Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type

Dmitri V. Alekseevsky

Dept. of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK D.V.Alekseevsky@maths.hull.ac.uk

Vicente Cortés

Institut Élie Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239,

F-54506 Vandœuvre-lès-Nancy Cedex, France cortes@iecn.u-nancy.fr

March 29, 2004

Dedicated to Arkadij L’vovich Onishchik on the occassion of his seventeenth birthday.

Abstract

A pseudo-Riemannian symmetric space M = G/H of dimension 4n > 4 is called quaternionic Kähler if its holonomy group Hol is a subgroup of Sp_{1} · Sp_{p,q}, where p + q = n, and it is called para-quaternionic Kähler if Hol ⊂ Sp_{1}(ℝ) · Sp_{n}(ℝ). We study the structure of pseudo-Riemannian quaternionic Kähler and para-quaternionic Kähler symmetric spaces with non-zero scalar curvature as homogeneous spaces. In particular, it is proven that they have simple isometry group G. The classification of such spaces is reduced to the description of appropriate involutive automorphisms of complex simple Lie algebras, which is given in terms of Kac diagrams.

This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft.

Contents

1 Introduction 1

2 Definitions, basic facts and statement of the classification problem 6 2.1 ϵ-quaternionic Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 ϵ-quaternionic Kähler symmetric spaces . . . . . . . . . . . . . . . . . . . . 8

3 Simplicity of the isometry group of ϵ-quaternionic Kähler symmetric spaces with non-zero scalar curvature 9

4 ϵ-quaternionic Kähler symmetric spaces 12

4.1 Reduction of the classification of ϵ-quaternionic Kähler symmetric spaces to an algebraic problem about involutive automorphisms . . . . . . . . . . 12

5 Classification of pseudo-quaternionic Kähler symmetric spaces 14

5.1 Classification of pseudo-quaternionic Kähler symmetric spaces of inner type 14 5.1.1 Involutive inner automorphisms of complex simple Lie algebras … 14 5.1.2 Inner real forms of pseudo-quaternionic type . . . . . . . . . . . . . 22

5.2 Classification of pseudo-quaternionic Kähler symmetric spaces of outer type 25 5.2.1 Involutive outer automorphisms of complex simple Lie algebras . . 25 5.2.2 Outer real forms of pseudo-quaternionic type . . . . . . . . . . . . . 28

1 Introduction

A quaternionic structure in a 4n-dimensional real vector space V is a three-dimensional linear Lie algebra Q ⊂ End(V) with a basis J_{1}, J_{2}, J_{3} which satisfies the quaternionic relations

J_{1}^{2} = J_{2}^{2} = J_{3}^{2} = −Id , J_{3} = J_{1}J_{2} = −J_{3}J_{2} .  (1.1)

Then H = ℝ Id + Q is an associative algebra, isomorphic to the algebra ℍ of quaternions and V has the structure of a left H-module.

A 4n-dimensional Riemannian manifold (M, g), for n > 1, is called a quaternionic Kähler manifold if it admits a parallel field

M ∋ x ↦ Q_{x} ⊂ so(T_{x}M)

of skew-symmetric quaternionic structures. This is equivalent to the condition that the holonomy group Hol of (M, g) is a subgroup of Sp_{1} · Sp_{n}.

Such a field Q is called a parallel quaternionic structure on (M, g). The generalization to pseudo-Riemannian case is straightforward:

A pseudo-Riemannian 4n-dimensional manifold (M, g), n > 1, is called a pseudoquaternionic Kähler manifold if it admits a parallel quaternionic structure Q ⊂ 𝔰𝔬(T M) or, equivalently, if its holonomy group Hol ⊂ Sp_{1} · Sp_{p,q}, p + q = n.

Similar to the quaternionic structure, one can define the notion of para-quaternionic structure Q in 4n-dimensional vector space V as three dimensional linear Lie algebra Q with a basis J_{1}, J_{2}, J_{3} which satisfies the para-quaternionic relations

J_{1}^{2} = −Id , J_{2}^{2} = J_{3}^{2} = Id , J_{3} = J_{1}J_{2} = −J_{2}J_{1} .  (1.2)

Then the associative algebra H = ℝ Id + Q is isomorphic to the algebra ℍ′ ≅ 𝔤𝔩_{2}(ℝ) of para-quaternions (or split quaternions).

A pseudo-Riemannian 4n-dimensional manifold (M, g), n > 1, with the holonomy group Hol ⊂ Sp_{1}(ℝ) · Sp_{n}(ℝ) is called a para-quaternionic Kähler manifold. It implies that (M, g) admits a parallel field Q ⊂ 𝔰𝔬(T M) of skew-symmetric para-quaternionic structures and that the metric g has signature (2n, 2n).

Pseudo-quaternionic Kähler and para-quaternionic Kähler manifolds have many common features. To unify the notions, we define the notion of an ϵ-quaternionic Kähler manifold, for ϵ = ±1, where a −1-quaternionic Kähler manifold is a pseudo-quaternionic Kähler manifold and 1-quaternionic Kähler manifold is a para-quaternionic Kähler manifold.

We will denote also the linear algebra 𝔰𝔭_{n}(ℝ) of the (reducible) linear group Sp_{n}(ℝ) ⊂ SO_{2n,2n} by 𝔰𝔭_{n}^{1} and the linear Lie algebra 𝔰𝔭_{p,q} of the group Sp_{p,q} ⊂ SO_{4p,4q} by 𝔰𝔭_{n}^{−1}.

Any ϵ-quaternionic Kähler 4n-dimensional manifold (M, g, Q) for n > 1 is an Einstein manifold. More precisely, the curvature tensor R has a decomposition

R = νR_{0} + W  (1.3)

where R_{0} is the curvature tensor of the standard metric of the projective space ℍ_{ϵ}P^{n} over the algebra of quaternions ℍ_{−1} = ℍ or para-quaternionic ℍ_{1} = ℍ′, ν is a constant, proportional to the scalar curvature and W is a traceless algebraic curvature tensor of type 𝔰𝔭_{n}^{ϵ}. We say that a 4-dimensional pseudo-Riemannian manifold (M, g) of signature (4, 0) (respectively, (2, 2)) is quaternionic Kähler (respectively, para-quaternionic Kähler) if its curvature tensor has the form (1.3). In other words, this means that (M, g) is an anti-selfdual Einstein manifold.

An ϵ-quaternionic Kähler manifold (M, g, Q) is called an ϵ-quaternionic Kähler symmetric space if (M, g) is a pseudo-Riemannian symmetric space. In this case, we can identify M with a homogeneous space M = G/K , where G is the connected group of isometries, generated by transvections, and K is the stabilizer of a point o ∈ M.

Quaternionic Kähler symmetric space were classified by J. Wolf [W]. The compact quaternionic Kähler symmetric space (called Wolf spaces) are in one-to-one correspondence with compact simple Lie groups G with trivial center. The Wolf space associated with the group G is given by M = G/N(𝔰_{μ}) where N(𝔰_{μ}) is the normalizer in G of the three dimensional subalgebra 𝔰_{μ} associated with the highest root μ of the complex Lie algebra 𝔤^{ℂ} corresponding to G.

The classification of pseudo-quaternionic Kähler symmetric spaces with zero scalar curvature (i.e. pseudo-hyperKähler symmetric spaces) was obtained in [AC]. These spaces M are in one-to-one correspondence with equivalence classes of some quartic polynomials. They have nilpotent group G generated by transvections and commutative holonomy group K and can be identified with the coset space M = G/K. A similar classification of para-quaternionic Kähler symmetric spaces with zero scalar curvature was given in [ABCV].

The main aim of the paper is to describe the structure of simply connected ϵ-qua-ternionic (i.e. pseudo-quaternionic and para-quaternionic) Kähler symmetric spaces with non-zero scalar curvature and classify pseudo-quaternionic Kähler symmetric spaces.

The problem is reduced to a classification of symmetric decompositions 𝔤 = 𝔨 + 𝔪 of simple real Lie algebras 𝔤 of ϵ-quaternionic type. This means that the stability subalgebra 𝔨 contains a three-dimensional ideal 𝔰 such that ad_{𝔰}|_{𝔪} is an ϵ-quaternionic (i.e. quaternionic or para-quaternionic) structure on 𝔪.

More precisely, let (M = G/K, g, Q) be a simply connected ϵ-quaternionic Kähler symmetric space with non-zero scalar curvature and 𝔤 = 𝔨 + 𝔪 the corresponding symmetric decomposition of the Lie algebra 𝔤 . We identify 𝔪 with the tangent space T_{o}M at the canonical base point o = eK. Then the isotropy representation of K is identified with the adjoint representation Ad_{K}|_{𝔪} and the isotropy group Ad_{K}|_{𝔪} coincides with the holonomy group Hol. If the scalar curvature is not zero, the holonomy algebra 𝔥𝔬𝔩 = ad_{𝔨}|_{𝔪} contains the quaternionic structure Q_{o} ⊂ 𝔰𝔬(𝔪). This implies that the symmetric decomposition is of ϵ-quaternionic type.

In Section 2, we give the main definitions and formulate the classification problem. We prove that any ϵ-quaternionic Kähler symmetric space M = G/K has simple isometry group G in Section 3.

Let 𝔥^{ℂ} be a Cartan subalgebra of a complex simple Lie algebra 𝔤^{ℂ}, Π a system of simple roots of the root system R of (𝔤^{ℂ}, 𝔥^{ℂ}) and μ ∈ R the highest root. We denote by H_{μ} the associated vector from 𝔥^{ℂ}, defined by α(H_{μ}) = 2\frac{⟨α,μ⟩}{⟨μ,μ⟩} and by 𝔰_{μ} = span\{E_{−μ}, H_{μ}, E_{μ}\} the three-dimensional regular subalgebra of 𝔤^{ℂ} spanned by H_{μ} and the root vectors E_{±μ}. The adjoint operator ad_{H_{μ}} has eigenvalues k = ±2, ±1, 0 and the corresponding eigenspaces 𝔤_{k} give a depth two gradation

𝔤^{ℂ} = 𝔤_{−2} + 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} + 𝔤_{2}

with 𝔤_{±2} = ℂE_{±2}.

In Section 4, we reduce the problem of classification of ϵ-quaternionic Kähler symmetric spaces M = G/K, where G is a real form of a complex simple Lie group G^{ℂ} to the classification of involutive automorphisms θ of the complex simple Lie algebra 𝔤^{ℂ} = Lie(G^{ℂ}) which preserve the Cartan subalgebra 𝔥^{ℂ} and satisfy the following condition:

1. θ(H_{μ}) = −H_{μ} for ϵ = 1 and

2. θ = Id on 𝔰_{μ} for ϵ = −1.

In Section 5, we give a classification of all involutive automorphisms θ of 𝔤^{ℂ} which preserve 𝔥^{ℂ} and act trivially on 𝔰_{μ} in terms of involutive Kac diagrams.

Let Γ̃ be the extended Dynkin diagram of the complex simple Lie algebra 𝔤^{ℂ} which represents the extended simple root system

Π̃ = {α_{0} = −μ, α_{1}, ⋯, α_{ℓ}}

and μ = ∑_{i=1}^{ℓ} n_{i}α_{i} the decomposition of the highest root μ into a linear combination of simple roots. An involutive Kac diagram (of inner type) is defined as the extended Dynkin diagram Γ̃ with nodes α_{i} labeled by non-negative integers p_{i} such that

∑_{i=0}^{ℓ} n_{i}p_{i} = 2,

where n_{0} = 1. We will denote an involutive Kac diagram as (Γ̃, \overrightarrow{p}), where \overrightarrow{p} = (p_{0}, ⋯, p_{ℓ}) is the vector of labels.

An involutive Kac diagram (Γ̃, \overrightarrow{p}) defines an (inner) involutive automorphism θ = θ(\overrightarrow{p}) of 𝔤^{ℂ} with the ±1-eigenspace decomposition

𝔤^{ℂ} = 𝔤_{+}^{ℂ} + 𝔤_{−}^{ℂ} = (𝔥^{ℂ} + ∑_{α∈R_{0}} 𝔤_{α}) + ∑_{α∈R_{1}} 𝔤_{α}

where R_{a} = {α = ∑_{i=1}^{ℓ} m_{i}α_{i} | ∑_{i=1}^{ℓ} m_{i}p_{i} ≡ a (mod 2)}, for a = 0, 1.

The involution θ commutes with the standard antilinear involution τ of 𝔤^{ℂ}, which preserves 𝔥^{ℂ} and defines a compact real form of 𝔤^{ℂ}. The composition θ ∘ τ =: σ = σ(\overrightarrow{p}) is an antilinear involution. It defines a non-compact real form 𝔤(\overrightarrow{p}) of 𝔤^{ℂ}, if θ ≠ Id.

The map \overrightarrow{p} ↦ 𝔤(\overrightarrow{p}) gives a bijection between involutive Kac diagrams (of inner type) of a simple complex Lie algebra 𝔤^{ℂ} (up to an isomorphism of labelled graphs) and real forms of inner type of 𝔤^{ℂ} (up to a conjugation by an automorphism of 𝔤^{ℂ}) [K, GOV].

Let (Γ̃, \overrightarrow{p}) be an involutive Kac diagram. We denote by Π_{0} the subset of the system Π of simple roots which consists of roots α_{i} with label p_{i} = 0. Simply connected pseudoquaternionic Kähler symmetric spaces M = G/K where G is the simply connected Lie group with the Lie algebra 𝔤(\overrightarrow{p}) are described by the following theorem.

Theorem 1 There is a bijection between connected components of Π_{0} which contain a long root α of Π (up to the action of automorphisms of (𝔤^{ℂ}, 𝔤_{+}^{ℂ}) on the simple summands of 𝔤_{+}^{ℂ}) and pseudo-quaternionic Kähler symmetric spaces M = G/K of the group G (up to homothety). If α ∈ Π_{0} is a long root which belongs to a connected component Π′_{0} of Π_{0}, then the pseudo-quaternionic symmetric space associated with Π′_{0} is given by M(Π′_{0}) = G/K, where K = N_{G}(𝔰_{α}) is the normalizer of the tree-dimensional subalgebra 𝔰_{α} of 𝔤^{ℂ} associated with α.

We give a similar description of pseudo-quaternionic Kähler symmetric spaces M = G/K of a simple real Lie group G associated with an outer involutive automorphism θ ∈ ρAd_{G} of the complex simple Lie algebra 𝔤^{ℂ}, where ρ is the involutive automorphism associated with a non-trivial involution r of the Dynkin diagram and a system of canonical generators. The outer involutions are described in terms of the extended Dynkin diagram Γ̃^{ρ} of the ρ-root system, see [GOV]. In this case, the Kac diagrams, which determine an outer involutive automorphism, are completely determined by a choice of a simple ρ-root β_{i} with the label n_{i} = 2.

The final list of simply connected pseudo-quaternionic Kähler symmetric spaces is the following:

Theorem 2 Let M = G/K be a simply connected symmetric pseudo-quaternionic Kähler manifold. Then M is one of the following symmetric pseudo-quaternionic Kähler manifolds.

A)

\frac{SU(p + 2, q)}{S(U(2) × U(p, q))} , \frac{SL(n + 1, ℍ)}{S(GL(1, ℍ) × GL(n, ℍ))} ,

BD)

\frac{SO_{0}(p + 4, q)}{SO(4) × SO_{0}(p, q)} , \frac{SO^{∗}(2l + 4)}{SO^{∗}(4) × SO^{∗}(2l)} ,

C)

\frac{Sp(p + 1, q)}{Sp(1) × Sp(p, q)} ,

E6)

\frac{E_{6(−78)}}{SU(2)SU(6)} , \frac{E_{6(2)}}{SU(2)SU(6)} , \frac{E_{6(2)}}{SU(2)SU(2, 4)} , \frac{E_{6(−14)}}{SU(2)SU(2, 4)} ,

\frac{E_{6(6)}}{Sp(1)SL(3, ℍ)} , \frac{E_{6(−26)}}{Sp(1)SL(3, ℍ)} ,

E7)

\frac{E_{7(−133)}}{SU(2)Spin(12)} , \frac{E_{7(−5)}}{SU(2)Spin(12)} , \frac{E_{7(−5)}}{SU(2)Spin_{0}(4, 8)} , \frac{E_{7(7)}}{SU(2)SO^{∗}(12)} , \frac{E_{7(−25)}}{SU(2)SO^{∗}(12)} ,

E8)

\frac{E_{8(−248)}}{SU(2)E_{7(133)}} , \frac{E_{8(−24)}}{SU(2)E_{7(133)}} , \frac{E_{8(−24)}}{SU(2)E_{7(−5)}} , \frac{E_{8(8)}}{SU(2)E_{7(−5)}} ,

F4)

\frac{F_{4(−52)}}{Sp(1)Sp(3)} , \frac{F_{4(4)}}{Sp(1)Sp(3)} , \frac{F_{4(4)}}{Sp(1)Sp(1, 2)} , \frac{F_{4(−20)}}{Sp(2)Sp(1, 2)} ,

G2)

\frac{G_{2(−14)}}{SO(4)} , \frac{G_{2(2)}}{SO(4)} .

2 Definitions, basic facts and statement of the classification problem

2.1 ϵ-quaternionic Kähler manifolds

Definition 1

1. An ϵ-quaternionic structure (ϵ = ±1) on a pseudo-Euclidean vector space V is a three-dimensional Lie subalgebra Q = Q^{ϵ} ⊂ 𝔰𝔬(V) which admits a basis (J_{1}, J_{2}, J_{3}) satisfying the ϵ-quaternionic relations, i.e. J_{3} = J_{1}J_{2}, J_{1}^{2} = −Id and J_{2}^{2} = J_{3}^{2} = ϵId. In both cases we will use the notation J_{α}^{2} = ϵ_{α}Id and we denote by ℍ^{ϵ} = ℝ Id+Q^{ϵ} the associative algebra isomorphic to the algebra of quaternions for ϵ = −1 and paraquaternionions for ϵ = 1. We also denote by 𝔰𝔭_{n}^{ϵ} = N_{𝔰𝔬(V)}(Q^{ϵ}) the normalizer of Q^{ϵ} in the pseudo-orthogonal Lie algebra 𝔰𝔬(V), which is isomorphic to 𝔰𝔭_{n}(ℝ) = 𝔰𝔭(ℝ^{2n}) for ϵ = 1 and to 𝔰𝔭_{p,q} for ϵ = −1, where 4n = dim V = 4p + 4q.

2. A pseudo-Riemannian manifold (M, g) of signature (k, l) is called strongly oriented if it is endowed with a reduction of the bundle of orthonormal frames to the connected component SO_{0}(k, l) ⊂ SO(k, l).

3. An ϵ-quaternionic Kähler manifold is a strongly oriented pseudo-Riemannian manifold (M, g) endowed with a subbundle Q ⊂ End(T M) invariant under parallel transport such that Q_{p} is an ϵ-quaternionic structure on (T_{p}M, g_{p}) (p ∈ M) which preserves the curvature tensor of (M, g).

4. An ϵ-quaternionic Kähler manifold is called para-quaternionic Kähler if ϵ = +1 and pseudo-quaternionic Kähler if ϵ = −1.

Remarks:

1. The dimension of an ϵ-quaternionic Kähler manifold is always divisible by 4: dim M = 4n. The signature of the metric is (2n, 2n) for para-quaternionic Kähler manifolds (as we shall prove below) and is (4p, 4q) for pseudo-quaternionic Kähler manifolds.

2. If n > 1 the condition that Q preserves the curvature tensor is automatically satisfied.

3. The assumption that (M, g) is strongly oriented is automatically fulfilled for pseudoquaternionic Kähler manifolds.

Proposition 1 A pseudo-Riemannian manifold (M, g) admits a subbundle Q ⊂ End(T M) such that (M, g, Q) is an ϵ-quaternionic Kähler manifold if and only if the holonomy group

Hol ⊂ Sp(1) · Sp(p, q) ⊂ SO(V) , V = (ℂ^{2} ⊗ ℂ^{2n})^{ρ} = ℍ^{n} (ϵ = −1) or (2.1)
Hol ⊂ Sp(ℝ^{2}) · Sp(ℝ^{2n}) ⊂ SO(V) , V = ℝ^{2} ⊗ ℝ^{2n} (ϵ = +1) ,  (2.2)

where ρ = j_{H} ⊗ j_{E} is the standard Sp(1) · Sp(p, q)-invariant real structure of ℂ^{2} ⊗ ℂ^{2n}, which is the product of the invariant quaternionic structures j_{H} and j_{E} of the factors H = ℂ^{2} = ℍ and E = ℂ^{2n} = ℍ^{n}.

Proof: It is clear that a pseudo-Riemannian manifold with holonomy as above is ϵ-qua-ternionic Kähler. We prove the converse statement. Let (M, g, Q) be an ϵ-quaternio-nic Kähler manifold. Then the holonomy group belongs to the normalizer of Q in the connected pseudo-orthogonal group SO_{0}(k, l) which is Sp(1) · Sp(p, q) (k = 4p, l = 4q) if ϵ = −1 and Sp(ℝ^{2}) · Sp(ℝ^{2n}) if ϵ = +1. To see this in the second case, we first extend the representation of Q by Id to a representation of Q ⊕ ℝ Id ≅ ℝ(2), the two-by-two matrix algebra. Since any representation of ℝ(2) without trivial subrepresentation is a direct sum of standard representations, the holonomy representation can be factorized as ℝ^{2} ⊗ E′, where E′ is a real vector space. The metric g is a Q-invariant and Hol -invariant element of the Q-module

S^{2}(ℝ^{2} ⊗ E′)^{∗} = S^{2}ℝ^{2} ⊗ S^{2}(E′)^{∗} + ∧^{2}ℝ^{2} ⊗ ∧^{2}(E′)^{∗} .

Since S^{2}ℝ^{2} has no non-trivial Q-invariants, g can be written as g = ω_{H′} ⊗ ω_{E′}, where ω_{H′} and ω_{E′} are Q-invariant and Hol -invariant symplectic forms on the factors H′ = ℝ^{2} and E′ = ℝ^{2n}, respectively. Now it is clear that the group of pseudo-orthogonal transformations which normalize Q is of the form Sp(ℝ^{2}) ⊗ Sp(ℝ^{2n}) ⊂ GL(ℝ^{2} ⊗ ℝ^{2n}).

Theorem 3 Any ϵ-quaternionic Kähler manifold M is Einstein and its curvature tensor R admits the following decomposition

R = νR_{0} + W ,  (2.3)

where ν = \frac{scal}{4n(n+2)} is the reduced scalar curvature,

R_{0}(X, Y) := −\frac{1}{2} ∑_{α} ϵ_{α}⟨X, J_{α}Y ⟩J_{α} + \frac{1}{4} (X ∧ Y − ∑_{α} ϵ_{α}J_{α}X ∧ J_{α}Y ) , X, Y ∈ T_{p}M ,

is the curvature tensor of the ϵ-quaternionic projective space ℍ^{ϵ}P^{n} of the same signature as M and W is an algebraic curvature tensor of type 𝔰𝔭_{n}^{ϵ}.

Proof: One can easily prove that

trJ_{α}R(·, ·) = 2nνg(J_{α}·, ·)  (2.4)

(see [Be] for the quaternionic case) and that R_{0} is a curvature tensor of type Q^{ϵ} +𝔰𝔭_{n}^{ϵ} with the reduced scalar curvature 1. Then the curvature tensor W = R−νR_{0} has zero reduced scalar curvature and the above formula shows that it has type 𝔰𝔭_{n}^{ϵ}, i.e. W (X, Y) ∈ 𝔰𝔭_{n}^{ϵ}.

Corollary 1 Let M be an ϵ-quaternionic Kähler manifold with holonomy algebra 𝔥𝔬𝔩 and ν ≠ 0. Then 𝔥𝔬𝔩 ⊃ 𝔰𝔭_{1}^{ϵ}.

Proof: Using the decomposition (2.3) of the curvature tensor and the formula

J_{α} = \frac{1}{2} ∑_{i=1}^{4n} ϵ_{i}J_{α}e_{i} ∧ e_{i} ,  (2.5)

where ⟨e_{i}, e_{j}⟩ = ϵ_{i}δ_{ij}, ϵ_{i} = ±1, we calculate

𝔥𝔬𝔩 ∋ R(J_{α}) = R_{0}(J_{α}) = nνJ_{α} .

Theorem 4 An ϵ-quaternionic Kähler manifold (M, g, Q) with ν ≠ 0 is locally indecomposable, i.e. is not locally a product of two pseudo-Riemannian manifolds of positive dimension.

Proof: It is sufficient to show that the there is no 𝔥𝔬𝔩-invariant proper subspace 0 ≠ E ⊂ V = T_{p}M (p ∈ M) such that V = E ⊕ E^{⊥}. This follows from the following formula

νg(X, X)g(Y, Y) = −g(R(X, J_{1}X)Y, J_{1}Y) + ϵg(R(X, J_{1}X)J_{2}Y, J_{3}Y) .  (2.6)

In fact, it is sufficient to choose X ∈ E and Y ∈ E^{⊥} such that g(X, X)g(Y, Y) ≠ 0. Such vectors exist, since the scalar product is non-degenerate on E and E^{⊥}. Then the left-hand side of (2.6) is non-zero, since ν ≠ 0. We claim that the right-hand side of (2.6) is zero. The subspaces E, E^{⊥} ⊂ V are Q-invariant, by Corollary 1. In particular, we have that X, J_{1}X ∈ E and Y, J_{1}Y, J_{2}Y, J_{3}Y ∈ E^{⊥}. The right-hand side of (2.6) can now be computed using the Bianchi identity:

−g(R(X, J_{1}X)Y, J_{1}Y) = g(R(J_{1}X, Y)X, J_{1}Y) + g(R(Y, X)J_{1}X, J_{1}Y) = 0 ,

by the 𝔥𝔬𝔩-invariance of E. Similarly, one shows that g(R(X, J_{1}X)J_{2}Y, J_{3}Y) = 0. This proves that (M, g) is locally indecomposable.

Now it remains to prove the formula (2.6). Let us decompose the curvature operators, according to Theorem 3:

R(X, Y) = −\frac{1}{2} ∑ ρ_{α}(X, Y)J_{α} + W (X, Y) , X, Y ∈ V ,

where W (X, Y) commutes with Q and ρ_{α}(·, J_{α}·) = νg. This implies, in particular, that

νg(X, X)g(Y, Y) = ρ_{1}(X, J_{1}X)g(Y, Y) = −ϵρ_{1}(X, J_{1}X)g(J_{2}Y, J_{2}Y)
= −ϵg([R(X, J_{1}X), J_{3}]Y, J_{2}Y)
= −ϵg(R(X, J_{1}X)J_{3}Y, J_{2}Y) + ϵg(J_{3}R(X, J_{1}X)Y, J_{2}Y)
= ϵg(R(X, J_{1}X)J_{2}Y, J_{3}Y) − g(R(X, J_{1}X)Y, J_{1}Y) .

Here we have used that ϵ_{1} = −1 and ϵ_{2} = e_{3} = ϵ.

2.2 ϵ-quaternionic Kähler symmetric spaces

Definition 2 An ϵ-quaternionic Kähler symmetric space is an ϵ-quaternionic Kähler manifold (M, g) which is a pseudo-Riemannian symmetric space, that is for any point x ∈ M there is an isometry S_{x} of M (called the central symmetry) which fixes x and has the differential d(S_{x})|_{x} = −Id.

The connected component G of the group generated by central symmetries acts transitively on M and it is called the group generated by transvections. If K is the stabilizer of a point o ∈ M in this group, we can identify M with the quotient M = G/K.

The conjugation with respect to the central symmetry S_{o} of a point o ∈ M defines an involutive automorphism σ of the group G and its Lie algebra 𝔤. The eigenspace decomposition 𝔤 = 𝔨 + 𝔪 of 𝔤 with respect to σ is a symmetric decomposition. The +1eigenspace 𝔨 is the Lie algebra of the stability subgroup K of the point o. We identify M with the quotient space G/K and the Ad_{K}-invariant subspace 𝔪 with the tangent space T_{o}M = T_{eK}G/K. Then the isotropy representation of K in T_{o}M is identified with the adjoint representation Ad_{K}|_{𝔪}. It is known that the isotropy group Ad_{K}|_{𝔪} coincides with the holonomy group of M.

We will always identify an ϵ-quaternionic Kähler symmetric space M with the homogeneous space M = G/K where G is the group generated by transvections and K is the stabilizer of a point o and we will assume that the manifold M is simply connected.

For the rest of this paper, we will study the following classification problem.

Problem: Classify ϵ-quaternionic Kähler symmetric spaces.

3 Simplicity of the isometry group of ϵ-quaternionic Kähler symmetric spaces with non-zero scalar curvature

Lemma 1 Let M = G/K be an ϵ-quaternionic Kähler symmetric space and 𝔤 = 𝔨 + 𝔪 the corresponding symmetric decomposition. If M has non-zero scalar curvature then ad_{𝔨}|_{𝔪} ⊃ 𝔰𝔭_{1}^{ϵ}.

Proof: Since the holonomy Lie algebra 𝔥𝔬𝔩 coincides with the isotropy Lie algebra ad_{𝔨}|_{𝔪} the lemma follows from Corollary 1.

Corollary 2 Let M = G/K be an ϵ-quaternionic Kähler symmetric space and 𝔤 = 𝔨+𝔪 the corresponding symmetric decomposition. Then

𝔨 = 𝔰𝔭_{1}^{ϵ} ⊕ 𝔨′ , 𝔨′ = Z_{𝔤}(𝔰𝔭_{1}^{ϵ}) , [𝔰𝔭_{1}^{ϵ}, 𝔤] = 𝔰𝔭_{1}^{ϵ} + 𝔪 and ad_{𝔨′}|_{𝔪} ⊂ 𝔰𝔭_{n}^{ϵ} .  (3.1)

Theorem 5 Let M = G/K be an ϵ-quaternionic Kähler symmetric space with effective action of the Lie group G and reduced scalar curvature ν ≠ 0. Then G is simple.

Proof: We first prove that G is semisimple. Let 𝔤 = 𝔰+𝔯 be a Levi-Malcev decomposition such that 𝔰 ⊃ 𝔰𝔭_{1}^{ϵ}. Then we have the following decompositions

𝔰 = Z_{𝔰}(𝔰𝔭_{1}^{ϵ}) + [𝔰𝔭_{1}^{ϵ}, 𝔰] ,  (3.2)
𝔯 = Z_{𝔯}(𝔰𝔭_{1}^{ϵ}) + [𝔰𝔭_{1}^{ϵ}, 𝔯] ,  (3.3)
Z_{𝔰}(𝔰𝔭_{1}^{ϵ}) = 𝔨′ ∩ 𝔰 =: 𝔨′_{s} , [𝔰𝔭_{1}^{ϵ}, 𝔰] = 𝔰𝔭_{1}^{ϵ} + 𝔪_{s} , 𝔪_{s} := 𝔪 ∩ [𝔰𝔭_{1}^{ϵ}, 𝔰] = 𝔪 ∩ 𝔰 , (3.4)
Z_{𝔯}(𝔰𝔭_{1}^{ϵ}) = 𝔨 ∩ 𝔯 =: 𝔨_{r} , [𝔰𝔭_{1}^{ϵ}, 𝔯] = 𝔪 ∩ 𝔯 =: 𝔪_{r} .  (3.5)

So putting 𝔨_{s} := 𝔰𝔭_{1}^{ϵ} ⊕ 𝔨′_{s} = 𝔨 ∩ 𝔰 we have

𝔰 = 𝔨_{s} + 𝔪_{s} , 𝔯 = 𝔨_{r} + 𝔪_{r} , 𝔨 = 𝔨_{s} + 𝔨_{r} = (𝔰𝔭_{1}^{ϵ} ⊕ 𝔨′_{s}) + 𝔨_{r} and 𝔪 = 𝔪_{s} + 𝔪_{r} . (3.6)

Since 𝔪_{r} = [𝔰𝔭_{1}^{ϵ}, 𝔯] ⊂ nilrad(𝔤) =: 𝔫 the operators ad_{X} are nilpotent for all X ∈ 𝔪_{r}. For X ∈ 𝔪_{r} and Y ∈ 𝔪 we have that

a := [X, Y] ∈ 𝔨 ∩ 𝔫 ⊂ 𝔨 ∩ 𝔯 = 𝔨_{r} = Z_{𝔯}(𝔰𝔭_{1}^{ϵ})

and hence [ad_{a}|_{𝔪}, J_{α}] = 0. This implies tr (J_{α} ∘ ad_{a}|_{𝔪}) = 0 since J_{α} ∘ ad_{a}|_{𝔪} is the product of two commuting endomorphisms one of which is nilpotent. To finish the proof of the semisimplicity we use the next lemma.

Lemma 2 Any ϵ-quaternionic Kähler symmetric space of dimension 4n satisfies

tr (J_{α} ∘ ad_{[X,Y]}|_{𝔪}) = −2nν⟨J_{α}X, Y ⟩ for all X, Y ∈ 𝔪 .

Proof: This follows from (2.4) and Theorem 3 using the formula

R(X, Y) = −ad_{[X,Y]}|_{𝔪} for all X, Y ∈ 𝔪

for the curvature tensor of a symmetric space.

The lemma applied to X ∈ 𝔪_{r} now shows that

0 = tr (J_{α} ∘ ad_{[X,Y]}|_{𝔪}) = −2nν⟨J_{α}X, Y ⟩

for all Y ∈ 𝔪 and hence 𝔪_{r} = 0. Here we have again used that ν ≠ 0. Finally 𝔯 = 𝔨_{r} ⊂ 𝔨 has to vanish by effectivity. This finishes the proof of the semisimplicity of G.

It remains to prove that G is simple. For this we need some facts about semisimple graded Lie algebras. Let 𝔤^{ℂ} be a complex semisimple Lie algebra and

𝔤^{ℂ} = 𝔤_{−d}^{ℂ} + ⋯ + 𝔤_{−1}^{ℂ} + 𝔤_{0}^{ℂ} + 𝔤_{1}^{ℂ} + ⋯ + 𝔤_{d}^{ℂ}

a grading of depth d.

The endomorphism A of 𝔤^{ℂ} defined by A|_{𝔤_{k}^{ℂ}} = k Id is a derivation. Since all derivations of a semisimple Lie algebra are of inner type we have A = ad_{D} for some D ∈ 𝔤^{ℂ}. The element D determines the grading and is called the grading element.

Lemma 3

(i) Let 𝔤^{ℂ} be a complex simple Lie algebra, μ ∈ (𝔥^{ℂ})^{∗} a long root with respect to a Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤^{ℂ} and H_{μ} ∈ 𝔥^{ℂ} defined by the condition α(H_{μ}) = 2\frac{⟨α,μ⟩}{⟨μ,μ⟩} for all α ∈ (𝔥^{ℂ})^{∗}. Then H_{μ} is the grading element of a depth two grading of 𝔤^{ℂ} with 𝔤_{±2}^{ℂ} = ℂE_{±μ}, where E_{±μ} are the corresponding root vectors. Moreover, 𝔰_{μ} := span{E_{−μ}, H_{μ}, E_{μ}} is the three-dimensional subalgebra associated with μ and 𝔤_{0}^{ℂ} = ℂH_{μ} ⊕ Z_{𝔤^{ℂ}}(𝔰_{μ}). Conversely, any depth two grading of 𝔤^{ℂ} with dim 𝔤_{±2}^{ℂ} = 1 is of this form.

(ii) Let D be the grading element of a depth two grading of a complex semisimple Lie algebra 𝔤^{ℂ} such that dim 𝔤_{±2}^{ℂ} = 1, [𝔤_{−2}, 𝔤_{2}] = ℂD and 𝔤_{0}^{ℂ} does not contain any non-trivial ideal of 𝔤^{ℂ} (“effectivity assumption”). Then 𝔤^{ℂ} is simple and D = H_{μ}.

Proof: (i) It is well known that H_{μ} defines a depth two grading with stated properties. We prove only the converse statement. Any grading of a complex semisimple Lie algebra is defined by a choice of non-negative labels m_{i} for each simple root α_{i} such that deg(E_{α_{i}}) = m_{i} and deg(𝔥) = 0. If μ = ∑ n_{i}α_{i} is the decomposition of the highest root with respect to simple roots, then the depth of the grading is given by d = deg(E_{μ}) = ∑ n_{i}m_{i}. In the case d = 2 there are only three possibilities.

(a) m_{k} = 1, n_{k} = 2,

(b) m_{p} = m_{q} = 1, n_{p} = n_{q} = 1 and

(c) m_{k} = 2, n_{k} = 1, where all other m_{i} = 0. The condition dim 𝔤_{±2}^{ℂ} = 1 implies that deg(E_{μ−α_{i}}) < 2 for all α_{i} for which μ − α_{i} is a root. We can write μ = ∑ k_{i}π_{i} as linear combination of fundamental weights. Then μ − α_{i} is a root if and only if k_{i} > 0. In fact, from ⟨π_{j}, α_{i}⟩ = \frac{1}{2}⟨α_{i}, α_{i}⟩δ_{ij} we obtain

⟨μ, α_{i}⟩ = ∑ k_{j}⟨π_{j}, α_{i}⟩ = \frac{1}{2}⟨α_{i}, α_{i}⟩k_{i} > 0 if and only if k_{i} > 0 .

This implies that m_{i} > 0 if k_{i} > 0. With exception of the A series the highest root is always proportional to a fundamental weight π_{k} with n_{k} = 2 so we are in case (a). For A_{l} we have μ = π_{1} + π_{l} with all n_{i} = 1 and hence m_{1} = m_{l} = 1 and we are in case (b). The case (c) never occurs. In all cases we obtain a unique labeling which gives the standard grading by H_{μ}. (ii) Let 𝔥^{ℂ} be a Cartan subalgebra of 𝔤_{0}^{ℂ}. It contains D and is a Cartan subalgebra of 𝔤^{ℂ}. Since [𝔥^{ℂ}, 𝔤_{±2}^{ℂ}] = 𝔤_{±2}^{ℂ}, the complex lines 𝔤_{±2}^{ℂ} are root spaces of some simple ideal 𝔤̃^{ℂ} ⊂ 𝔤^{ℂ}. Hence [𝔤_{−2}, 𝔤_{2}] = ℂD ⊂ 𝔤̃^{ℂ}. This implies that all other ideals of 𝔤^{ℂ} belong to 𝔤_{0}^{ℂ} and hence vanish by effectivity.

Lemma 4 Let M = G/K be an ϵ-quaternionic Kähler symmetric space as above and h ∈ 𝔰𝔭_{1}^{ϵ} such that the corresponding isotropy operator I = ad_{h}|_{𝔪} satisfies I^{2} = Id or I^{2} = −Id. Then D = h or D = ih, respectively, is the grading element of a depth two grading

𝔤^{ℂ} = 𝔤_{−2}^{ℂ} + 𝔤_{−1}^{ℂ} + 𝔤_{0}^{ℂ} + 𝔤_{1}^{ℂ} + 𝔤_{2}^{ℂ} ,  (3.7)

of the complexified Lie algebra 𝔤^{ℂ} such that dim 𝔤_{±2}^{ℂ} = 1, [𝔤_{−2}, 𝔤_{2}] = ℂD, 𝔨^{ℂ} = 𝔤_{ev}^{ℂ} = 𝔤_{−2}^{ℂ} + 𝔤_{0}^{ℂ} + 𝔤_{2}^{ℂ} and 𝔪^{ℂ} = 𝔤_{odd}^{ℂ} = 𝔤_{−1}^{ℂ} + 𝔤_{1}^{ℂ}. In the case D = h the grading is defined over ℝ, i.e. on 𝔤. In both cases 𝔰𝔭_{1}(ℂ) = 𝔰𝔭_{1}^{ϵ} ⊗ ℂ = 𝔤_{−2}^{ℂ} + ℂD + 𝔤_{2}^{ℂ}.

This lemma shows that the grading associated with h ∈ 𝔰𝔭_{1}^{ϵ} satisfies the assumptions of Lemma 3 (ii). So we obtain that G is simple. This finishes the proof of Theorem 5.

Corollary 3 Under the assumptions of the previous lemma, there exists a Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤_{0}^{ℂ} of 𝔤^{ℂ} and a long root μ of (𝔤^{ℂ}, 𝔥^{ℂ}) such that D = H_{μ} and 𝔤_{±2}^{ℂ} = ℂE_{±μ}.

Proof: This follows from Lemma 3 (i).

4 ϵ-quaternionic Kähler symmetric spaces

4.1 Reduction of the classification of ϵ-quaternionic Kähler symmetric spaces to an algebraic problem about involutive automorphisms

Now we reduce the classification problem for ϵ-quaternionic Kähler symmetric spaces to the description of involutive automorphisms of a complex simple Lie algebra 𝔤^{ℂ} which preserve the three-dimensional subalgebra 𝔰_{μ} ⊂ 𝔤^{ℂ} associated with a highest root μ.

Let M = G/K be an ϵ-quaternionic Kähler symmetric space and 𝔤 = 𝔨 + 𝔪 = (𝔰𝔭_{1}^{ϵ} ⊕ 𝔨′) + 𝔪 the corresponding symmetric decomposition. By Lemma 4 any element h ∈ 𝔰𝔭_{1}^{ϵ} such that the corresponding isotropy operator I = ad_{h}|_{𝔪} satisfies I^{2} = Id or I^{2} = −Id defines a depth two grading of 𝔤^{ℂ} with grading element D = h or ih. Thanks to Corollary 3, we can identify D = H_{μ}, 𝔤_{±2}^{ℂ} = ℂE_{±μ} and 𝔰𝔭_{1}^{ϵ} ⊗ ℂ = 𝔰_{μ}. The antilinear involution (i.e. involutive automorphism) σ of the Lie algebra 𝔤^{ℂ} defined by complex conjugation with respect to 𝔤 preserves 𝔰_{μ} and σH_{μ} = H_{μ} if H_{μ} = h ∈ 𝔰𝔭_{1}^{ϵ} and σH_{μ} = −H_{μ} if H_{μ} = ih ∈ i𝔰𝔭_{1}^{ϵ}. This implies that σ𝔤_{j}^{ℂ} = 𝔤_{j}^{ℂ} or σ𝔤_{j}^{ℂ} = 𝔤_{−j}^{ℂ}, respectively. In both cases σ preserves 𝔤_{ev}^{ℂ} = 𝔨^{ℂ} and 𝔤_{odd}^{ℂ} = 𝔪^{ℂ}. Conversely, any antilinear involution σ of 𝔤^{ℂ}, which preserves 𝔰_{μ} defines an ϵ-quaternionic Kähler symmetric space. We denote by Aut(𝔤^{ℂ}, 𝔰_{μ}) the group of automorphisms which preserves 𝔰_{μ}.

Proposition 2 The above correspondence M = G/K ↦ σ induces a bijection between simply connected ϵ-quaternionic Kähler symmetric spaces (up to homothety) and antilinear involutions σ of 𝔤^{ℂ} (up to conjugation by elements of Aut(𝔤^{ℂ}, 𝔰_{μ})) which preserve the threedimensional subalgebra 𝔰_{μ} ⊂ 𝔤^{ℂ} associated with a long root μ and have the fixed point set 𝔰_{μ}^{σ} ≅ 𝔰𝔭_{1}^{ϵ}.

Proof: We prove that any antilinear involution σ as above defines an ϵ-quaternionic Kähler symmetric space. By Lemma 3 (i) there exists a depth two grading 𝔤^{ℂ} = 𝔤_{−2}^{ℂ} + 𝔤_{−1}^{ℂ} + 𝔤_{0}^{ℂ} + 𝔤_{1}^{ℂ} + 𝔤_{2}^{ℂ} defined by H_{μ} such that 𝔰_{μ} = 𝔤_{−2}^{ℂ} + ℂH_{μ} + 𝔤_{2}^{ℂ}. Since σ is an automorphism of 𝔰_{μ} we may assume, without loss of generality, that σH_{μ} = ±H_{μ}. This implies that σ preserves 𝔨^{ℂ} := 𝔤_{ev}^{ℂ} = 𝔰_{μ} ⊕ Z_{𝔤^{ℂ}}(𝔰_{μ}), (𝔨′)^{ℂ} := Z_{𝔤^{ℂ}}(𝔰_{μ}) and 𝔪^{ℂ} := 𝔤_{odd}^{ℂ}. We have 𝔰_{μ}^{σ} = 𝔰𝔭_{1}^{ϵ} for ϵ = ±1. We denote by 𝔤, 𝔨, 𝔨′ and 𝔪 the fixed point sets of σ on 𝔤^{ℂ}, 𝔨^{ℂ}, (𝔨′)^{ℂ} and 𝔪^{ℂ}, respectively, and by M = G/K the simply connected symmetric space associated with the symmetric decomposition 𝔤 = 𝔨 + 𝔪. Here G is the simply connected Lie group with Lie algebra 𝔤 and K ⊂ G is the connected Lie subgroup generated by 𝔨. We will prove that M is an ϵ-quaternionic Kähler symmetric space. The Killing form of 𝔤 induces a 𝔨-invariant scalar product g on 𝔪. Up to scale, g is the restriction of the Killing form B^{𝔤^{ℂ}}|_{𝔪^{ℂ}} =: g^{ℂ} to 𝔪^{ℂ}. The latter is invariant under 𝔰_{μ}. This shows that the pseudo-Riemannian metric g is invariant under 𝔰𝔭_{1}^{ϵ}.

Since ad_{H_{μ}}|_{𝔪^{ℂ}} has eigenvalues ±1, we can identify 𝔪^{ℂ} = ℂ^{2} ⊗ ℂ^{2n} such that 𝔰_{μ} ≅ 𝔰𝔭_{1}(ℂ) acts in the standard way on the first factor ℂ^{2} Then it is clear the real subalgebra 𝔰𝔭_{1}^{ϵ} ⊂ End(𝔪^{ℂ}) has a basis (J_{1}, J_{2}, J_{3}) which satisfies the ϵ-quaternion relations. Moreover 𝔰𝔭_{1}^{ϵ} preserves the real form 𝔪 of 𝔪^{ℂ} and hence defines on 𝔪 an ϵ-quaternionic structure Q = 𝔰𝔭_{1}^{ϵ} ⊂ End(𝔪). Since the isotropy representation of the symmetric space M = G/K preserves Q, this defines on M the structure of an ϵ-quaternionic Kähler symmetric space (M, g, Q).

It is clear that antilinear involutions σ and σ′ = ϕ ∘ σ ∘ ϕ^{−1} where ϕ ∈ Aut(𝔤^{ℂ}, 𝔰_{μ}) define homothetic ϵ-quaternionic Kähler symmetric spaces. Conversely, if 𝔤 = 𝔰𝔭_{1}^{ϵ} +𝔨′ +𝔪 and 𝔤′ = 𝔰𝔭_{1}^{ϵ}′ + 𝔨′′ + 𝔪′ are two symmetric decompositions of ϵ-quaternionic type, which define homothetic ϵ-quaternionic Kähler symmetric spaces, then there is an isomorphism ϕ : 𝔤 → 𝔤′ which is consistent with these decompositions. It is extended to a complex linear automorphism ϕ : 𝔤^{ℂ} → 𝔤^{ℂ} which preserves 𝔰_{μ} = (𝔰𝔭_{1}^{ϵ})^{ℂ} = (𝔰𝔭_{1}^{ϵ}′)^{ℂ}.

Next we reduce the description of antilinear involutions as in Proposition 2 to complex linear involutions.

Lemma 5 Let σ be an antilinear involution of a complex simple Lie algebra 𝔤^{ℂ} which preserves 𝔰_{μ}. Then there exists a compact (i.e. such that the real form (𝔤^{ℂ})^{τ} is compact) antilinear involution τ which commutes with σ, satisfies τH_{μ} = −H_{μ} and, in the case ϵ = −1, coincides with σ on 𝔰_{μ}. Moreover, there exists a Cartan subalgebra 𝔥 ⊂ (𝔤^{ℂ})^{τ}, which is invariant under σ. Its complexification 𝔥^{ℂ} ⊂ 𝔤^{ℂ} is a Cartan subalgebra invariant under σ and τ and contains H_{μ}.

Proof: We can assume that σH_{μ} = ϵH_{μ}. There exists a Cartan decomposition 𝔤 := (𝔤^{ℂ})^{σ} = 𝔨 + 𝔪, where 𝔨 ⊂ 𝔤 is a maximal compact subalgebra, such that H_{μ} ∈ 𝔪 if ϵ = +1 and iH_{μ} ∈ 𝔨 if ϵ = −1. In the case ϵ = −1 we choose 𝔨 such that 𝔨 ⊃ (𝔰_{μ})^{σ} = 𝔰𝔭_{1}^{ϵ} = 𝔰𝔭_{1}. We define τ by (𝔤^{ℂ})^{τ} = 𝔨 + i𝔪. Then τ commutes with σ and τH_{μ} = −H_{μ}. This implies that τ preserves the three-dimensional subalgebra 𝔰_{μ} = 𝔤_{−2} + ℂH_{μ} + 𝔤_{2}. Moreover, in the case ϵ = −1, we have τ = σ on 𝔰_{μ}. Now we can choose a Cartan subalgebra 𝔥 ⊂ (𝔤^{ℂ})^{τ} of the form 𝔥 = 𝔥_{k} + i𝔥_{m}, where 𝔥_{k} = 𝔥 ∩ 𝔨 and 𝔥_{m} = 𝔥 ∩ 𝔪, such that H_{μ} ∈ 𝔥_{m} if ϵ = +1 and iH_{μ} ∈ 𝔥_{k} if ϵ = −1. It has the claimed properties.

Let us fix a Cartan subalgebra 𝔥^{ℂ} and a long root μ of the simple Lie algebra 𝔤^{ℂ}. We denote by τ the standard compact antilinear involution of (𝔤^{ℂ}, 𝔥^{ℂ}). Notice that any involutive automorphism θ of the complex Lie algebra 𝔤^{ℂ} which commutes with τ defines a real form 𝔤 = 𝔤(θ) := (𝔤^{ℂ})^{σ}, where σ = θ ∘ τ. The real form 𝔤(θ) is called of inner (respectively, outer) type if θ is an inner (respectively, outer) automorphism.

Theorem 6 There is one-to-one correspondence between ϵ-quaternionic Kähler symmetric spaces M = G/K (up to homothety) and involutive automorphisms θ (up to conjugation by admissible elements of Aut(𝔤^{ℂ})) of 𝔤^{ℂ} which preserve 𝔥^{ℂ}, commute with τ and satisfy θH_{μ} = −ϵH_{μ} and, in the case ϵ = −1, θ|_{𝔰_{μ}} = Id, i.e. 𝔰_{μ} ⊂ (𝔤^{ℂ})^{θ}. (An element g ∈ Aut(𝔤^{ℂ}) is admissible if θ^{g} = g ∘ θ ∘ g^{−1} satisfies the same conditions as θ, i.e. θ^{g}𝔥^{ℂ} = 𝔥^{ℂ}, [θ^{g}, τ] = 0, etc.)

Proof: It is sufficient to establish a correspondence between anti-linear involutions σ as in Proposition 2 and involutive automorphisms θ as in the statement of the theorem. Let σ be an anti-linear involution of 𝔤^{ℂ} which preserves the three-dimensional subalgebra 𝔰_{μ} ⊂ 𝔤^{ℂ} and has the fixed point set 𝔰_{μ}^{σ} = 𝔰𝔭_{1}^{ϵ}. Thanks to the previous Lemma, up to conjugation of σ by an element of Aut(𝔤^{ℂ}), we can assume that σ preserves the Cartan subalgebra 𝔥^{ℂ}, commutes with τ, satisfies σH_{μ} = ϵH_{μ} and, in the case ϵ = −1, coincides with τ on 𝔰_{μ}. Then θ := στ is an involutive automorphism which commutes with τ, preserves 𝔥^{ℂ}, satisfies θH_{μ} = −ϵH_{μ} and, in the case ϵ = −1, verifies θ|_{𝔰_{μ}} = Id. Conversely, let θ be an involutive automorphism of 𝔤^{ℂ} which preserves 𝔥^{ℂ}, commutes with τ, satisfies θH_{μ} = −ϵH_{μ} and, in the case ϵ = −1, θ|_{𝔰_{μ}} = Id. Then σ := θτ is an anti-linear involution which preserves 𝔰_{μ} and has the fixed point set 𝔰_{μ}^{σ} = 𝔰𝔭_{1}^{ϵ}. In fact, if ϵ = −1 then σ = τ on 𝔰_{μ}, which shows that (𝔰_{μ})^{σ} = (𝔰_{μ})^{τ} = 𝔰𝔭_{1}. If ϵ = +1 then σH_{μ} = H_{μ}, which shows that (𝔰_{μ})^{σ} is non-compact, since it contains the non-compact element H_{μ}.

This Theorem reduces the problem of classification of ϵ-quaternionic Kähler symmetric spaces to the problem of classification of involutive automorphisms of a simple complex Lie algebra 𝔤^{ℂ} which preserve the Cartan subalgebra 𝔥^{ℂ} and satisfy the following conditions: ϵ = 1: θH_{μ} = −H_{μ} ϵ = −1: θ = Id on 𝔰_{μ}. As we shall see, this can be done using the Kac diagram of θ.

5 Classification of pseudo-quaternionic Kähler symmetric spaces

5.1 Classification of pseudo-quaternionic Kähler symmetric spaces of inner type

5.1.1 Involutive inner automorphisms of complex simple Lie algebras

We recall the description of involutive automorphisms of inner type of a complex simple Lie algebra 𝔤^{ℂ} in terms of Kac diagrams.

Let R be the root system of a complex simple Lie algebra 𝔤^{ℂ} with respect to a Cartan subalgebra 𝔥^{ℂ}, R^{+} a system of positive roots and Π = {α_{1}, …, α_{l}} the corresponding system of simple roots of R. We denote by Π̃ := Π ∪ {α_{0}} the extended system of simple roots, where α_{0} = −μ is the lowest root. We denote by n_{i} the coordinates of μ with respect to Π,

μ = ∑_{i=1}^{l} n_{i}α_{i} ,

and put n_{0} = 1. The extended system of simple roots is represented by the extended Dynkin diagram with the labels n_{i} over the nodes associated with α_{i}, see the table 3 [GOV].

Definition 3 An involutive Kac diagram of a semisimple Lie algebra 𝔤^{ℂ} is the extended Dynkin diagram of 𝔤^{ℂ} labeled by non-negative integers p_{i} over the nodes α_{i}, which satisfy the relation

∑_{i=0}^{l} n_{i}p_{i} = 2 .

An involutive Kac diagram defines a decomposition R = R_{0} ∪ R_{1}, where

R_{a} := {α = ∑_{i=1}^{l} k_{i}α_{i} ∈ R | ∑ k_{i}p_{i} ≡ a (mod 2)} ,

for a = 0, 1.

Notice that R_{0} is a root system and that its Dynkin diagramm Π_{0} is obtained from the Kac diagramm by deleting all nodes which have non-zero labels. We denote by \overrightarrow{p} = (p_{0}, ⋯, p_{l}) the (l + 1)-vector with the components p_{i}.

Note that any semi-simple inner automorphism of a complex semisimple Lie algebra 𝔤^{ℂ} is of the form Ad_{g}, g = exp x, where x is an element of a Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤^{ℂ}. V. Kac described all x ∈ 𝔥^{ℂ} which define an automorphism of finite order using Kac diagrams, see [GOV]. The following theorem is a special case.

Theorem 7 There exists a natural one-to-one correspondence between involutive Kac diagrams, up to automorphisms of the extended Dynkin diagram, and involutive inner automorphisms of 𝔤^{ℂ}, up to conjugation by automorphisms of 𝔤^{ℂ}. More precisely, an involutive Kac diagram with labels p_{i} defines an involutive automorphism θ with eigenspace decomposition

𝔤^{ℂ} = 𝔤_{+}^{ℂ} + 𝔤_{−}^{ℂ} ,

such that θ|_{𝔤_{±}^{ℂ}} = ± Id, where

𝔤_{+}^{ℂ} = 𝔥^{ℂ} + ∑_{α∈R_{0}} ℂE_{α}  (5.1)
𝔤_{−}^{ℂ} = ∑_{α∈R_{1}} ℂE_{α} .  (5.2)

Applying this theorem, for each simple complex Lie algebra we will now give the list of involutive automorphisms. We indicate also the corresponding non-compact Riemannian symmetric space S(θ) = G/G_{+} associated with the Cartan decomposition

𝔤 = 𝔤_{+} + 𝔤_{−} ,  (5.3)

where 𝔤 = 𝔤(θ) = (𝔤^{ℂ})^{σ} is the real form of 𝔤^{ℂ} associated with the antilinear involution σ = θ ∘ τ and 𝔤_{±} = 𝔤_{±}^{ℂ} ∩ 𝔤. Recall that τ is the standard compact antilinear involution associated with the Cartan subalgebra 𝔥^{ℂ}.

Corollary 4 The non-trivial involutive automorphisms of the simple complex Lie algebras 𝔤^{ℂ} (up to conjugation in Aut(𝔤^{ℂ})) are defined by the following involutive Kac diagrams. For each case, the corresponding system of positive roots R_{0}^{+} = R_{0} ∩ R^{+} of 𝔤_{+}^{ℂ} is indicated together with the symmetric space S(θ).

A_{l}) System of positive roots: R^{+} = {ϵ_{i} − e_{j} | 1 ≤ i < j ≤ l + 1}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, Highest root: μ = ϵ_{1} − ϵ_{l+1} = ∑_{i=1}^{l} α_{i}, i.e. \overrightarrow{n} = (1, …, 1), Extended Dynkin diagram Π̃:

[figure: •; α_{1}; 1; •; 1; α_{0}; •; α_{l}; 1; •; α_{l−1}; 1; ⋯; •; 1; α_{2}; ⋯]

AIII.m) Kac diagram:

[figure: •; •; 1; •; •; ⋯; •; ⋯; •; α_{m}; 1]

\overrightarrow{p}(m) = (\underbrace{1, 0, … 0, 1}_{m+1}, 0, …, 0), where 1 ≤ m ≤ l,

R_{0}^{+} = {ϵ_{i} − ϵ_{j} | i < j ≤ m or j > i ≥ m + 1},

S(θ) = SU(m, n − m)/S(U(m) × U(n − m)), where n = l + 1.

B_{l}) System of positive roots: R^{+} = {ϵ_{i} ± e_{j} | 1 ≤ i < j ≤ l + 1} ∪ {ϵ_{i} | i = 1, …, l}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, l − 1, α_{l} = ϵ_{l}, Highest root: μ = ϵ_{1} + ϵ_{2} = α_{1} + 2 ∑_{i=2}^{l} α_{i}, i.e. \overrightarrow{n} = (1, 1, 2, …, 2), Extended Dynkin diagram Π̃:

[figure: •; α_{0}; 1; •; α_{1}; 1; •; α_{2}; 2; •; α_{3}; 2; ⋯; •; α_{l−2}; 2; ⋯; •; α_{l−1}; 2; •; >; •; α_{l}; 2]

BI.1) Kac diagram:

[figure: •; 1; •; 1; •; •; ⋯; •; ⋯; •; •; >; •]
\overrightarrow{p} = (1, 1, 0 … 0),
R_{0}^{+} = {ϵ_{i} ± ϵ_{j} | 2 ≤ i < j} ∪ {ϵ_{i} | i = 1, …, l},
S(θ) = SO_{0}(2, 2l − 1)/SO(2) × SO(2l − 1).

BI.m) Kac diagram:

[figure: •; •; •; •; ⋯; •; •; 1; α_{m}; ⋯; •; •; >; •]

\overrightarrow{p}(m) = (\underbrace{0, …, 0, 1}_{m+1}, 0, …, 0), where 2 ≤ m ≤ l,

R_{0}^{+} = {ϵ_{i} ± ϵ_{j} | i < j ≤ m or j > i ≥ m + 1} ∪ {ϵ_{i} | i ≥ m + 1}, S(θ) = SO_{0}(2m, 2(l − m) + 1)/SO(2m) × SO(2(l − m) − 1).

C_{l}) System of positive roots: R^{+} = {ϵ_{i} ± e_{j} | 1 ≤ i < j ≤ l} ∪ {2ϵ_{i} | i = 1, …, l}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, l − 1, α_{l} = 2ϵ_{l}, Highest root: μ = 2ϵ_{1} = 2 ∑_{i=1}^{l−1} α_{i} + α_{l}, i.e. \overrightarrow{n} = (1, 2, …, 2, 1), Extended Dynkin diagram Π̃:

• α^{0} > 1  • α^{1} 2 • α^{2} 2⋯  • α^{l−2} 2 ⋯  • α^{l−1} 2• < • α^{l} 1

CI) Kac diagram:

• > 1  • •⋯  • ⋯  • • < •1

\overrightarrow{p} = (1, 0 … 0, 1), R_{0}^{+} = {ϵ_{i} − ϵ_{j} | i < j}, S(θ) = Sp(2l, ℝ)/U(l).

CII.m) Kac diagram:

• > • •⋯  • •1 α^{m}  ⋯  • • < •

\overrightarrow{p}(m) = (\underbrace{0, …, 0, 1}_{m+1}, 0, …, 0), where 1 ≤ m ≤ l − 1,

R_{0}^{+} = {ϵ_{i} ± ϵ_{j} | i < j ≤ m or j > i ≥ m + 1} ∪ {2ϵ_{i} | i ≥ l}, S(θ) = Sp(m, l − m)/Sp(m) × Sp(l − m).

D_{l}) System of positive roots: R^{+} = {ϵ_{i} ± ϵ_{j} | 1 ≤ i < j ≤ l}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, l − 1, α_{l} = ϵ_{l−1} + ϵ_{l}, Highest root: μ = ϵ_{1} + ϵ_{2} = α_{1} + 2 ∑_{i=2}^{l−2} α_{i} + α_{l−1} + α_{l}, i.e. \overrightarrow{n} = (1, 1, 2, …, 2, 1, 1), Extended Dynkin diagram Π̃:

[figure: •; α_{0}; 1; •; α_{1}; 1; •; α_{2}; 2; •; α_{3}; 2; ⋯; •; α_{l−3}; 2; ⋯; •; α_{l−2}; 2; •; •; α_{l−1}; 1; •; α_{l}; 1]

DI.1) Kac diagram:

[figure: •; 1; •; 1; •; •; ⋯; •; ⋯; •; •; •; •]

\overrightarrow{p} = (1, 1, 0, …, 0), R_{0}^{+} = {ϵ_{i} ± ϵ_{j} | 2 ≤ i < j}, S(θ) = SO_{0}(2, 2l − 2)/SO(2) × SO(2l − 2).

DI.m) Kac diagram:

[figure: •; •; •; •; ⋯; •; •; 1; α_{m}; ⋯; •; •; •; •]

\overrightarrow{p}(m) = (\underbrace{0, …, 0, 1}_{m+1}, 0, …, 0), where 2 ≤ m ≤ l − 2,

R_{0}^{+} = {ϵ_{i} ± ϵ_{j} | i < j ≤ m or j > i ≥ m + 1}, S(θ) = SO_{0}(2m, 2l − 2m)/SO(2m) × SO(2l − 2m).

DIII) Kac diagram:

[figure: •; 1; •; •; •; ⋯; •; ⋯; •; •; •; •; 1]

\overrightarrow{p} = (1, 0, …, 0, 1), R_{0}^{+} = {ϵ_{i} − ϵ_{j} | i < j}, S(θ) = SO^{∗}(2l)/U(l).

E_{6}) System of positive roots: R^{+} = {ϵ_{i} − ϵ_{j} , ϵ_{i} + ϵ_{j} + ϵ_{k} + ϵ , 2ϵ | 1 ≤ i < j < k ≤ 6}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, 5, α_{6} = ϵ_{4} + ϵ_{5} + ϵ_{6} + ϵ, Highest root: μ = 2ϵ = α_{1} + 2α_{2} + 3α_{3} + 2α_{4} + α_{5} + 2α_{6}, i.e. \overrightarrow{n} = (1, 1, 2, 3, 2, 1, 2), Extended Dynkin diagram Π̃:

• α_{1} 1  • α_{2} 2  α_{3} 3• •α_{6} 2 •α_{0} 1  α_{4} 2•  • α_{5} 1

EII) Kac diagram:

•  •1  • • •  •  •

\overrightarrow{p} = (0, 0, 1, 0, 0, 0, 0), R_{0}^{+} = {ϵ_{1} − ϵ_{2} , ϵ_{i} − ϵ_{j} | 3 ≤ i < j ≤ 6} ∪ {ϵ_{1} + ϵ_{2} + ϵ_{k} + ϵ | 3 ≤ k ≤ 6} ∪ {ϵ_{i} + ϵ_{j} + ϵ_{k} + ϵ | 3 ≤ i < j < k ≤ 6} ∪ {2ϵ}, S(θ) = E_{6(2)}/SU(2)SU(6).

EIII) Kac diagram:

•1  •  • • •  •  •1

\overrightarrow{p} = (0, 1, 0, 0, 0, 1, 0), R_{0}^{+} = {ϵ_{1}−ϵ_{6} , ϵ_{i}−ϵ_{j} , ϵ_{i}+ϵ_{j} +ϵ_{6}+ϵ , 2ϵ | 2 ≤ i < j ≤ 5}∪{ϵ_{1}+ϵ_{j} +ϵ_{k} +ϵ | 2 ≤ j < k ≤ 5}, S(θ) = E_{6(−14)}/SO(2)Spin(10).

E_{7}) System of positive roots: R^{+} = {ϵ_{i} − ϵ_{j} | 1 ≤ i < j ≤ 7} ∪ {−ϵ_{i} + ϵ_{8} | i < 8} ∪{ϵ_{i} + ϵ_{j} + ϵ_{k} + ϵ_{8} | 1 ≤ i < j < k ≤ 7}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, 6, α_{7} = ϵ_{5} + ϵ_{6} + ϵ_{7} + ϵ_{8}, Highest root: μ = −ϵ_{7} + ϵ_{8}, \overrightarrow{n} = (1, 1, 2, 3, 4, 3, 2, 2), Extended Dynkin diagram Π̃:

• α_{1} 1  • α_{2} 2  α_{3} 3•  •α_{7} 2 α_{4} 4•  •  • α_{6} 2  • α_{0} 1 α_{5} 3

EV) Kac diagram:

•  •  •  • 1 •  •  •  •

\overrightarrow{p} = (0, 0, 0, 0, 0, 0, 0, 1), R_{0}^{+} = {ϵ_{i} − ϵ_{j} | 1 ≤ i < j ≤ 7} ∪ {−ϵ_{i} + ϵ_{8} | i < 8}, S(θ) = E_{7(7)}/SU(8).

EVI) Kac diagram:

•  •1  •  • •  •  •  •

\overrightarrow{p} = (0, 0, 1, 0, 0, 0, 0, 0), R_{0}^{+} = {ϵ_{1} −ϵ_{2} , ϵ_{i} −ϵ_{j} | 3 ≤ i < j ≤ 7}∪{−ϵ_{k} +ϵ_{8} , ϵ_{1} +ϵ_{2} +ϵ_{k} +ϵ_{8} | 3 ≤ k ≤ 7} ∪{ϵ_{i} + ϵ_{j} + ϵ_{k} + ϵ_{8} | 3 ≤ i < j < k ≤ 7}, S(θ) = E_{7(−5)}/SU(2)Spin(12).

EVII) Kac diagram:

•1  •  •  • •  •  •  •1

\overrightarrow{p} = (1, 1, 0, 0, 0, 0, 0, 0), R_{0}^{+} = {ϵ_{1} − ϵ_{7} , ϵ_{i} − ϵ_{j} | 2 ≤ i < j ≤ 7} ∪ {ϵ_{i} + ϵ_{j} + ϵ_{k} + ϵ_{8} | 2 ≤ i < j < k ≤ 7}, S(θ) = E_{7(−25)}/SO(2)E_{6}.

E_{8}) R^{+} = {ϵ_{i} − ϵ_{j} | 1 ≤ i < j ≤ 9} ∪ {ϵ_{i} + ϵ_{j} + ϵ_{k} | 1 ≤ i < j < k ≤ 9}, Simple roots: α_{i} = ϵ_{i} − ϵ_{i+1}, i = 1, …, 7, α_{8} = ϵ_{6} + ϵ_{7} + ϵ_{8}, Highest root: μ = ϵ_{1} − ϵ_{9}, \overrightarrow{n} = (1, 2, 3, 4, 5, 6, 4, 2, 3), Extended Dynkin diagram Π̃:

• α_{0} 1  • α_{1} 2  α_{2} 3•  •α_{8} 3 α_{3} 4•  •  •α_{5} 6  • α_{6} 4  • α_{7} 2 α_{4} 5

EVIII) Kac diagram:

•  •  •  • •  •  •  •  •1

\overrightarrow{p} = (0, 0, 0, 0, 0, 0, 0, 1, 0), R_{0}^{+} = {ϵ_{i} − ϵ_{j} , ϵ_{i} + ϵ_{j} + ϵ_{8} | 1 ≤ i < j ≤ 7} ∪ {ϵ_{i} − ϵ_{9} , ϵ_{i} + ϵ_{8} + ϵ_{9} | i ≤ 7}, S(θ) = E_{8(8)}/Spin(16).

EIX) Kac diagram:

•  •1  •  • •  •  •  •  •

\overrightarrow{p} = (0, 1, 0, 0, 0, 0, 0, 0, 0), R_{0}^{+} = {ϵ_{1} − ϵ_{9} , ϵ_{i} − ϵ_{j} | 2 ≤ i < j ≤ 8} ∪ {ϵ_{i} + ϵ_{j} + ϵ_{k} | 2 ≤ i < j < k ≤ 8} ∪{ϵ_{1} + ϵ_{j} + ϵ_{9} | 2 ≤ j ≤ 8}, S(θ) = E_{8(−24)}/SU(2)E_{7}.

F_{4}) System of positive roots: R^{+} = {ϵ_{i} | 1 ≤ i ≤ 4} ∪ {ϵ_{i} ± ϵ_{j} | 2 ≤ i < j ≤ 4} ∪ {ϵ_{1} ± ϵ_{j} | 2 ≤ j ≤ 4} ∪ {(ϵ_{1} + s_{2}ϵ_{2} + s_{3}ϵ_{3} + s_{4}e_{4})/2 | s_{i} = ±1}, Simple roots: α_{1} = (ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4})/2, α_{2} = ϵ_{4}, α_{3} = ϵ_{3} − ϵ_{4}, α_{4} = ϵ_{2} − ϵ_{3}, Highest root: μ = ϵ_{1} + ϵ_{2}, \overrightarrow{n} = (1, 2, 4, 3, 2),, Extended Dynkin diagram Π̃:

• α_{1} 2  • α_{2} 4 <  • α_{3} 3  • α_{4} 2  • α_{0} 1

F I) Kac diagram:

•  • <  •  •1  •

\overrightarrow{p} = (0, 0, 0, 0, 1), R_{0}^{+} = {ϵ_{3} , ϵ_{4} , ϵ_{3} ± ϵ_{4} , ϵ_{1} ± ϵ_{2} , (ϵ_{1} − ϵ_{2} ± ϵ_{3} ± ϵ_{4})/2}, S(θ) = F_{4(4)}/Sp(3)SU(2).

F II) Kac diagram:

•1  • <  •  •  •

\overrightarrow{p} = (0, 1, 0, 0, 0), R_{0}^{+} = {ϵ_{i} | 1 ≤ i ≤ 4} ∪ {ϵ_{i} ± ϵ_{j} | 2 ≤ i < j ≤ 4},

S(θ) = F_{4(−20)}/Spin(9).

G_{2}) System of positive roots: R^{+} = {ϵ_{1} , −ϵ_{2} , −ϵ_{3}} ∪ {ϵ_{i} − ϵ_{j} | 1 ≤ i < j ≤ 3}, Simple roots: α_{1} = −ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3} Highest root: μ = ϵ_{1} − ϵ_{3}, \overrightarrow{n} = (1, 3, 2), Extended Dynkin diagram Π̃:

• α_{1} 3 <  • α_{2} 2  • α_{0} 1

G) Kac diagram:

• <  •1  •

\overrightarrow{p} = (0, 0, 1),

R_{0}^{+} = {−ϵ_{2} , ϵ_{1} − ϵ_{3}},

S(θ) = G_{2(2)}/SO(4).

5.1.2 Inner real forms of pseudo-quaternionic type

Here we give a necessary and sufficient condition for a real form 𝔤 = 𝔤(θ) of inner type of a complex simple Lie algebra 𝔤^{ℂ} to admit a symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type, i.e. such that the corresponding pseudo-Riemannian symmetric space M = G/K is pseudo-quaternionic Kähler. Moreover, we describe all such symmetric decompositions in terms of the Kac diagramm of θ. Recall that an involutive Kac diagram defines an involution of 𝔤^{ℂ} which acts trivially on the fixed Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤^{ℂ}.

Theorem 8 Let θ = θ(\overrightarrow{p}) be an inner involutive automorphism of a complex simple Lie algebra 𝔤^{ℂ} associated with a Kac diagram with labels \overrightarrow{p}. The corresponding real form 𝔤 = 𝔤(θ) = (𝔤^{ℂ})^{σ}, σ = θ ∘ τ, of 𝔤^{ℂ} admits a symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type if and only if the root system R_{0} ⊂ R of 𝔤_{+}^{ℂ} = (𝔤^{ℂ})^{θ} contains a simple long root ν of R. Any such root ν defines a compact three-dimensional subalgebra

𝔰_{ν}^{σ} = 𝔰_{ν} ∩ 𝔤 = span{iH_{ν}, i(E_{ν} + E_{−ν}), E_{ν} − E_{−ν}}

and a symmetric decomposition of pseudo-quaternionic type:

𝔤 = 𝔨 + 𝔪 , 𝔨 = N_{𝔤}(𝔰_{ν}^{σ}) .  (5.4)

Conversely, any symmetric decomposition of 𝔤 of pseudo-quaternionic type is conjugate by an automorphism of 𝔤 to one of the above decompositions. Two such decompositions are conjugate by an automorphism of 𝔤 if and only if the corresponding roots are related by an automorphism of the root system R which preserves R_{0}.

Proof: Assume that ν ∈ R_{0} is a simple long root. Then the corresponding threedimensional subalgebra 𝔰_{ν} ⊂ 𝔤^{ℂ} satisfies the assumptions of Theorem 6 for ϵ = −1 and hence defines a symmetric decomposition (5.4) of pseudo-quaternionic type.

Conversely, assume that 𝔤 = 𝔤(θ) = (𝔤^{ℂ})^{σ} admits a symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type. Then according to Proposition 2 and Lemma 5, 𝔨 = N_{𝔤}(𝔰_{μ}), where 𝔰_{μ} ⊂ 𝔤^{ℂ} is a σ-invariant three-dimensional subalgebra associated with a long root μ, with respect to some σ-invariant Cartan subalgebra, such that 𝔰_{μ}^{σ} = span{iH_{μ}, i(E_{μ}+E_{−μ}), E_{μ}−E_{−μ}} is compact. Since any two maximal compact subalgebras of 𝔤 are conjugate we can assume that 𝔰_{μ}^{σ} is contained in the maximal compact subalgebra 𝔤_{+} ⊂ 𝔤, see (5.3). There exists a Cartan subalgebra 𝔥′ of 𝔤_{+} which contains iH_{μ}. It normalizes 𝔰_{μ}^{σ}, since ℂE_{±μ} are the ±2 eigenspaces of ad_{H_{μ}} on 𝔤^{ℂ}.

Notice that H_{μ} ∈ (𝔥′)^{ℂ} is dual to a root μ′ with respect to the Cartan subalgebra (𝔥′)^{ℂ} of 𝔤_{+}^{ℂ}, which is also a Cartan subalgebra of 𝔤^{ℂ}, since rk 𝔤_{+}^{ℂ} = rk 𝔤^{ℂ}. The root μ′ of 𝔤^{ℂ} is again long, because ⟨μ′, μ′⟩ = 4/⟨H_{μ}, H_{μ}⟩ = ⟨μ, μ⟩. Here we used the formula ⟨H_{μ}, ·⟩ = \frac{2}{⟨μ,μ⟩}μ. To sum up, we have proven that the root system R′_{0} of 𝔤_{+}^{ℂ} with respect to the Cartan subalgebra (𝔥′)^{ℂ} contains a long root. Since any two Cartan subalgebras of 𝔤_{+}^{ℂ} are conjugate, also R_{0} contains a long root of 𝔤^{ℂ} with respect to 𝔥^{ℂ}.

Now we prove the last statement. Let ν, ν′ be two long roots in R_{0} which are related by an automorphism φ of (R, R_{0}). It extends to an automorphism of 𝔤^{ℂ}, which preserves the Cartan subalgebra 𝔥^{ℂ}. This implies that it commutes with θ, τ and σ = θ ∘ τ. So it defines an automorphism of 𝔤 which maps the symmetric decomposition of 𝔤 defined by ν to the one defined by ν′.

Conversely, let φ be an automorphism of 𝔤 which maps the symmetric decomposition 𝔤 = 𝔨 + 𝔪, 𝔨 = N_{𝔤}(𝔰_{ν}^{σ}), of pseudo-quaternionic type to another such decomposition 𝔤 = 𝔨′ + 𝔪′, 𝔨′ = N_{𝔤}(𝔰_{ν′}^{σ}). Without loss of generality, we may assume that φ𝔰_{ν}^{σ} = 𝔰_{ν′}^{σ}. We denote by 𝔤_{+}(ν) the simple summand of 𝔤_{+} which contains 𝔰_{ν}^{σ}.

If 𝔰_{ν′}^{σ} belongs to the same component, then there exists an element of the Weyl group W (R_{0}) ⊂ O(i𝔥^{∗}) of R_{0} which transforms ν to ν′. Remark that W (R_{0}) ⊂ W (R). This follows from the fact that any element of W (R_{0}) is induced by an inner automorphism Ad_{g}, g ∈ G_{+}^{ℂ} ⊂ G^{ℂ}, of 𝔤_{+}^{ℂ} which preserves the Cartan subalgebra 𝔥^{ℂ}. The corresponding inner automorphism of 𝔤^{ℂ} induces an automorphism of R ⊂ i𝔥^{∗}.

If 𝔰_{ν′}^{σ} ⊄ 𝔤_{+}(ν) then 𝔰_{ν′}^{σ} belongs to an other simple summand φ𝔤_{+}(ν) ⊂ 𝔤_{+}. Composing φ with an inner automorphism of 𝔤^{+}, we obtain an automorphism φ′ of 𝔤 which preserves the Cartan subalgebra 𝔥 ⊂ 𝔤 and maps 𝔤_{+}(ν) into φ𝔤_{+}(ν). The automorphism φ′ induces an automorphism α of (R, R_{0}). The root ν′′ = αν′ is again a long root of R which belongs to the indecomposable root system of the simple Lie algebra 𝔤_{+}(ν)^{ℂ}. As above, we conclude that there exists an element w ∈ W (R_{0}) ⊂ W (R) such that wν′′ = ν. Then w ∘ α is an automorphism of (R, R_{0}) such that wαν′ = ν.

Corollary 5 Let 𝔤 = 𝔤(θ) be a real form of inner type of a simple complex Lie algebra 𝔤^{ℂ}. There is a natural one-to-one correspondence between symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type (up to conjugation by an automorphism) and connected components of the Dynkin diagram Π_{0} which contain a long root of Π (up to the action of automorphisms of (𝔤^{ℂ}, 𝔤_{+}^{ℂ}) on the simple summands of 𝔤_{+}^{ℂ}).

Corollary 6 Under the assumptions of the previous theorem, a real form 𝔤 = 𝔤(θ) admits a unique symmetric decomposition of pseudo-quaternionic type (up to conjugation by an automorphism) if and only if R_{0} contains only one indecomposable component with a long root (up to the action of automorphisms of (𝔤^{ℂ}, 𝔤_{+}^{ℂ}) on the simple summands of 𝔤_{+}^{ℂ}).

As a corollary of the previous results we obtain:

Theorem 9 Let M = G/K be a simply connected symmetric pseudo-quaternionic Kähler manifold, where G is a (non-compact) real form of inner type of a complex simple Lie group. Then M is one of the following symmetric pseudo-quaternionic Kähler manifolds.

A)

\frac{SU(p + 2, q)}{S(U(2) × U(p, q))} ,

BD)

\frac{SO_{0}(p + 4, q)}{SO(4) × SO_{0}(p, q)} , where pq ≡ 0 (mod 2) , \frac{SO^{∗}(2l + 4)}{SO^{∗}(4) × SO^{∗}(2l)} ,

C)

\frac{Sp(p + 1, q)}{Sp(1) × Sp(p, q)} ,

E6)

\frac{E_{6(2)}}{SU(2)SU(6)} , \frac{E_{6(2)}}{SU(2)SU(2, 4)} , \frac{E_{6(−14)}}{SU(2)SU(2, 4)} ,

E7)

\frac{E_{7(−5)}}{SU(2)Spin(12)} , \frac{E_{7(−5)}}{SU(2)Spin_{0}(4, 8)} , \frac{E_{7(7)}}{SU(2)SO^{∗}(12)} , \frac{E_{7(−25)}}{SU(2)SO^{∗}(12)} ,

E8)

\frac{E_{8(−24)}}{SU(2)E_{7(133)}} , \frac{E_{8(−24)}}{SU(2)E_{7(−5)}} , \frac{E_{8(8)}}{SU(2)E_{7(−5)}} ,

F4)

\frac{F_{4(4)}}{Sp(1)Sp(3)} , \frac{F_{4(4)}}{Sp(1)Sp(1, 2)} , \frac{F_{4(−20)}}{Sp(1)Sp(1, 2)} ,

G2)

\frac{G_{2(2)}}{SO(4)} .

Proof: To obtain the above list, one has to determine the normalizer N_{G}(SU(2)) of the three-dimensional compact subgroup SU(2) ⊂ G_{+} associated to a long root (for each of the connected components of the Dynkin diagramm Π_{0} of G_{+}). This is not difficult, using the fact that the normalizer N_{G_{+}}(SU(2)) in the maximal compact subgroup G_{+} ⊂ G is a maximal compact subgroup of N_{G}(SU(2)) and that we know the complexification of N_{G}(SU(2)).

5.2 Classification of pseudo-quaternionic Kähler symmetric spaces of outer type

5.2.1 Involutive outer automorphisms of complex simple Lie algebras

Any involutive automorphism r of the Dynkin diagram of a complex simple Lie algebra 𝔤^{ℂ} has an extension to an automorphism ρ of 𝔤^{ℂ}, which preserves the Cartan subalgebra 𝔥^{ℂ}, the system of simple roots Π and acts as the permutation r on Π. We will always assume that dim(𝔤^{ℂ})^{ρ} =: l(ρ) is minimal, i.e. dim(𝔤^{ℂ})^{ρ} ≤ dim(𝔤^{ℂ})^{ρ′} for any other such extension ρ′.

Let 𝔥^{ℂ} = 𝔥_{+}^{ℂ} + 𝔥_{−}^{ℂ} be the ρ-eigenspace decomposition of 𝔥^{ℂ}. Then we have the root decomposition

(𝔤^{ℂ})^{ρ} = 𝔥_{+}^{ℂ} + ∑_{ᾱ∈R̄_{0}} 𝔤ᾱ  (5.5)

of the Lie algebra (𝔤^{ℂ})^{ρ} with respect to the Cartan subalgebra 𝔥_{+}^{ℂ} and the weight decomposition

(𝔤^{ℂ})^{ρ=−1} = 𝔥_{−}^{ℂ} + ∑_{β̄∈R̄_{1}} 𝔤β̄  (5.6)

of the (−1)-eigenspace (𝔤^{ℂ})^{ρ=−1}. We can write 𝔤ᾱ = ℂE0 ; ᾱ where E0 ; ᾱ is the root vector with the root ᾱ ∈ R̄_{0}, and 𝔤β̄ = ℂE1 ; β̄ where E1 ; β̄ is the weight vector with the weight β̄ ∈ R̄_{1}. It is known that R̄^{(ρ)} = R̄_{0} ∪ R̄_{1} is a root system in the space 𝔥_{+}^{ℂ}.

Following [GOV] , we define the system of ρ-roots R^{(ρ)} by

R^{(ρ)} = {(ᾱ, a), ᾱ ∈ R̄_{a}, a ∈ ℤ_{2} = {0, 1}}.

We will identify roots α ∈ R̄_{0} with ρ-roots (α, 0) and define the extended system of simple ρ-roots

Π̃^{(ρ)} = {β_{0}, ⋯, β_{l(ρ)}}

as a system Π^{ρ} = {β_{1}, ⋯, β_{l(ρ)}} ⊂ R^{(ρ)} of simple roots of the root system R̄_{0} together with the ρ-root β_{0} := (−η, 1), where η is the highest weight of the irreducible representation of (𝔤^{ℂ})^{ρ} on (𝔤^{ℂ})^{ρ=−1}.

The vectors β_{i} ∈ Π̃^{(ρ)}, i = 0, …, l(ρ) satisfy a unique linear relation

∑_{i=0}^{l(ρ)} n_{i}β_{i} = 0  (5.7)

normalized such that n_{0} = 2. The system Π̃^{(ρ)} is represented by the corresponding (extended) Dynkin diagram with the label n_{i} over the node β_{i}, see [GOV] p. 119, where these numbers are tabulated.

Remark that any semisimple automorphism of a complex semisimple Lie algebra 𝔤^{ℂ} is a product θ = ρ ∘ Ad_{g} of an automorphism ρ which preserves a Cartan subalgebra 𝔥^{ℂ} and a system of simple roots Π with an inner automorphism Ad_{g}, g = exp x, x ∈ 𝔥^{ℂ}. It is an outer involution if and only if it induces a nontrivial involutive permutation r = r(ρ) of Π.

Kac gave the classification of outer automorphisms of finite order of the simple complex Lie algebras. The special case of involutive automorphisms (short: involutions) can be stated as follows.

Theorem 10 There exists a natural one-to-one correspondence between elements β_{j} ∈ Π̃^{(ρ)} with label n_{j} = 2, up to automorphisms of the extended Dynkin diagram Π̃^{(ρ)}, and outer involutions θ = θ(j) of a simple complex Lie algebra 𝔤^{ℂ}, up to automorphisms of 𝔤^{ℂ}. The eigenspace decomposition 𝔤^{ℂ} = 𝔤_{+}^{ℂ} + 𝔤_{−}^{ℂ} of θ is given by

𝔤_{+}^{ℂ} = (𝔤^{ℂ})^{θ} = 𝔥_{+}^{ℂ} + ∑_{α∈R_{0}(θ)} 𝔤_{α}  (5.8)
𝔤_{−}^{ℂ} = 𝔥_{−}^{ℂ} + ∑_{α∈R_{1}(θ)} 𝔤_{α} ,  (5.9)

where

R_{0}(θ) = {±α ∈ R^{(ρ)} | α = ∑_{i=1}^{l(ρ)} k_{i}β_{i} , k_{i} ≥ 0 and k_{j} = 0} ,

R_{1}(θ) = {±α ∈ R^{(ρ)} | α = ∑_{i=1}^{l(ρ)} k_{i}β_{i} , k_{i} ≥ 0 and k_{j} = 1} ,

and

𝔤_{α} = ℂEa ; ᾱ if α = (ᾱ, a). The projection Π_{0}|_{𝔥_{+}^{ℂ}} of the subset Π_{0} = Π̃^{(ρ)} − {β_{j}} on the first component is a system of simple roots of the Lie algebra 𝔤_{+}^{ℂ} with respect to the Cartan subalgebra 𝔥_{+}^{ℂ}.

Note that R_{0}(θ)|_{𝔥_{+}^{ℂ}} is the root system of 𝔤_{+}^{ℂ}.

Since the standard compact involution τ associated with the Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤^{ℂ} commutes with θ, the product σ = θ ∘ τ is an antilinear involution. It defines the real form 𝔤 := (𝔤^{ℂ})^{σ} with the Cartan decomposition

𝔤 = 𝔤_{+} + 𝔤_{−} , 𝔤_{±} := 𝔤_{±}^{ℂ} ∩ 𝔤 .  (5.10)

The corresponding Riemannian symmetric space is denoted by S(θ) = G/G_{+}.

Corollary 7 The outer involutions of the complex simple Lie algebras 𝔤^{ℂ} are defined by the following Kac diagrams, up to automorphisms of 𝔤^{ℂ}. For each case, the corresponding extended Dynkin diagram with the labels n_{i} (see (5.7)) over the ρ-roots β_{i} and the symmetric space S(θ) is indicated.

A) 𝔤^{ℂ} = A_{2n−1}, n ≥ 3. Extended Dynkin diagram Π̃^{(ρ)} = C̃_{n}:

[figure: •; β_{n}; 2; •; β_{n−1}; 2; •; β_{n−2}; 4; •; β_{n−3}; 4; ⋯; •; β_{2}; 4; ⋯; •; β_{1}; 4; •; <; •; β_{0}; 2]

AI) Kac diagram:

[figure: •; •; •; •; ⋯; •; ⋯; •; •; <; •; 1]

S(θ) = SL(2n, ℝ)/SO(2n). AII) Kac diagram:

[figure: •; 1; •; •; •; ⋯; •; ⋯; •; •; <; •]

S(θ) = SL(n, ℍ)/Sp(n).

𝔤^{ℂ} = A_{2n}, n ≥ 2. Extended Dynkin diagram Π̃^{(ρ)} = B̃C_{n}:

β^{0} 2• > • β^{1} 4 • β^{2} 4⋯  • β^{n−2} 4 ⋯  • β^{n−1} 4• > • β^{n} 4

AI) Kac diagram:

1• > • •⋯  • ⋯  • • > •

S(θ) = SL(2n + 1, ℝ)/SO(2n + 1).

𝔤^{ℂ} = A_{2}. Extended Dynkin diagram Π̃^{(ρ)} = B̃C_{1}:

2 β^{0} • > •4 β^{1}

AI) Kac diagram:

1• > •

S(θ) = SL(3, ℝ)/SO(3).

D) 𝔤^{ℂ} = D_{n+1}, n ≥ 2. Extended Dynkin diagram Π̃^{(ρ)} = B̃_{n}:

β^{0} 2• < • β^{1} 2 • β^{2} 2⋯  • β^{n−2} 2 ⋯  • β^{n−1} 2• > • β^{n} 2

DI.0) Kac diagram:

1• < • •⋯  • ⋯  • • > •
S(θ) = SO_{0}(2n + 1, 1)/SO(2n + 1).

DI.m) Kac diagram (1 ≤ m ≤ n − 1):

• < • •⋯  •1 β^{m}  • ⋯  • • > •
S(θ) = SO_{0}(2m + 1, 2(n − m) + 1)/SO(2m + 1) × SO(2(n − m) + 1).

E_{6}) Extended Dynkin diagram Π̃^{(ρ)} = F̃_{4}:

β^{0} 2• • β^{1} 4  • β^{2} 6 < • β^{3} 4  • β^{4} 2

EI) Kac diagram:

• • • < • •1
S(θ) = E_{6(6)}/Sp(4).

EIV) Kac diagram:

1• • • < • •
S(θ) = E_{6(6)}/F_{4}.

5.2.2 Outer real forms of pseudo-quaternionic type

Here we give a necessary and sufficient condition for a real form 𝔤 = 𝔤(θ) of outer type of a complex simple Lie algebra 𝔤^{ℂ} to admit a symmetric decomposition 𝔤 = 𝔨+𝔪 of pseudoquaternionic type, i.e. such that the corresponding pseudo-Riemannian symmetric space M = G/K is pseudo-quaternionic Kähler. Moreover, we describe all such symmetric decompositions in terms of the extended Dynkin diagram Π̃^{(ρ)}. Recall that an outer involution θ = θ(j) of a simple complex Lie algebra 𝔤^{ℂ} exists only if the Dynkin diagram of 𝔤^{ℂ} has a non-trivial involutive symmetry and is then defined (up to conjugation in Aut(𝔤^{ℂ}) by a choice of node β_{j} ∈ Π̃^{(ρ)} with label n_{j} = 2. This involution preserves the Cartan subalgebra 𝔥^{ℂ} with the ±1-eigenspaces 𝔥_{±}^{ℂ}. The standard compact involution τ associated with the Cartan subalgebra 𝔥^{ℂ} ⊂ 𝔤^{ℂ} commutes with θ and the product σ = θ∘τ is an antilinear involution, which defines the real form 𝔤 = (𝔤^{ℂ})^{σ}.

Definition 4 We say that a ρ-root (ᾱ, a) ∈ R^{(ρ)} is the restriction to 𝔥_{+}^{ℂ} of a root α ∈ R if the corresponding root or weight vector Ea ; ᾱ is a root vector of (𝔤^{ℂ}, 𝔥^{ℂ}) with the root α.

Notice that any ρ-root (ᾱ, a) ∈ R_{0}(θ) which is the restriction of a root α ∈ R determines a regular three-dimensional subalgebra

𝔰_{α} = span{H_{α}, E_{α}, E_{−α}} ∩ (𝔤^{ℂ})^{θ}

of (𝔤^{ℂ})^{θ}.

Theorem 11 Let θ = θ(j) be an outer involutive automorphism of a complex simple Lie algebra 𝔤^{ℂ} associated with the node β_{j} ∈ Π̃^{(ρ)}. The corresponding real form 𝔤 = 𝔤(θ) = (𝔤^{ℂ})^{σ}, σ = θ∘τ, of 𝔤^{ℂ} admits a symmetric decomposition 𝔤 = 𝔨+𝔪 of pseudo-quaternionic type if and only if the root system R_{0}(θ) ⊂ R^{(ρ)} contains a simple root (ν̄, a) ∈ Π̃^{(ρ)} which is the restriction of a long root ν of R. Any such root ν defines a compact three-dimensional subalgebra

𝔰_{ν}^{σ} = 𝔰_{ν} ∩ 𝔤 = span{iH_{ν}, i(E_{ν} + E_{−ν}), E_{ν} − E_{−ν}}

and a symmetric decomposition of pseudo-quaternionic type:

𝔤 = 𝔨 + 𝔪 , 𝔨 = N_{𝔤}(𝔰_{ν}^{σ}) .  (5.11)

Conversely, any symmetric decomposition of 𝔤 of outer pseudo-quaternionic type is conjugate by an automorphism of 𝔤 to one of the above decompositions. Two such decompositions are conjugate by an automorphism of 𝔤 if and only if the corresponding roots ν, ν′ ∈ R are related by an automorphism of the root system R which preserves 𝔥_{+}^{ℂ}.

Proof: Assume that ν̄ ∈ Π_{0} ⊂ R̄ is the restriction of a long root ν ∈ R_{0} of R. Then the corresponding three-dimensional subalgebra 𝔰_{ν} ⊂ 𝔤^{ℂ} satisfies the assumptions of Theorem 6 for ϵ = −1 and hence defines a symmetric decomposition (5.11) of pseudo-quaternionic type.

Conversely, let 𝔤 = 𝔤(θ) = (𝔤^{ℂ})^{σ} be a real form of outer type with the Cartan decomposition 𝔤 = 𝔤_{+} + 𝔤_{−}. Assume that it admits a symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type. Then according to Proposition 2 and Lemma 5, 𝔨 = N_{𝔤}(𝔰_{μ}), where 𝔰_{μ} ⊂ 𝔤^{ℂ} is a σ-invariant three-dimensional subalgebra associated with a long root μ, with respect to some σ-invariant and τ-invariant Cartan subalgebra 𝔥^{ℂ} = 𝔥_{+}^{ℂ} +𝔥_{−}^{ℂ}, such that 𝔥_{+}^{ℂ} is a Cartan subalgebra of 𝔤_{+}^{ℂ} and 𝔰_{μ}^{σ} = span{iH_{μ}, i(E_{μ} + E_{−μ}), E_{μ} − E_{−μ}} ⊂ 𝔤_{+} is compact. There exists a Cartan subalgebra 𝔥′_{+} of 𝔤_{+} which contains iH_{μ}. It normalizes 𝔰_{μ}^{σ}, since ℂE_{±μ} are the ±2 eigenspaces of ad_{H_{μ}} on 𝔤^{ℂ}. We extend 𝔥′_{+} to a Cartan subalgebra (𝔥′)^{ℂ} = (𝔥′_{+})^{ℂ} + (𝔥′_{−})^{ℂ} of 𝔤^{ℂ}. The same argument as before, shows that (𝔥′)^{ℂ} normalizes 𝔰_{μ} and, moreover, E_{±μ} are root vectors of 𝔰_{μ} with root μ′ with respect to the Cartan subalgebra (𝔥′)^{ℂ}.

The root μ′ of 𝔤^{ℂ} is again long, because ⟨μ′, μ′⟩ = 4/⟨H_{μ}, H_{μ}⟩ = ⟨μ, μ⟩. To sum up, we have proven that the root system R′_{0}(θ) of 𝔤_{+}^{ℂ} with respect to the Cartan subalgebra (𝔥′_{+})^{ℂ} contains a root μ′|(𝔥′_{+})^{ℂ} which is the restriction of a long root μ′ of 𝔤^{ℂ} with respect to the Cartan subalgebra (𝔥′)^{ℂ}. Since any two Cartan subalgebras of 𝔤_{+}^{ℂ} are conjugate, also R_{0} contains the restriction of a long root of 𝔤^{ℂ} with respect to 𝔥^{ℂ}.

To finish the proof, we have to show that the long ρ-invariant root μ′ ∈ R can be included in an extended system of simple ρ-roots Π̃^{′(ρ)}. But this is clear, since the restriction μ′|_{𝔥_{+}^{ℂ}} is either a root from R̄_{0} or is the lowest weight of the representation of 𝔤_{+}^{ℂ} on 𝔤_{−}^{ℂ}.

Now we prove the last statement. Let ν̄, ν̄′ ∈ R̄ be restrictions of two long ρ-invariant roots ν, ν′ of R which are related by an automorphism φ of R which preserves 𝔥_{+}^{ℂ}. It extends to an automorphism of 𝔤^{ℂ}, which preserves the Cartan subalgebra 𝔥^{ℂ} and the decomposition 𝔤^{ℂ} = 𝔤_{+}^{ℂ} + 𝔤_{−}^{ℂ}. This implies that it commutes with θ, τ and σ = θ ∘ τ. So it defines an automorphism of 𝔤 which maps the symmetric decomposition of 𝔤 defined by ν to the one defined by ν′.

Conversely, let φ be an automorphism of 𝔤 which maps the symmetric decomposition 𝔤 = 𝔨 + 𝔪, 𝔨 = N_{𝔤}(𝔰_{ν}^{σ}), of pseudo-quaternionic type to another such decomposition 𝔤 = 𝔨′ + 𝔪′, 𝔨′ = N_{𝔤}(𝔰_{ν′}^{σ}). Without loss of generality, we may assume that φ𝔰_{ν}^{σ} = 𝔰_{ν′}^{σ}. We denote by 𝔤_{+}(ν) the simple summand of 𝔤_{+} which contains 𝔰_{ν}^{σ}.

If 𝔰_{ν′}^{σ} belongs to the same component, then there exists an element of the Weyl group W (R_{0}) ⊂ O(i𝔥^{∗}) of R_{0} which transforms ν to ν′. Since W (R_{0}) ⊂ W (R), this shows that ν and ν′ are related by an automorphism of (R, 𝔥_{+}^{ℂ}).

If 𝔰_{ν′}^{σ} ⊄ 𝔤_{+}(ν) then 𝔰_{ν′}^{σ} belongs to an other simple summand φ𝔤_{+}(ν) ⊂ 𝔤_{+}. Composing φ with an inner automorphism of 𝔤^{+}, we obtain an automorphism φ′ of 𝔤 which preserves the Cartan subalgebra 𝔥 ⊂ 𝔤 and maps 𝔤_{+}(ν) into φ𝔤_{+}(ν). The automorphism φ′ induces an automorphism α of R which preserves 𝔥_{+}^{ℂ}. The root ν′′ = αν′ is again a long root of R which belongs to the indecomposable root system of the simple Lie algebra 𝔤_{+}(ν)^{ℂ}. As above, we conclude that there exists an element w of the Weyl group W (R) which preserves 𝔥_{+}^{ℂ} such that wν′′ = ν. Then w ∘ α is an automorphism of R which preserves 𝔥_{+}^{ℂ} such that wαν′ = ν.

Corollary 8 Let 𝔤 = 𝔤(θ) be a real form of outer type of a simple complex Lie algebra 𝔤^{ℂ}. There is a natural one-to-one correspondence between symmetric decomposition 𝔤 = 𝔨 + 𝔪 of pseudo-quaternionic type (up to conjugation by an automorphism) and connected components of the Dynkin diagram Π_{0} which contain the restriction of a long root of R (up to the action of automorphisms of (𝔤^{ℂ}, 𝔤_{+}^{ℂ}) on the simple summands of 𝔤_{+}^{ℂ}).

Corollary 9 Under the assumptions of the previous theorem, a real form 𝔤 = 𝔤(θ) admits a unique symmetric decomposition of pseudo-quaternionic type if and only if R_{0} contains only one indecomposable component with a long root (up to the action of automorphisms of (𝔤^{ℂ}, 𝔤_{+}^{ℂ}) on the simple summands of 𝔤_{+}^{ℂ}).

Now we can prove the following classification theorem.

Theorem 12 Let M = G/K be a simply connected symmetric pseudo-quaternionic Kähler manifold, where G is a (non-compact) real form of outer type of a complex simple Lie group. Then M is one of the following symmetric pseudo-quaternionic Kähler manifolds.

A)

\frac{SL(n + 1, ℍ)}{S(GL(1, ℍ) × GL(n, ℍ))} ,

\frac{SO_{0}(p + 4, q)}{SO(4) × SO_{0}(p, q)} , where pq ≡ 1 (mod 2) ,

E6)

\frac{E_{6(6)}}{Sp(1)SL(3, ℍ)} , \frac{E_{6(−26)}}{Sp(1)SL(3, ℍ)} .

Proof: Corollary 7 shows that in all cases all roots R of a simple Lie algebra 𝔤^{ℂ} which admits an outer automorphism have the same length. Now the theorem follows from Theorem 11 and the following lemma.

Lemma 6 A root β ∈ Π̃^{(ρ)} is the restriction of a root from R if and only if it is a long root of the root system R̄ = R̄_{0} ∪ R̄_{1} and all such roots β are equivalent with respect to the transformations from the Weyl group W (R) which preserve 𝔥_{+}^{ℂ}.

Proof: This can be checked easily, using the explicit description of the root system R̄_{0} and the weight system R̄_{1}.

Consider the case 𝔤^{ℂ} = A_{2n−1}.

Let e_{1}, ⋯, e_{2n} be the standard basis of V = ℂ^{2n} and E_{ij} = e_{i} ⊗e_{j}^{∗} the associated basis of 𝔤𝔩(V). The involution ρ of 𝔤^{ℂ} = 𝔰𝔩(V) is defined by

ρ(E_{ij}) = −E_{j′i′}, ρ(E_{ij′}) = −E_{ji′} , ρ(E_{i′j}) = −E_{j′i} ,

where i, j ∈ {1, ⋯, n}, i′ = 2n + 1 − i, j′ = 2n + 1 − j. We have

𝔥_{+}^{ℂ} = {diag(x_{1}, ⋯, x_{n}, −x_{n}, ⋯, −x_{1})}

R̄_{0} = {±ϵ_{i} ± ϵ_{j} (i ≠ j)}

(a root system of type D_{n}) and

R̄_{1} = {±ϵ_{i} ± ϵ_{j} (i ≠ j), ±2ϵ_{i}}.

The corresponding root vectors are

E_{ij}^{0} = E_{ij} − E_{j′i′} , E_{ij′}^{0} = E_{ij′} − E_{ji′} , E_{i′j}^{0} = E_{i′j} − E_{j′i} , i, j = 1, ⋯, n, (i ≠ j)

and the weight vectors are

E_{ij}^{1} = E_{ij} +E_{j′i′} , E_{ij′}^{1} = E_{ij′} +E_{ji′} (i ≠ j) , E_{ii′}^{1} = E_{ii′} , E_{i′i}^{1} = E_{i′i} , i, j = 1, ⋯, n .

Note that the long weights ±2ϵ_{i} are restrictions of roots from R, since E_{ii′} and E_{i′i} are root vectors of 𝔤^{ℂ}.

The automorphism ρ corresponds to the Kac diagram AI of Corollary 7. In this case there is no long roots in Π̃^{(ρ)} − β_{0}. Diagram AII corresponds to the second involutive automorphism. In this case, there is a long root β_{0} in Π̃^{(ρ)} − β_{n} and it is a restriction of a root from R.

In the case D_{n+1} we consider a bases e_{i}, e_{−i}, i = 1, ⋯, n + 1 of the complex Euclidean space V = ℂ^{2n+2} with the Gram matrix ⟨e_{i}, e_{−j}⟩ = δ_{ij}, ⟨e_{i}, e_{j}⟩ = ⟨e_{−i,e_{−j}}⟩ = 0. The involution ρ of 𝔰𝔬(V) is defined by the automorphism ρ of V given by

ρ(e_{±(n+1)}) = e_{∓(n+1)} , ρ(e_{±i}) = e_{±i} , i ≤ n .

We have 𝔥_{+}^{ℂ} = span{e_{i} ∧ e_{−i}, i ≤ n} and

R̄_{0} = {±ϵ_{i} ± ϵ_{j} (i ≠ j) , ±ϵ_{i}}

with the corresponding root vectors

e_{±i} ∧ e_{±j} (i ≠ j) , e_{±i} ∧ e_{n+1} + e_{±i} ∧ e_{−(n+1)}

where i, j = {1, ⋯ n}. The weights are

R̄_{1} = {±ϵ_{i}}

with weight vectors

e_{±i} ∧ e_{n+1} − e_{±i} ∧ e_{−(n+1)}.

Again the long vectors from R̄ = R̄_{0} ∪ R̄_{1} are restrictions of roots from R.

In the case 𝔤^{ℂ} = E_{6} there are two involutive automorphisms θ(β_{0}) and θ(β_{5}) which define the symmetric spaces E_{6(−26)}/F_{4} and E_{6(6)}/Sp(4), respectively. In the first case

R̄_{0} = \{±ϵ_{i}, ±ϵ_{i} ± ϵ_{j}, \frac{1}{2}(±ϵ_{1} ± ϵ_{2} ± ϵ_{3} ± ϵ_{4})\}, R̄_{1} = \{±ϵ_{i}, \frac{1}{2}(±ϵ_{1} ± ϵ_{2} ± ϵ_{3} ± ϵ_{4})\}

and the (long) simple roots β_{3} = ϵ_{3} − ϵ_{4}, β_{4} = ϵ_{2} − ϵ_{3} are the restriction of roots from R.

In the second case,

R̄_{0} = {±2ϵ_{i}, ±ϵ_{i} ± ϵ_{j}}, R̄_{1} = {±ϵ_{i} ± ϵ_{j}, ±ϵ_{1} ± ϵ_{2} ± ϵ_{3} ± ϵ_{4}}

and the (long) simple root β_{3} = 2ϵ_{4} is the restriction of a root from R.

(Here we use the standard numeration of root and weight vectors of the types F_{4} and Sp(4) from [GOV].)

References

[AC] D. V. Alekseevsky and V. Cortés, Classification of indefinite hyper-Kähler symmetric spaces, Asian J. Math. 5 (2001), no. 4, 663-684.

[ABCV] D. V. Alekseevsky, N. Blažić, V. Cortés and S. Vukmirović, A class of Osserman spaces, preprint no. 3, SFB 611, University Bonn.

[Be] A. L. Besse, Einstein manifolds, Springer 1987.

[K] V. G. Kac Automorphisms of finite order of semisimple Lie algebras, Func. Analys and its Appl., 3 (1969), no. 3, 94-96.

[OV] A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, Springer 1990.

[GOV] V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Lie groups and Lie algebras III, Encylopaedia of Mathematical Sciences Vol. 41, Springer 1994.

[W] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech., 14 (1965), 1033-1048.

[W1] J. A. Wolf, Complex forms of quaternionic symmetric spaces, preprint, Sept. 2003.