Y) = −g(R(X, J Y)V, W),

and therefore R(J X, Y) = −R(X, J Y) since g is non-degenerate.

(2): If M is semisimple, the Ricci-form g(X, Y) = tr R(X, ·, Y) is non-degenerate and symmetric, and g(J X, Y) = tr R(J X, ·, Y) = − tr(R(X, ·, Y) ∘ J) = − tr R(X, ·, J Y) = −g(X, J Y).

Theorem 3.1.3. Let M = G/H be an irreducible symmetric space. Then an invariant almost complex structure is straight or twisted.

Proof. This follows from [Koh65, Th.8].

Proposition 3.1.4. Every semisimple pseudo-Hermitian symmetric space has a real form.

Proof. By the general theory of Cartan involutions, the (straight) complexification of a semisimple symmetric space M has a non-compact Riemannian real form M^{r}. If M is pseudo-Hermitian, then M^{r} is Hermitian and has, according to [Sa80, Lemma 8.4], a real form. The restriction of the complexification of this real form to M is a real form of M.

3.2. Non-degenerate symmetric spaces with twist.

Definition 3.2.1. A bilinear form (·, ·) is called invariant w.r.t. a JTS T if

(T (X, Y)U, W) = (U, T (Y, X)W)

holds for all X, Y, U, W ∈ V .

Theorem 3.2.2. Under the equivalence given by Theorem 2.2.6, real forms of pseudo-Hermitian symmetric spaces correspond to Jordan triple systems with a non-degenerate invariant symmetric bilinear form.

Proof. If (M, J , g) is a pseudo-Hermitian symmetric space, then

g(T (X, Y)U, V) = \frac{1}{2}(g(R(X, Y)U, V) + g(J R(X, J Y)U, V))

= \frac{1}{2}(−g(U, R(X, Y)V) + g(U, J R(X, J Y)V)

= g(U, T (Y, X)V),

thus g is invariant for T. This relation also holds when we restrict g and T to a real form (of course, it is understood that conjugations are isometries w.r.t g), and the restriction is again non-degenerate.

Conversely, given an invariant form b on V , we let b̃ be the real part of the Hermitian extension of b onto V_{ℂ} × V_{ℂ}; it is characterized by the conditions b̃(X, JY) = 0, b̃(JX, JY) = b̃(X, Y) (X, Y ∈ V ). Then J is skew-symmetric for b̃, and the invariance property for b ensures that the operators R_{T_{hℂ}}(X, Y) are skew-symmetric for b̃. Therefore b̃ defines an invariant tensor field on M_{hℂ} which turns M_{hℂ} into a pseudo-Hermitian symmetric space of which M is a real form.

Definition 3.2.3. A JTS T is called non-degenerate if the trace-form

b(X, Y) := Tr T (X, Y)

is non-degenerate.

It is easily verified that the trace form is invariant and that it is symmetric if it is nondegenerate (cf. [Sa80, Section I.6]).

Theorem 3.2.4. Under the equivalence given by Theorem 2.2.6, real forms of semisimple pseudo-Hermitian symmetric spaces correspond to non-degenerate Jordan triple systems. Proof. If T (X, Y) = \frac{1}{2}(R(X, ·, Y) + J ∘R(X, ·, Y) ∘ J^{−1} is the structure tensor of a twisted complex symmetric space, then

b(X, Y) = Tr R(X, ·, Y)

is just the Ricci-tensor. Thus non-degenerate Hermitian JTS correspond to semisimple pseudo-Hermitian symmetric spaces. One proves easily that a real JTS T is non-degenerate if and only if its Hermitian complexification T_{hℂ} is non-degenerate, and the claim follows. (Note that one cannot conclude that the Ricci-tensor of R_{T} is the trace form of T.)

Ideals in Jordan- or Lie triple systems are kernels of homomorphisms, and a triple system is called simple if it has no proper ideals.

Proposition 3.2.5. Let T be a real JTS. The following are equivalent:

(1) T is a simple JTS.

(2) R_{T} or R_{T_{hℂ}} is a simple LTS.

(3) Either R_{T} is a simple twisted complex JTS or R_{T_{hℂ}} is a simple LTS. Proof. Clearly (3) implies (2). In order to prove that (2) implies (1), note that an ideal of T is also an ideal of R_{T} ; therefore T is simple if R_{T} is simple. If R_{T_{hℂ}} is simple, then T_{hℂ} is simple, and this implies that T is simple.

Before proving that (1) implies (3), let us remark that a Hermitian JTS T is simple if and only if R_{T} is simple. This is an immediate consequence of the equivalence of categories Prop. 1.3.5 and the fact that J is in the simple case always an inner derivation.

Now let T be a simple JTS and assume that R_{T_{hℂ}} is not simple; then by the preceding remark T_{hℂ} is not simple. We have to show that then R_{T} is simple. By assumption there exists a proper T_{hℂ} -ideal W ⊂ V_{ℂ}. Using that (V, T) is simple, one deduces that (V_{ℂ}, T_{hℂ}) decomposes as a direct sum W ⊕ τ(W) (where τ is the conjugation w.r.t. V ). One deduces that T must have been already a Hermitian JTS, isomorphic to W. Again by the above remark, for such systems simplicity of T and R_{T} is equivalent; thus R_{T} is simple.

Lifted to a local level, the preceding proposition says that simple JTS correspond to germs (M, T) of symmetric spaces with twist such that M or M_{hℂ} is simple, and more precisely, either M is a simple twisted complex symmetric space or M_{hℂ} is simple.

The fact that M_{hℂ} is simple does not imply that M is simple: for example, the twisted complexification of Gl(n, ℝ) constructed in Ex. 2.3.2 is a simple symmetric space, although Gl(n, ℝ) has a one-dimensional center. For the sake of completeness we quote the following result due to E. Neher ([N85, Th.1.11]) which classifies the cases where T is simple, but R_{T} is not simple:

Theorem 3.2.6. If T is a simple real JTS, then either R_{T} is simple or

(1) R_{T} is the direct sum of a simple ideal and a one-dimensional center; this case appears precisely if T or −T is associated to a simple Jordan algebra via T (x, y, z) = x(yz) − y(xz) + (xy)z; or (2) R_{T} is the direct sum of two simple ideals; this case appears precisely in the cases of a JTS

associated to a quadratic form and a non-trivial reflection (Ch. 4, Table 4.2.4, Type 4.a and Table 4.2.5, Type 4.a).

4. Classification

The purpose of this chapter is to give a complete list of simple non-exceptional symmetric spaces with twist and to specify for each such space structure tensor and complexification diagram. Since there are nearly hundred series of simple spaces, we will not write down all complexification diagrams as in Example 2.3.2, but we organize our list in the following way:

– Table 4.2.1: first we list spaces having invariant structures of all kinds: they are para-Hermitian, straight complex and pseudo-Hermitian. They have been classified in various ways, cf. [KanKo85, p.97] for a Lie theoretic approach and [Lo75] for a Jordan theoretic approach. The complexification diagram of such a space is rather trivial: it contains only direct products.

– Table 4.2.2: second we list straight real forms of the spaces from Table 4.2.1, namely para-Hermitian spaces. As for 4.2.1, their classification is well known. For convenience of the reader, we include in Tables 4.2.2 and 4.2.1 also the exceptional cases; cf. [KanKo85]. The complexification diagrams are easy to calculate: straight and Hermitian complexification are isomorphic, and the double complexification reduces to a direct product of straight complexifications.

– Table 4.2.3: third we list para-real forms of spaces from Table 4.2.1 which are pseudo-Hermitian symmetric spaces. As above, their complexification diagram is easily calculated since now twisted para- and straight complexification are isomorphic.

– Table 4.2.4: fourth we list para-real forms of spaces from Table 4.2.1 which are complex symmetric spaces. Again, the complexification diagram is easily determined.

– Table 4.2.5: finally we list para-real forms of spaces from Table 4.2.2 (on the left in the examples). This is the “generic” case, having non-trivial complexification diagrams which have to be calculated case by case.

The para-real forms appearing in the last three tables are determined by using the following lemma:

Lemma 4.0. For a given JTS T, all para-real forms of T_{phℂ} are of the form

T^{α}(X, Y, Z) = T (X, αY, Z)

for a suitable α satisfying condition (∗) of Lemma 2.2.3.

Proof. If ν is a para-conjugation of T_{phℂ}, we write ν = τβ where τ is the standardconjugation. Then, since β ∘ J = J ∘ β, β preserves the eigenspaces of J and can be written in the form β = (α_{+}, α_{−}). The condition id = ν^{2} = τβτβ reads α_{+} = α_{−}^{−1} because τ exchanges the eigenspaces. Let α := α_{+} . Then a short calculation shows that the condition that (α, α^{−1}) be an automorphism of T_{phℂ} is equivalent to condition (∗) of Lemma 2.2.3.

We call the variety of invertible endomorphisms α satisfying condition (∗) of Lemma 2.2.3 the structure variety. It can be shown that the isomorphism class of T^{α} depends only on the connected component of the structure variety belonging to α (cf. [Be96b, Lemma 2.2.11]). Thus our strategy is to determine representatives of the connected component of the structure variety. For JTS belonging to simple Jordan algebras this has been carried out (although stated without proof) by B.O. Makarevič ([Ma73]). However, comparison with the classification of simple real JTS by E. Neher ([N80], [N81]) shows that his list is (up to some minor errors) complete. The method used by E. Neher is different from the one explained above.

4.1. Algebraic part: classification of simple Jordan triple systems

4.1.0. Notation. The space of p × q -matrices over 𝔽 = ℝ, ℂ or ℍ is denoted by M(p, q; 𝔽), and we let M(n, 𝔽) := M(n, n; 𝔽). If A is an invertible n × n-matrix over 𝔽 and ε an anti-automorphism of 𝔽 we denote by

Herm(A, ε, 𝔽) := {X ∈ M(n, 𝔽)| A^{−1}X^{t}A = εXε^{−1}}

the corresponding space of Hermitian matrices. The corresponding space of skew-Hermitian matrices is denoted by Aherm(A, ε, 𝔽). If 𝔽 = ℝ or ℂ and ε = id_{𝔽}, one writes instead of Herm and Aherm also Sym and Asym, and if 𝔽 = ℂ or ℍ and ε(w) = \overline{w} is the standardconjugation, one drops the parameter ε in the notation. A second conjugation of ℍ is given by the composition φ of the standard conjugation with conjugation by the quaternion j. Then Herm(I, φ, ℍ) and Aherm(I, ℍ) are isomorphic (cf. [Be96b,1.2.1]). If A = I_{n}, we just note the rank n instead of A, and if A = I_{p,q} = ({1_{p} \atop 0} {0 \atop −1_{q}} ), we note the signature (p, q) instead of A. If not otherwise specified, p, q, m, k, l are integers such that n = 2m, p + q = n (p > q), k + l = p. We frequently use the matrix F := F_{m} := ({0 \atop −1_{m}} {1_{m} \atop 0} ).

4.1.1. Classification of simple complex Jordan pairs (cf. [Lo77,4.11]) These are para-complexifications of complex Jordan triple systems T defined on a complex vector space V_{ℂ}. We specify V_{ℂ} and a (non-unique, but natural) choice for T. For type 4 we use the notation xy = (x|e)y + (y|e)x − (x|y)e, where e is the first canonical basis vector in ℂ^{n} and (u|v) = ∑_{i} u_{i}v_{i}.

underlying space V_{ℂ}  Jordan triple product on V_{ℂ}
1. M(n, ℂ)  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)
2. Sym(n, ℂ)  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)
3. Sym(F, ℂ) (≅ Asym(2m, ℂ))  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)
4. ℂ^{n}  T (x, y, z) = x(yz) − y(xz) + (xy)z
5. Herm(3, 𝕆_{ℂ})  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)
6. M(p, q; ℂ), p > q  T (X, Y, Z) = \frac{1}{2}(XY^{t}Z + ZY^{t}X)
7. Asym(2m + 1, ℂ)  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)
8. M(1, 2; 𝕆_{ℂ})  T (X, Y, Z) = \frac{1}{2}(XY Z + ZY X)

The first five pairs are of the first kind: they have invertible elements and thus come from Jordan algebras. The Jordan algebra structure is given by X · Y = \frac{1}{2}(XY + Y X) in cases 1,2,3 and 5 and by xy in case 4. The other pairs do not have invertible elements, but we have natural embeddings into Jordan algebras: 7 → 1 is clear; 6 → 2 is of the form X ↦ ({0 \atop X} {X^{t} \atop 0} ) and 8 →

5 of the form X ↦ ({0 \atop X} \begin{matrix}\overline{X} & ^{t} \\ 0 \end{matrix} ).

4.1.2. Classification of real Jordan pairs. Real forms of the above complex Jordan pairs are given by conjugations τ of the complex JTS (V_{ℂ}, T) specified above. We indicate the space and the corresponding conjugation τ; the JTS- and with it the Jordan pair structure is given by restricting the triple product from the corresponding complex space.

real form V = V_{ℂ}^{τ}  conjugation τ of V_{ℂ}
1.1. Herm(n, ℂ)  τ(Z) = \overline{Z}^{t}
1.2. M(n, ℝ)  τ(Z) = \overline{Z}
1.3. n = 2m: M(m, ℍ)  τ(Z) = F \overline{Z}F^{−1}
2.1. Sym(n, R)  τ(Z) = \overline{Z}
2.2. n = 2m: Herm(m, φ, ℍ)  τ(Z) = F \overline{Z}F^{−1}
3.1. Herm(m, ℍ)  τ(Z) = \overline{Z}^{t} = F \overline{Z}F^{−1}
3.2. Sym(F, ℝ) (≅ Asym(2m, ℝ))  τ(Z) = \overline{Z}
4.1. ℝ^{p} × ℝ^{q}, p + q = n, p ≥ 1  τ(z) = \overline{x} − \overline{y} (z = x + y, x ∈ ℂ^{p}, y ∈ ℂ^{q})

5.1. Herm(3, 𝕆)  τ(Z) = \overline{Z}
5.2. Herm(3, 𝕆_{s})  τ(Z) = Z̃
6.1. M(p, q; ℝ), p > q  τ(Z) = \overline{Z}
6.2. p = 2r, q = 2s: M(r, s; ℍ)  τ(Z) = F_{r}\overline{Z}F_{s}^{−1}
7.1. Asym(2m + 1, ℝ)  τ(Z) = \overline{Z}
8.1. M(1, 2; 𝕆)  τ(Z) = \overline{Z}
8.2. M(1, 2; 𝕆_{s})  τ(Z) = Z̃

Construction of Jordan triple systems from Jordan pairs. To each polarized JTS T_{phℂ} with T from one of the tables above we determine all para-real forms T^{α} by giving representatives α of the structure variety of T as explained after Lemma 4.0.

c-duality. On the level of LTS and JTS, the well-known duality betweeen compact and noncompact symmetric spaces generalizes to the duality given by the functor R ↦ −R, T ↦ −T. It is clear that T^{−α} = −T^{α}. We just list one of the endomorphisms α and −α, but in general there is no canonical choice between them. However, if T comes from a Jordan algebra, in many cases it is possible to choose α to be a Jordan algebra automorphism; then −α is just an automorphism of triple systems, but not of algebras, and the choice becomes canonical. If T^{α} is straight complex or twisted polarized, then T^{α} and −T^{α} are always isomorphic, where the isomorphism is given by the invariant (para-) complex structure J.

4.1.3. Complex-antilinear modifications of complex Jordan pairs.

1. A. α(X) = \overline{X}

1. B. α(X) = I_{p,q}\overline{X}^{t}I_{u,v}

2. A. α(X) = I_{p,q}\overline{X}I_{p,q}

3. A. α(X) = F I_{p,q}\overline{X}^{t}F I_{p,q} (p + q = 2m)

4. A. α(z) = I_{p,q}\overline{z}

4. B. (n even) α(z) = F \overline{z}

6. A. α(X) = I_{k,l}\overline{X}I_{i,j} (k + l = p, i + j = q)

7. A. α(X) = I_{p,q}\overline{X}I_{p,q} (p + q = 2m + 1) 4.1.4. Complex-linear modifications of complex Jordan pairs.

1. a. α = id

1. b. α(X) = X^{t}

1. c. n = 2m: α(X) = F X^{t}F^{−1}

2. a. α = id

2. b. n = 2m: α(X) = F XF^{−1}

3. a. α = id

3. b. α(X) = F XF^{−1}

4. a. α = I_{q+1,p−1} , p > 0

6. a. α = id

6. b. (p = 2r, q = 2s): α(X) = F_{r}XF_{s}^{−1}

7. a. α = id 4.1.5. Modifications of real (non-complex) Jordan pairs. 1.1. a. α(X) = I_{p,q}XI_{p,q} 1.1. b. α(X) = X^{t} 1.1. c. n = 2m: α(X) = F X^{t}F^{−1} 1.2. a. α = id 1.2. b. n = 2m: α(X) = F X^{t}F^{−1} 1.2. c. α(X) = I_{p,q}X^{t}I_{u,v} 1.3. a. α = id 1.3. b. α(X) = X^{t} 1.3. c. α(X) = I_{p,q}\overline{X}^{t}I_{u,v} 2.1. a. α(X) = I_{p,q}XI_{p,q} 2.1. b. n = 2m: α = F XF^{−1} 2.2. a. α = id 2.2. b. α(X) = I_{2r,2s}X^{t}I_{2r,2s} (r + s = m) 3.1. a. α(X) = F I_{p,q}XF I_{p,q} 3.1. b. α(X) = \overline{X} = X^{t} 3.2. a. α = id 3.2. b. α(X) = F I_{p,q}XF I_{p,q} 4.1. a. α(x + y) = I_{r,p−r}x + I_{s,q−s}y (x ∈ ℝ^{p}, y ∈ ℝ^{q}) 4.1. b. n even, p = q: α = F 6.1. a. α(X) = I_{k,l}XI_{i,j} 6.1. b. p = 2r, q = 2s: α(X) = F_{r}XF_{s}^{−1} 6.2. a. α(X) = I_{k,l}XI_{i,j} 6.2. b. α(X) = \overline{X} 7.1. a. α(X) = I_{p,q}XI_{p,q}

4.2. Geometric part 4.2.0. Notation. The para-Hermitian symmetric space associated to a simple Jordan

pair V is denoted by G/H = Co(V)/ Str(V). The group Str(V) is is isomorphic to the structure group of T and Co(V) to the conformal group of T. (This will be explained in [Be98].)

We denote by X^{(α)} = G^{(α)}/H and X^{(−α)} = G^{(−α)}/H the (c-dual) symmetric spaces associated to the modifications T^{α} and T^{−α} of a JTS T as described in the algebraic classification. This agrees with the notation introduced in [Be96b] and [Be96c] where we have defined these spaces by a construction using Jordan algebras. The determination of the conformal group and of the spaces X^{(±α)} from the data T and α uses basically only linear algebra; we have given details for the classical matrix spaces in [Be96b]. A symmetric space of group type is denoted by X = H × H/H where a Lie group H is identified with the diagonal in H × H; its c-dual space is given by H_{ℂ}/H. The symbol ℙ in front of a group means the quotient group w.r.t. scalars.

4.2.1. Spaces having invariant structures of all three types. These are the spaces G/H = Co(V_{ℂ})_{o}/ Str(V_{ℂ})_{o} where V_{ℂ} is a complex Jordan pair. From Table 4.1.1 we get:

V_{ℂ}  G = Co(V_{ℂ})_{o}  H = Str(V_{ℂ})_{o}
1. ℙ Gl(2n, ℂ)  ℙ(Gl(n, ℂ) × Gl(n, ℂ))
2. Sp(n, ℂ)  Gl(n, ℂ)
3. SO(4m, ℂ)  Gl(2m, ℂ)
4. SO(n + 2, ℂ)  SO(n, ℂ) × SO(2, ℂ)
5. E_{7}  E_{6} × ℂ^{∗}
6. ℙ Gl(p + q, ℂ)  ℙ(Gl(p, ℂ) × Gl(q, ℂ))
7. SO(4m + 2, ℂ)  Gl(2m + 1, ℂ)
8. E_{6}  SO(10, ℂ) × ℂ^{∗}

Any of the three complexification functors, applied to one of these spaces, yields a direct product of the space with itself. The space of type 4 admits a second Jordan-extension, given by Table 4.2.4, 6.a. (q = 2).

4.2.2 Para-Hermitian symmetric spaces without invariant complex structure. These are the spaces Co(V)_{o}/ Str(V)_{o} with V a Jordan pair from Table 4.1.2.

V  G = Co(V)_{o}  H = Str(V)_{o}

1.1. SU(n, n)  Gl(n, ℂ)/ℝ^{∗}
1.2. ℙ Gl(2n, ℝ)  ℙ(Gl(n, ℝ) × Gl(n, ℝ))
1.3. n = 2m: ℙ Gl(2m, ℍ)  ℙ(Gl(m, ℍ) × Gl(m, ℍ))
2.1. Sp(n, ℝ)  Gl(n, ℝ)
2.2. n = 2m: Sp(m, m)  Gl(m, ℍ)
3.1. SO^{∗}(4m)  Gl(m, ℍ)
3.2. n = 2m: SO(n, n)  Gl(n, ℝ)
4.1. SO(p + 1, q + 1)  SO(p, q) × SO(1, 1), p + q = n
5.1. E_{7(−25)}  E_{6(−25)} × ℝ^{+}
5.2. E_{7(7)}  E_{6(6)} × ℝ^{+}
6.1. ℙ Gl(p + q, ℝ)  ℙ(Gl(p, ℝ) × Gl(q, ℝ))
6.2. p = 2r, q = 2s: ℙ Gl(r + s, ℍ)  ℙ(Gl(r, ℍ) × Gl(s, ℍ))
7.1. n = 2m + 1: SO(n, n)  Gl(n, ℝ)
8.1. E_{6(6)}  SO(5, 5) × ℝ^{+}
8.2. E_{6(−26)}  SO(1, 9) × ℝ^{+}

The straight and the twisted complexification of the space labelled by i.j. is given by the space in Table 4.2.1. labelled by i. The space of type 4.1. has another Jordan extension given by Table 4.2.5, 6.1.a (i = j = 1). The spaces 1.1., 2.1., 3.1., 4.1. and 5.1. have invariant causal structures; they are known as causal symmetric spaces of Cayley type.

4.2.3. Pseudo-Hermitian symmetric spaces without invariant straight complex structure. These are the spaces G^{(α)}/H and G^{(−α)}/H belonging to the Hermitian JTS’s T^{α} and T^{−α} from Table 4.1.3.

α  G^{(α)}  G^{(−α)}  H
1. A. Gl(n, ℍ)  Gl(2n, ℝ)  Gl(n, ℂ)
1. B. U(p + v, q + u)  U(p + u, q + v)  U(u, v) × U(p, q)
2. A. Sp(p, n − p)  Sp(n, ℝ)  U(p, n − p)
3. A. SO(2p, 4m − 2p)  SO^{∗}(4m)  U(p, 2m − p)
4. A. SO(p − 1, q + 3)  SO(p + 1, q + 1)  SO(p − 1, q + 1) × SO(2)
4. B. SO^{∗}(2 + 2q)  SO^{∗}(2 + 2q)  SO^{∗}(2) × SO^{∗}(2q)
6. A. U(l + i, k + j)  U(l + j, k + i)  U(k, l) × U(i, j)
7. A. SO^{∗}(4m − 2)  SO(2p, 4m + 2 − 2p)  U(p, 2m + 1)

The straight complexification and twisted para-complexification of space i.X. is given by space i. in Table 4.2.1. The space of type 4.A. carries a second Jordan extension, given by Table 4.2.5, 6.1.a (i = 2, j = 0), and the space of type 4.B. carries a second structure given by Table 4.2.5, 6.2.b (p = 1). The following are pseudo-Hermitian symmetric spaces of tube type in the sense of [FG96]: 1.A.+, 1.A.-, 1.B.+(p = u, q = v), 2.A.+, 2.A.-, 3.A.+, 3.A.-, 4.A.+ .

4.2.4. Straight complex symmetric spaces with twist having no invariant twisted complex structure. They correspond to Table 4.1.4; here X^{(α)} is isomorphic to its c-dual X^{(−α)} .

α  G^{(α)} ≅ G^{(−α)}  H
1. a. H × H  H = Gl(n, ℂ)
1. b. SO(2n, ℂ)  SO(n, ℂ) × SO(n, ℂ)
1. c. Sp(2m, ℂ)  Sp(m, ℂ) × Sp(m, ℂ)
2. a. Gl(n, ℂ)  SO(n, ℂ)
2. b. H × H  H = Sp(m, ℂ)
3. a. Gl(2m, ℂ)  Sp(m, ℂ)
3. b. H × H  H = SO(2m, ℂ)
4. a. SO(p + 1, ℂ) × SO(q + 1, ℂ)  SO(p, ℂ) × SO(q, ℂ)
6. a. SO(p + q, ℂ)  SO(p, ℂ) × SO(q, ℂ)

6. b. p = 2r, q = 2s : Sp(r + s, ℂ)  Sp(r, ℂ) × Sp(s, ℂ)
7. a. H × H  H = SO(2m + 1, ℂ)

The twisted complexification and para-complexification of space i.x. is given by space i. in Table 4.2.1. The complex spheres carry three Jordan extensions: they appear in the above table as types 6.a. p = 1 and 4.a. q = 0, and also in Table 4.2.5, 4.1.b. Further, the spaces of type 4.a. carry also structures of reducible Jordan extensions, namely those given by direct product of the structures on the complex spheres. Finally, the space of type 6.a. q = 2 carries a second Jordan extension given by Table 4.2.1, 4.

4.2.5 Symmetric spaces with twist having no invariant complex or paracomplex structure. These are the spaces X^{(α)} and X^{(−α)} corresponding to Table 4.1.5. In the last column we list the twisted complexification of the space: it is given by the line of Table 4.2.3 with the indicated label. The additional parameter “–” means that in the corresponding line of Table 4.2.3, α and −α have to be exchanged. For example, the twisted complexification of the group Sp(m, ℝ) (line 2.1.b, first and third column) can be found in Table 4.2.3, line 2.A, second and third column with the indicated parameters: it is the space Sp(2m, ℝ)/ U(m, m). For some symplectic series one has to take twice the parameters as given in Table 4.2.3 in order to get the twisted complexification; this is indicated by “double param.”

The twisted para-complexification of space i.j.x. is given by space i.j. of Table 4.2.2, and the “double complexification” is given by space i. of Table 4.2.1.

α G^{(α)}  G^{(−α)}  H  hℂ
1.1.a. H × H  H_{ℂ} = Gl(n, ℂ)  H = U(p, q)  1.B, v = p, u = q
1.1.b. SO^{∗}(2n)  SO(n, n)  SO(n, ℂ)  1.A
1.1.c. Sp(2m, ℝ)  Sp(m, m)  Sp(m, ℂ)  1.A.-
1.2.a. H_{ℂ} = Gl(n, ℂ) H × H  H = Gl(n, ℝ)  1.A
1.2.b. Sp(2m, ℝ)  Sp(2m, ℝ)  Sp(m, ℝ) × Sp(m, ℝ) 1.B.-, p = q = u = v = m
1.2.c. SO(p + v, q + u) SO(p + u, q + v)  SO(u, v) × SO(p, q) 1.B
1.3.a. H_{ℂ} = Gl(2m, ℂ) H × H  H = Gl(m, ℍ)  1.A.-
1.3.b. SO^{∗}(4m)  SO^{∗}(4m)  SO^{∗}(2m) × SO^{∗}(2m) 1.B.-, p = q = u = v = m
1.3.c. Sp(p + v, q + u) Sp(p + u, q + v)  Sp(u, v) × Sp(p, q)  1.B, double param.
2.1.a. U(p, q)  Gl(n, ℝ)  O(p, q)  2.A
2.1.b. H × H  H_{ℂ} = Sp(m, ℂ)  H = Sp(m, ℝ)  2.A.-, p = m = n/2
2.2.a. U(m, m)  Gl(m, ℍ)  SO^{∗}(2m)  2.A.-, p = m = n/2
2.2.b. H × H  H_{ℂ} = Sp(m, ℂ)  H = Sp(r, m − r)  2.A, p = 2r, n = 2m
3.1.a. U(2p, 2q)  Gl(m, ℍ)  Sp(p, q)  3.A
3.1.b. H × H  H_{ℂ} = SO(2m, ℂ)  H = SO^{∗}(2m)  3.A.-, p = m
3.2.a. U(m, m)  Gl(2m, ℝ)  Sp(m, ℝ)  3.A.-, p = m
3.2.b. H × H  H_{ℂ} = SO(2m, ℂ)  H = SO(p, 2m − p) 3.A
4.1.b. SO(p + 1, ℂ)  SO(p + 1, ℂ)  SO(p, ℂ)  4.B, p = q
6.1.a. SO(l + i, k + j) SO(l + j, k + i)  SO(k, l) × SO(i, j)  6.A
6.1.b. Sp(r + s, ℝ)  Sp(r + s, ℝ)  Sp(r, ℝ) × Sp(s, ℝ)  6.A, k = l = r, i = j = s
6.2.a. Sp(l + i, k + j) Sp(l + j, k + i)  Sp(k, l) × Sp(i, j)  6.A, double param.
6.2.b. SO^{∗}(2p + 2q) SO^{∗}(2p + 2q)  SO^{∗}(2p) × SO^{∗}(2q)  6.A, k = l = 2p, i = j = 2q
7.1.a. H × H  H_{ℂ} = SO(2m + 1, ℂ) H = SO(p, 2m + 1 − p) 7.A.-

Finally, in case 4.1.a we have G^{(α)} = SO(r − 1, s + 1) × SO(q − r + 2, p − s), G^{(−α)} = SO(r − 1, s) × SO(q − r + 1, p − s), H = SO(r, s) × SO(q − r − 1, p − s + 1), and the twisted complexification is given by 4.A.

Remarks. 1. The spaces with numbers 1.1, 2.1, 3.1 and 4.1 (p = 1) are causal symmetric spaces with a causal structure given by a symmetric cone (cf. [Be96b]).

2. The following spaces have several Jordan-extensions: 6.2.b. (cf. Table 4.2.3), 6.1.a. for i = 2, j = 0 (cf. Table 4.2.3) and for i = j = 1 (cf. Table 4.2.2), 4.1.b. (cf. Table 4.2.4), and 4.1.a. for r = 1, s = 0 which appears also in 6.1.a. with i = 1, j = 0. Among the latter are the real hyperbolic spaces. Finally, the spaces of 4.1.a. also have structures of reducible Jordan-extensions. The reader may have noticed that all multiple structures come out of the relation between types 4 (“classical” conformal space) and 6, p = 1 (projective space).

3. The trivial one-dimensional LTS has a continious family of Jordan-extensions – all scalar multiples of T (x, y, z) = xyz. The corresponding complexifications of the real line can be understood as the hyperbolic plane which is deformed into the flat complex plane as the positive scalar factor tends to zero, and similarly with the Riemann sphere if the scalar factor is negative. The corresponding para-complexfication is the one-sheeted hyperboloid.

4.2.6. A semi-exceptional space. It can be deduced from [Ma73, p.416], that the symmetric spaces of classical type M = Sl(4, ℍ)/ Sp(3, 1) and M_{ℂ} = Sl(8, ℂ)/ Sp(4, ℂ) have exceptional Hermitian complexifications, namely M_{hℂ} = E_{7(2)}/(E_{6(2)} × S^{1}) and M_{phℂ} = Co(Herm(3, 𝕆))/ Str(Herm(3, 𝕆)) (cf. Table 4.2.2). Remark that central extensions of M and M_{ℂ} have classical Hermitian complexifications, cf. Table 4.2.5, 3.1.a and 4.2.4, 3.a, corresponding to the equation of dimensions dim Herm(3, 𝕆) = dim Herm(4, ℍ) − 1. The fact that classical LTS’s may have exceptional Jordan-extensions has also been observed by Loos [Lo77, 11.19] and Neher [N85].

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