A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

Vicente Cortés

Author address:

E-mail address: vicente@math.uni-bonn.de

Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany Supported by SFB 256 (Bonn University).

Contents

Abstract viii Introduction 1 1. Extended Poincaré algebras 7 2. The homogeneous quaternionic manifold (M, Q) associated to an extended

Poincaré algebra 17 3. Bundles associated to the quaternionic manifold (M, Q) 34 4. Homogeneous quaternionic supermanifolds associated to superextended

Poincaré algebras 39 Appendix. Supergeometry 44 Bibliography 59

Abstract

Let V = ℝ^{p,q} be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cℓ^{0}(V). A homogeneous quaternionic manifold (M, Q) is constructed for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map Π : ∧^{2}W → V. If the skew symmetric vector valued bilinear form Π is nondegenerate then (M, Q) is endowed with a canonical pseudo-Riemannian metric g such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold. If the metric g is positive definite, i.e. a Riemannian metric, then the quaternionic Kähler manifold (M, Q, g) is shown to admit a simply transitive solvable group of automorphisms. In this special case (p = 3) we recover all the known homogeneous quaternionic Kähler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If p > 3 then M does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kähler manifolds. Then it is shown that for q = 0 the noncompact quaternionic manifold (M, Q) can be endowed with a Riemannian metric h such that (M, Q, h) is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if p > 3.

The twistor bundle Z → M and the canonical SO(3)-principal bundle S → M associated to the quaternionic manifold (M, Q) are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution D of complex codimension one, which is a complex contact structure if and only if Π is nondegenerate. Moreover, an equivariant open holomorphic immersion Z → Z̄ into a homogeneous complex manifold Z̄ of complex algebraic group is constructed.

Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map Π : ∨^{2}W → V a homogeneous quaternionic supermanifold (M, Q) is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler supermanifold (M, Q, g) if the symmetric vector valued bilinear form Π is nondegenerate.

Key words: Quaternionic Kähler manifolds, twistor spaces, complex contact manifolds, homogeneous spaces, supermanifolds

1991 Mathematics Subject Classification. Primary: 53C30, Secondary: 53C25

Introduction

Let us start this introduction by recalling the notion of quaternionic manifold, see [A-M2]. A hypercomplex structure on a real vector space E consists of 3 complex structures (J_{1}, J_{2}, J_{3}) on E satisfying J_{1}J_{2} = J_{3}. It defines on E the structure of (left-) vector space over the quaternions ℍ = {1, i, j, k} such that multiplication by i, j and k is given, respectively, by J_{1}, J_{2} and J_{3}. The 3-dimensional subspace Q = span{J_{1}, J_{2}, J_{3}} ⊂ End(E) is what is called a quaternionic structure on E. A Euclidean scalar product ⟨·, ·⟩ on (E, Q) is called (Q-) Hermitian if Q consists of skew symmetric endomorphisms of (E, ⟨·, ·⟩). Now let M be a smooth manifold, dim M > 4. An almost quaternionic structure Q on M is a smooth field m ↦ Q_{m} whose value at m ∈ M is a quaternionic structure on T_{m}M. Q is called a quaternionic structure and (M, Q) a quaternionic manifold if there exists a torsionfree connection on T M preserving the rank 3 subbundle Q ⊂ End(T M). Now let g be a Riemannian metric on M, Hermitian with respect to an (almost) quaternionic structure Q. Then M with the structure (Q, g) is called an (almost) quaternionic Hermitian manifold. If, moreover, the Levi-Civita connection preserves Q then (M, Q, g) is said to be a quaternionic Kähler manifold. We remark that quaternionic Kähler manifolds represent one of the few basic Riemannian geometries, as defined by Berger’s list of possible Riemannian holonomy groups, see [A1], [Bes],

[Br1] and [S2]. For the possible holonomy groups of, not necessarily Riemannian, torsionfree connections see [Br2], [Schw] and references therein.

Next we review what is known about homogeneous quaternionic Kähler manifolds. First of all, quaternionic Kähler manifolds (M, Q, g) are Einstein manifolds, i.e. Ric = cg, see e.g. [A1], [Bes], [S2] and [Kak]. We discuss 3 cases depending on the sign of the constant c (which is the sign of the scalar curvature).

c = 0) Simply connected Ricci-flat quaternionic Kähler manifolds are hyper-Kähler manifolds, see [Bes] Ch. 14. These are Kähler manifolds with respect to 3 complex structures J_{1}, J_{2} and J_{3} = J_{1}J_{2}. It is a general fact that any homogeneous Ricci-flat Riemannian manifold is necessarily flat, see [A-K]. In particular, all homogeneous hyper-Kähler manifolds are flat. For hyper-Kähler manifolds of small cohomogeneity see [Bi1], [Bi2], [B-G], [D-S1], [D-S2], [K-S2], [Sw1], [Sw2] and [C4].

c > 0) It follows from Myer’s theorem that any complete Einstein manifold of positive scalar curvature is necessarily compact. In particular, any complete quaternionic Kähler manifold of positive scalar curvature is compact. It was proven in [L-S] that for every n > 1 there is only a finite number of such manifolds of dimension 4n up to homothety, cf. [Bea], [G-S], [L1], [L4], [L5], [P-S]. The only known examples are, up to now, the Wolf spaces [W1]. These are precisely the homogeneous quaternionic Kähler manifolds of positive scalar curvature and are all symmetric of compact type, cf. [A2]. More generally, the Wolf spaces can be characterized as the compact quaternionic Kähler manifolds which admit an action of cohomogeneity ≤ 1 by a compact semisimple group of isometries and which are not scalar-flat, see [D-S3], cf. [A-P].

c < 0) Complete noncompact quaternionic Kähler manifolds of negative scalar curvature exist in abundance. In fact, it was proven in [L3] that the moduli space of complete quaternionic Kähler metrics on ℝ^{4n}, n > 1, is infinite dimensional. For explicit constructions of, in general not complete, quaternionic Kähler manifolds see e.g. [G1], [G-L] and [L2].

Received by the editors March 1, 1999.

What about homogeneous examples? First of all, the noncompact duals of the Wolf spaces are symmetric (and hence homogeneous) quaternionic Kähler manifolds of negative Ricci and nonpositive sectional curvature. Moreover, like any Riemannian symmetric space of noncompact type, these manifolds admit smooth quotients by discrete cocompact groups of isometries, see [Bo], [Fi-S]. The first examples of quaternionic Kähler manifolds which are not locally symmetric were found by D.V. Alekseevsky in [A3]. An Alekseevsky space is a homogeneous quaternionic Kähler manifold of negative scalar curvature which admits a simply transitive splittable solvable group of isometries. It follows from Iwasawa’s decomposition theorem that the noncompact duals of the Wolf spaces are precisely the symmetric Alekseevsky spaces. Besides these there are 3 series of nonsymmetric Alekseevsky spaces, see [A3], [dW-VP2] and [C2]. In [A3] it was conjectured that any noncompact homogeneous quaternionic Kähler manifold admits a transitive solvable group of isometries. This conjecture is still open: up to now, the only known examples of homogeneous quaternionic Kähler manifolds of negative scalar curvature are the Alekseevsky spaces. However, by the construction presented in this work we obtain many homogeneous quaternionic pseudo-Kähler manifolds (with indefinite metric) which do not admit any transitive action of a solvable Lie group. Moreover, in 2.6 we construct a family of noncompact homogeneous quaternionic Hermitian manifolds (with positive definite metric) with no transitive solvable group of isometries.

It is natural to ask for examples of compact locally homogeneous quaternionic Kähler manifolds. The following negative result was proven in [A-C4]. Let M be a compact quaternionic Kähler manifold or, more generally, a quaternionic Kähler manifold of finite volume. If the universal cover M̃ is a homogeneous quaternionic Kähler manifold then it is necessarily symmetric. In particular, the only Alekseevsky spaces which admit smooth quotients of finite volume by discrete groups of isometries are the symmetric ones, this was as well proven in [A-C1] by a simpler method. Additionally, the symmetric Alekseevsky spaces can be characterized by the property of having nonpositive curvature, see [C2].

Given a simply transitive Lie group L of isometries acting on a Riemannian manifold M, there exists an algorithm to compute the full isometry group of M [A-W], cf. [Wo]. However, this algorithm involves the covariant derivatives (of all orders) of the curvature tensor and hence can only be applied effectively in very simple situations. If the simply transitive group L is splittable solvable and unimodular then the full isometry group is easily computed, see [G-W]. Unfortunately, the splittable solvable groups of isometries acting simply transitively on the Alekseevsky spaces are not unimodular. For that reason in [A-C1] a new algorithm was developed which completely avoids the curvature tensor and works also for nonunimodular splittable solvable groups. Using it the full isometry group of the nonsymmetric Alekseevsky spaces was determined [A-C1]. The Lie algebra of the full isometry group was previously described by the theoretical physicists de Wit, Vanderseypen and Van Proeyen by a different method, see [dW-V-VP].

In the following, we shortly comment on the attention paid to Alekseevsky’s spaces in the physical literature. There is a concept of special geometry, which evolved in the theory of strings and supergravity, see e.g. [Z], [B-W], [dW-VP1], [G-S-T]. More specifically, special Kähler geometry is the geometry associated to N = 2 supergravity in d = 4 space time dimensions coupled to vector multiplets and was first described in [dW-VP1], cf. [C-D-F], [St], [C3] and [Fr]. The (Kuranishi) moduli space of a Calabi-Yau 3-fold bears this particular geometry. Moreover, there is a construction (the “c-map”), related to Mirror Symmetry, which to any special Kähler manifold associates a quaternionic Kähler manifold, see [C-F-G] and [F-S]; for the general framework see [Cr]. Using the c-map, Cecotti [Ce] was the first to relate the classification problem for homogeneous special Kähler manifolds to Alekseevsky’s classification [A3]. In addition, he introduced Vinberg’s theory of T -algebras [V2] to describe the first nonsymmetric homogeneous special Kähler manifolds (the symmetric special Kähler manifolds were described in terms of Jordan algebras, see [C-VP] and [G-S-T]). Cecotti’s classification of homogeneous special Kähler manifolds was extended in [dW-VP2], [C3] and [A-C5]. In [C4], the hyper-Kählerian version of the c-map was used to construct a natural (pseudo-) hyper-Kähler structure on the bundle of intermediate Jacobians over the moduli space of gauged Calabi-Yau 3-folds.

In the last part of the introduction we describe the main components, results and the global structure of the present paper. The basic algebraic data of our construction are a pseudo-Euclidean vector space V, a module W over the even Clifford algebra Cℓ^{0}(V) and a 𝔰𝔭𝔦𝔫(V)-equivariant linear map Π : ∧^{2}W → V. Any such Π defines a ℤ_{2}-graded Lie algebra 𝔭 = 𝔭(Π) = 𝔭_{0} + 𝔭_{1}, where 𝔭_{0} = Lie Isom(V) = 𝔬(V) + V and 𝔭_{1} = W. Here V acts trivially on W and 𝔬(V) ≅ 𝔰𝔭𝔦𝔫(V) acts via the inclusion 𝔰𝔭𝔦𝔫(V) ⊂ Cℓ^{0}(V) on the Cℓ^{0}(V)-module W. The Lie bracket on 𝔭_{1} × 𝔭_{1} is given by Π. The Lie algebras 𝔭(Π) were introduced in [A-C2], where a basis for the vector space of 𝔰𝔭𝔦𝔫(V)-equivariant linear maps Π : ∧^{2}W → V was explicitly constructed. The Lie algebra 𝔭(Π) is called an extended Poincaré algebra of signature (p, q) if V ≅ ℝ^{p,q} has signature (p, q). A mirror symmetric version of this construction is obtained replacing Π : ∧^{2}W → V by a 𝔰𝔭𝔦𝔫(V)-equivariant linear map Π : ∨^{2}W → V. Here ∨^{2}W = Sym^{2}W denotes the symmetric square of W. The corresponding algebraic structure 𝔭(Π) = 𝔭_{0} +𝔭_{1} is now a super Lie algebra. It is called a superextended Poincaré algebra of signature (p, q). For special signatures (p, q) of space time V these super Lie algebras play an important role in the physical literature since the early days of supersymmetry and supergravity, see e.g. [Go-L] and [O-S]; for more recent contributions see e.g. [F] and references therein. Notice that if (p, q) = (1, 3) then V = ℝ^{1,3} is Minkowski space and the even subalgebra 𝔭_{0} = Lie Isom(ℝ^{1,3}) ⊂ 𝔭(Π) is the classical Poincaré algebra. The construction of all superextended Poincaré algebras (of arbitrary signature) was carried out in [A-C2]. In [A-C-D-S] the twistor equation is interpreted as the differential equation satisfied by infinitesimal automorphisms of a geometric structure modelled on the linear Lie supergroup associated to a superextended Poincaré algebra (for more on the twistor equation see e.g [K-R] and references therein). In contrast with the case of superextended Poincaré algebras, to our knowledge, extended Poincaré algebras do not occur in the physical literature before the publication of [A-C2]: The first occurrence is [D-L].

Any extended Poincaré algebra 𝔭(Π), as above, admits a derivation D with eigenspace decomposition 𝔭(Π) = 𝔬(V) + V + W and corresponding eigenvalues (0, 1, 1/2). We can extend 𝔭(Π) by D obtaining a new ℤ_{2}-graded Lie algebra 𝔤 = 𝔤(Π) = 𝔤_{0} + 𝔤_{1}, where 𝔤_{0} = ℝD + 𝔭_{0} = ℝD + 𝔬(V) + V and 𝔤_{1} = 𝔭_{1} = W. The adjoint representation of the Lie algebra 𝔤 is faithful and hence defines on 𝔤 the structure of linear Lie algebra. Let G = G(Π) ⊂ Aut 𝔤 denote the corresponding connected linear Lie group. It is the numerator of the homogeneous quaternionic manifolds M = M(Π) = G/K we are going to construct. To define the denominator K let E ⊂ V be a 3-dimensional Euclidean subspace and 𝔨 = 𝔨(E) ⊂ 𝔬(V) ⊂ 𝔤 the maximal subalgebra which preserves E, i.e. 𝔨 = 𝔬(E) ⊕ 𝔬(E^{⊥}). Now we define K = K(E) ⊂ G to be the connected linear Lie group with Lie algebra 𝔨. Our main theorem is the following, see Thm. 8:

Theorem A Let V be any pseudo-Euclidean vector space, E ⊂ V a Euclidean 3-dimensional subspace, W any Cℓ^{0}(V)-module and Π : ∧^{2}W → V any 𝔰𝔭𝔦𝔫(V)- equivariant linear map. Let M = G/K be the homogeneous manifold associated to the Lie groups G = G(Π) and K = K(E) ⊂ G constructed above. Then the following is true:

1) There exists a G-invariant quaternionic structure Q on M.

2) If Π is nondegenerate (i.e. if W ∋ s ↦ Π(s ∧ ·) ∈ W^{∗} ⊗ V is injective) then there exists a G-invariant pseudo-Riemannian metric g on M such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold.

We remark that for V = ℝ^{3,q} and W arbitrary the map Π can always be chosen such that the metric g in 2) becomes positive definite. In this special case we recover the 3 series of Alekseevsky spaces by a simple and unified construction, which completely avoids the technicalities of constructing complicated representations of Kählerian Lie algebras, see [A3] and [C2].

If in all constructions Π : ∧^{2}W → V is replaced by a 𝔰𝔭𝔦𝔫(V)-equivariant linear map Π : ∨^{2}W → V then we obtain the following analogue of Theorem A in the category of supermanifolds, see Thm. 17:

Theorem B Let V be any pseudo-Euclidean vector space, E ⊂ V a Euclidean 3-dimensional subspace, W any Cℓ^{0}(V)-module and Π : ∨^{2}W → V any 𝔰𝔭𝔦𝔫(V)- equivariant linear map. Let M = G/K be the homogeneous supermanifold associated to the Lie supergroups G = G(Π) and K = K(E) ⊂ G constructed in 4. Then the following holds:

1) There exists a G-invariant quaternionic structure Q on M.

2) If Π is nondegenerate (i.e. if W ∋ s ↦ Π(s ∨ ·) ∈ W^{∗} ⊗ V is injective) then there exists a G-invariant pseudo-Riemannian metric g on M such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler supermanifold.

(For the definition of quaternionic structure, pseudo-Riemannian metric etc. on a supermanifold see the appendix.)

Furthermore, we remark that replacing Euclidean 3-space E ≅ R^{3,0} by Lorentzian 3-space E ≅ ℝ^{1,2} one obtains a para-quaternionic version of our construction.

Finally, we outline the structure of the paper: In section 1 we discuss extended Poincaré algebras. The basic definitions and formulas are given in 1.1. To our fundamental map Π : ∧^{2}W → V and to an oriented Euclidean subspace E ⊂ V, dim E ≡ 3 (mod 4), we associate a canonical symmetric bilinear form b on W and study its properties in 1.2. Using the form b, in 1.3 we classify extended Poincaré algebras of signature (p, q), p ≡ 3 (mod 4), up to isomorphism.

In section 2 we construct the homogeneous quaternionic manifolds of Theorem A. The basic notions of quaternionic geometry are recalled in 2.2. First, see 2.1, we describe the structure of the Lie group G = G(Π) and the coset spaces M = G/K, K = K(E), where E ⊂ V is any pseudo-Euclidean subspace. The proof of Theorem A is given in 2.4. A crucial observation is that M = G/K contains the locally symmetric quaternionic pseudo-Kähler submanifold M_{0} = G_{0}/K, where G_{0} ⊂ G is the connected linear Lie group associated to 𝔤_{0} ⊂ 𝔤 = 𝔤_{0} + 𝔤_{1}. The first step consists in extending the G_{0}-invariant quaternionic and pseudo-Riemannian structures on M_{0} to G-invariant structures on M. Using the canonical symmetric bilinear form b on W introduced in 1.2 the pseudo-Riemannian metric is extended, in the nondegenerate case, to a G-invariant pseudo-Riemannian metric g on M. The quaternionic structure is always extended by the beautifully simple formula (13) to a G-invariant almost quaternionic structure Q on M. To prove that Q is a quaternionic structure, we construct a G-invariant torsionfree connection ∇ on M which preserves Q. In the nondegenerate case ∇ is simply the Levi-Civita connection of the pseudo-Riemannian metric g. We use the description of invariant connections on homogeneous manifolds in terms of Nomizu maps, see 2.3.

Then we concentrate on the Riemannian case, see 2.5. The Riemannian manifolds M(Π) are classified up to isometry using results of 1.3. We show that all these manifolds admit a non-Abelian simply transitive splittable solvable group of isometries and hence are Alekseevsky spaces, see Thm. 9. Moreover, we explain how to obtain the 3 series of Alekseevsky spaces by specifying V, W, and Π, see Thm. 10.

The quaternionic pseudo-Kähler manifolds (M(Π), Q, g) of Theorem A do not admit any transitive action of a solvable group if V = ℝ^{p,q} with p > 3, see Thm.

12. Moreover, if q = 0, then we can replace g by a G-invariant Riemannian and Q-Hermitian metric h such that (M(Π), h, g) are homogeneous quaternionic Hermitian manifolds with no transitive solvable group of isometries, see Thm. 11.

In section 3 we study the various bundles associated to the quaternionic manifolds M = M(Π) = G/K: The twistor bundle Z(M), the canonical SO(3)-principal bundle S(M) and the Swann bundle U(M). We show that G acts transitively on Z(M) and S(M) and with cohomogeneity one on U(M). In particular, Z = Z(M) is a homogeneous complex manifold of the group G = G(Π). We exhibit a Ginvariant holomorphic tangent hyperplane distribution D on Z and prove that D defines a complex contact structure on Z if and only if Π is nondegenerate. Moreover, we construct an open G-equivariant holomorphic immersion Z → Z̄ of Z into a homogeneous complex manifold of the complex algebraic group Gℂ ⊂ Aut(𝔤ℂ), see Thm. 14. This immersion is a finite covering over an open G-orbit. In the nondegenerate case Z̄ is a homogeneous complex contact manifold of the group Gℂ and the immersion is a morphism of complex contact manifolds.

In the final section 4 we extend our construction to the category of supermanifolds, proving Theorem B. We have aimed at a straightforward presentation, summarizing the needed supergeometric background in the appendix.

Acknowledgements It is a pleasure to thank my friend and coauthor Dmitry V. Alekseevsky for many intensive discussions during our productive collaboration. Also I am very grateful to Werner Ballmann and Ursula Hamenstädt for their encouragement and support. For reading the manuscript I thank Gregor Weingart. Finally, I am indebted to many collegues for hospitality and conversations, especially to Oliver Baues, Robert Bryant, Shiing-Shen Chern, Jost Eschenburg, Phillip A. Griffiths, Ernst Heintze, Alan Huckleberry, Enrique Macías, Robert Osserman, Hans-Bert Rademacher, Gudlaugur Thorbergsson and Joseph A. Wolf.

My research has received important support from the following institutions: SFB 256 (Bonn University), Alexander von Humboldt Foundation and Mathematical Sciences Research Institute (Berkeley).

1. Extended Poincaré algebras

1.1. Basic facts. Let V be a pseudo-Euclidean vector space with scalar product ⟨·, ·⟩. There exists an orthonormal basis (e_{i}), i = 1, …, n = dim V = p + q, of V such that ⟨x, x⟩ = ∑_{i=1}^{p}(x^{i})^{2} − ∑_{j=p+1}^{n}(x^{j})^{2} for all x = ∑_{i=1}^{n} x^{i}e_{i} ∈ V. Any such basis defines an isometry between V and the standard pseudo-Euclidean vector space ℝ^{p,q} of signature (p, q). The isometry group of V is the semidirect product

Isom(V) = O(V)⋉V .

Definition 1. The Lie group P(V) := Isom(V) is called the Poincaré group of V. Its Lie algebra 𝔭(V) = 𝔬(V) + V is called the Poincaré algebra of V.

Next we recall some basic facts concerning the Clifford algebra Cℓ(V) = Cℓ^{0}(V)+ Cℓ^{1}(V), see [L-M]. Any unit vector x ∈ V, ⟨x, x⟩ = ±1, defines an invertible element x ∈ Cℓ(V). The group Pin(V) ⊂ Cℓ(V) generated by all unit vectors is called the pin group. Its subgroup Spin(V) := Pin(V) ∩ Cℓ^{0}(V) consisting of even elements is called the spin group. The adjoint representation

Ad : Pin(V) ⟶ O(V) ,
Ad(x)y = xyx^{−1} ∈ V , x ∈ Pin(V) , y ∈ V ,

induces a two-fold covering of the special orthogonal group:

Spin(V) ⟶^{2:1} SO(V) .

In fact, if r_{x} ∈ O(V) denotes the reflection in the hyperplane x^{⊥} ⊂ V orthogonal to a unit vector x ∈ V then the following formula

Ad(xy) = r_{x} ∘ r_{y} (1)

holds for any two unit vectors x, y ∈ V. The groups Pin(V) and Spin(V) are Lie groups whose Lie algebra is the Lie subalgebra 𝔰𝔭𝔦𝔫(V) ⊂ Cℓ^{0}(V) generated by the commutators [x, y] = xy − yx of all elements x, y ∈ V. It is canonically isomorphic to the orthogonal Lie algebra 𝔬(V) via the adjoint representation

ad = Ad_{∗} : 𝔰𝔭𝔦𝔫(V) ⟶^{∼} 𝔬(V) , ad(x)y = [x, y] , x ∈ 𝔰𝔭𝔦𝔫(V) , y ∈ V . (2)

In fact, if we identify ∧^{2}V = 𝔬(V) by

(x ∧ y)(z) := ⟨y, z⟩x − ⟨x, z⟩y , x, y, z ∈ V (3)

then ad^{−1} : ∧^{2}V = 𝔬(V) ⟶^{∼} 𝔰𝔭𝔦𝔫(V) is given by the following equation:

ad^{−1}(x ∧ y) = −\frac{1}{4}[x, y] , x, y ∈ V . (4)

In particular, ad(xy) = −2x ∧ y if x and y are orthogonal.

Any Cℓ(V)-module W can be decomposed into irreducible submodules. Depending on the signature (p, q) of V, there exist one or two irreducible Cℓ(V)- modules up to equivalence. In case there are two, they are related by the unique automorphism of Cℓ(V) which preserves V and acts on V as −Id. The restriction of an irreducible Cℓ(V)-module S to Cℓ^{0}(V) (respectively, to Spin(V) and 𝔰𝔭𝔦𝔫(V)) is, up to equivalence, independent of W and is called the spinor module of Cℓ^{0}(V) (respectively, of Spin(V) and 𝔰𝔭𝔦𝔫(V)). The spinor module S is either irreducible or S = S^{+} ⊕ S^{−} is the sum of two irreducible semispinor modules S^{±}, which may be equivalent or not, depending on the signature of V. If S^{+} and S^{−} are not equivalent, they are related by an automorphism of Cℓ^{0}(V) which preserves V and acts as an isometry on V. In the following we will freely use standard notations such as Cℓ_{p,q} = Cℓ(ℝ^{p,q}), Spin(p, q) = Spin(ℝ^{p,q}), Spin(p) = Spin(p, 0) etc., cf. [L-M].

Now let W be a module of the even Clifford algebra Cℓ^{0}(V) and Π : ∧^{2}W → V a 𝔰𝔭𝔦𝔫(V)-equivariant linear map. Given these data we extend the Lie bracket on 𝔭(V) to a Lie bracket [·, ·] on the vector space 𝔭(V) + W by the following requirements:

1) W is a 𝔭(V)-submodule with trivial action of V and action of 𝔬(V) defined by ad^{−1} : 𝔬(V) → 𝔰𝔭𝔦𝔫(V) ⊂ Cℓ^{0}(V), see equation (2),

2) [s, t] = Π(s ∧ t) for all s, t ∈ W.

The reader may observe that the Jacobi identity follows from 1) and 2). The resulting Lie algebra will be denoted by 𝔭(Π). Note that 𝔭 = 𝔭(Π) has a ℤ_{2}-grading

𝔭 = 𝔭_{0} + 𝔭_{1} , 𝔭_{0} = 𝔭(V) , 𝔭_{1} = W ,

compatible with the Lie bracket, i.e. [𝔭_{a}, 𝔭_{b}] ⊂ 𝔭_{a+b}, a, b ∈ ℤ_{2} = ℤ/2ℤ. In other words, 𝔭(Π) is a ℤ_{2}-graded Lie algebra.

Definition 2. Any ℤ_{2}-graded Lie algebra 𝔭(Π) = 𝔭(V) + W as above is called an extended Poincaré algebra (of signature (p, q) if V ≅ ℝ^{p,q}). 𝔭(Π) is called nondegenerate if Π is nondegenerate, i.e. if the map W ∋ s ↦ Π(s ∧ ·) ∈ W^{∗} ⊗ V is injective.

The structure of extended Poincaré algebra on the vector space 𝔭(V) + W is completely determined by the 𝔬(V)-equivariant map Π : ∧^{2}W → V (𝔬(V) acts on W via ad^{−1} : 𝔬(V) → 𝔰𝔭𝔦𝔫(V) ⊂ Cℓ^{0}(V)). The set of all 𝔬(V)-equivariant linear maps ∧^{2}W → V is naturally a vector space. In fact, it is the subspace (∧^{2}W^{∗} ⊗V)^{𝔬(V)} of 𝔬(V)-invariant elements of the vector space ∧^{2}W^{∗} ⊗V of all linear maps ∧^{2}W → V. In the classification [A-C2] an explicit basis for the vector space (∧^{2}W^{∗} ⊗ V)^{𝔬(V)} of extended Poincaré algebra structures on 𝔭(V) + W is constructed for all possible signatures (p, q) of V and any Cℓ^{0}(V)-module W.

1.2. The canonical symmetric bilinear form b. Let V = ℝ^{p,q} be the standard pseudo-Euclidean vector space with scalar product ⟨·, ·⟩ of signature (p, q). From now on we fix a decomposition p = p′ + p′′ and assume that p′ ≡ 3 (mod 4), see Remark 2 below. Remark 1: Notice that p and q are on equal footing, since any extended Poincaré algebra of signature (q, p) is isomorphic to an extended Poincaré algebra of signature (p, q). In fact, the canonical antiisometry which maps the standard orthonormal basis of ℝ^{q,p} to that of ℝ^{p,q} induces an isomorphism of the corresponding Poincaré algebras which is trivially extended to an isomorphism of extended Poincaré algebras.

We denote by (e_{i}) = (e_{1}, …, e_{p′}) the first p′ basis vectors of the standard basis of V and by (e′_{i}) = (e′_{1}, …, e′_{p′′+q}) the remaining ones. The two complementary orthogonal subspaces of V spanned by these bases are denoted by E = ℝ^{p′} = ℝ^{p′,0} and E′ = E^{⊥} = ℝ^{p′′,q} respectively. The vector spaces V, E and E′ are oriented by their standard orthonormal bases. E.g, the orientation of Euclidean p′-space E defined by the basis (e_{i}) is e_{1}^{∗} ∧ ⋯ ∧ e_{p′}^{∗} ∈ ∧^{p′}E^{∗}. Here (e_{i}^{∗}) denotes the basis of E^{∗} dual to (e_{i}). Now let 𝔭(Π) = 𝔭(V) + W be an extended Poincaré algebra of signature (p, q) and (ẽ_{i}) any orhonormal basis of E. Then we define a ℝ-bilinear form b_{Π,(ẽ_{i})} on the Cℓ^{0}(V)-module W by:

b_{Π,(ẽ_{i})}(s, t) = ⟨ẽ_{1}, [ẽ_{2} … ẽ_{p′}s, t]⟩ = ⟨ẽ_{1}, Π(ẽ_{2} … ẽ_{p′}s ∧ t)⟩ , s, t ∈ W . (5)

We put b = b(Π) := b_{Π,(e_{i})} for the standard basis (e_{i}) of E. Remark 2: Equation (5) defines a skew symmetric bilinear form on W if p′ ≡ 1 (mod 4). For even p′ the above formula does not make sense, unless one assumes that W is a Cℓ(V)-module rather than a Cℓ^{0}(V)-module. Here we are only interested in the case p′ ≡ 3 (mod 4). Moreover, later on, for the construction of homogeneous quaternionic manifolds we will put p′ = 3.

Theorem 1. The bilinear form b has the following properties:

1) b_{Π,(ẽ_{i})} = ±b if ẽ_{1} ∧ ⋯ ∧ ẽ_{p′} = ±e_{1} ∧ ⋯ ∧ e_{p′}. In particular, b_{Π,(ẽ_{i})} = b for any positively oriented orthonormal basis (ẽ_{i}) of E.

2) b is symmetric.

3) b is invariant under the maximal connected subgroup K(p′, p′′) = Spin(p′) · Spin_{0}(p′′, q) ⊂ Spin(p, q) which preserves the orthogonal decomposition V = E + E′ (and is not Spin_{0}(p, q)-invariant, unless p′′ + q = 0).

4) Under the identification 𝔬(V) = ∧^{2}V = ∧^{2}E + ∧^{2}E′ + E ∧ E′, see equation (3), the subspace E ∧ E′ acts on W by b-symmetric endomorphisms and the subalgebra ∧^{2}E ⊕ ∧^{2}E′ ≅ 𝔬(p′) ⊕ 𝔬(p′′, q) acts on W by b-skew symmetric endomorphisms.

Proof: Obviously 4) ⇒ 3). We show first that 3) ⇒ 1). If (ẽ_{i}) is a positively oriented orthonormal basis then there exists φ ∈ Spin(p′) such that Ad(φ)e_{i} = ẽ_{i}, i = 1, …, p′. Now Π = [·, ·]| ∧^{2} W : ∧^{2}W → V is 𝔰𝔭𝔦𝔫(V)-equivariant and hence equivariant under the connected group Spin_{0}(V) ⊂ Spin(V). In particular, Π is Spin(p′)-equivariant. Under the condition 3), this implies that

b(s, t) = b(φ^{−1}s, φ^{−1}t) = ⟨e_{1}, [e_{2} … e_{p′}φ^{−1}s, φ^{−1}t]⟩
= ⟨e_{1}, [φ^{−1}Ad_{φ}e_{2} … Ad_{φ}e_{p′}s, φ^{−1}t]⟩
= ⟨e_{1}, Ad(φ^{−1})[ẽ_{2} … ẽ_{p′}s, t]⟩ = ⟨ẽ_{1}, [ẽ_{2} … ẽ_{p′}s, t]⟩
= b_{Π,(ẽ_{i})}(s, t) , s, t ∈ W .

Here we have used the notation Ad_{φ} = Ad(φ). The case of negatively oriented orthonormal basis (ẽ_{i}) follows now from the Clifford relation ẽ_{i}ẽ_{j} = −ẽ_{j}ẽ_{i}, i ≠ j.

Next we prove 2) using first the 𝔰𝔭𝔦𝔫(V)-equivariance of Π then equation (4) and eventually p′ ≡ 3 (mod 4):

b(t, s) = ⟨e_{1}, [e_{2} … e_{p′}t, s]⟩
= −⟨e_{1}, [e_{4} … e_{p′}t, e_{2}e_{3}s]⟩ + ⟨e_{1}, ad(e_{2}e_{3})[e_{4} … e_{p′}t, s]⟩
= −⟨e_{1}, [e_{4} … e_{p′}t, e_{2}e_{3}s]⟩ = ⋯ = −⟨e_{1}, [t, e_{2} … e_{p′}s]⟩
= ⟨e_{1}, [e_{2} … e_{p′}s, t]⟩ = b(s, t) .

Finally we prove 4). By equation (4) we have to check that e_{i}e_{j}, e′_{k}e′_{l} ∈ 𝔰𝔭𝔦𝔫(V) (i ≠ j and k ≠ l) act by b-skew symmetric endomorphisms and e_{i}e′_{k} by a bsymmetric endomorphism on W. This is done in the next computation, in which we use again the equivariance of Π and equations (1) and (4) to express the adjoint representations Ad and ad respectively:

b(e_{1}e_{2}s, t) = ⟨e_{1}, [e_{2} … e_{p′}e_{1}e_{2}s, t]⟩ = ⟨e_{1}, [e_{1}e_{2}e_{2} … e_{p′}s, e_{1}e_{2}e_{1}e_{2}t]⟩
= ⟨e_{1}, Ad(e_{1}e_{2})[e_{2} … e_{p′}s, e_{1}e_{2}t]⟩ = −⟨e_{1}, [e_{2} … e_{p′}s, e_{1}e_{2}t]⟩
= −b(s, e_{1}e_{2}t) ,
b(e_{2}e_{3}s, t) = ⟨e_{1}, [e_{2} … e_{p′}e_{2}e_{3}s, t]⟩ = ⟨e_{1}, [e_{2}e_{3}e_{2} … e_{p′}s, t]⟩
= ⟨e_{1}, ad(e_{2}e_{3})[e_{2} … e_{p′}s, t]⟩ − ⟨e_{1}, [e_{2} … e_{p′}s, e_{2}e_{3}t]⟩
= −b(s, e_{2}e_{3}t) ,
b(e′_{k}e′_{l}s, t) = ⟨e_{1}, [e_{2} … e_{p′}e′_{k}e′_{l}s, t]⟩ = ⟨e_{1}, [e′_{k}e′_{l}e_{2} … e_{p′}s, t]⟩
= −⟨e_{1}, [e_{2} … e_{p′}s, e′_{k}e′_{l}t]⟩ + ⟨e_{1}, ad(e′_{k}e′_{l})[e_{2} … e_{p′}s, t]⟩
= −b(s, e′_{k}e′_{l}t) + 0 = −b(s, e′_{k}e′_{l}t) .

This already proves 3) and hence 1); in particular we have:

b_{Π,(e_{1},e_{2},... ,e_{p′})} = b_{Π,(e_{2},... ,e_{p′} ,e_{1})} .

Due to this symmetry, it is sufficient to check that e_{2}e′_{k} acts as b-symmetric endomorphism on W:

b(e_{2}e′_{k}s, t) = ⟨e_{1}, [e_{2} … e_{p′}e_{2}e′_{k}s, t]⟩ = ⟨e_{1}, [e_{3} … e_{p′}e′_{k}s, t]⟩
=^{(∗)} −⟨e_{1}, [s, e_{3} … e_{p′}e′_{k}t]⟩ = −⟨e_{1}, [s, e_{2} … e_{p′}e_{2}e′_{k}t]⟩
= ⟨e_{1}, [e_{2} … e_{p′}e_{2}e′_{k}t, s]⟩ = b(e_{2}e′_{k}t, s)
= b(s, e_{2}e′_{k}t) .

At (∗) we have used (p′ − 1)/2 times the 𝔰𝔭𝔦𝔫(V)-equivariance of Π and the fact that (p′ − 1)/2 is odd if p′ ≡ 3 (mod 4). □

Definition 3. The bilinear form b = b(Π) = b_{Π,(e_{1},... ,e_{p′})} defined above is called the canonical symmetric bilinear form on W associated to the 𝔰𝔭𝔦𝔫(V)- equivariant map Π : ∧^{2}W → V = ℝ^{p,q} and the decomposition p = p′ + p′′.

Proposition 1. The kernels of the linear maps Π : W → W^{∗} ⊗ V and b = b(Π) : W → W^{∗} coincide: ker Π = ker b.

Proof: It follows from Thm. 1, 4) that W_{0} := ker b ⊂ W is 𝔬(V)-invariant. This implies that Π(W_{0} ∧ W) ⊂ V is an 𝔬(V)-submodule. The definition of W_{0} implies that Π(W_{0} ∧ W) ⊂ E′ = E^{⊥} and hence by Schur’s lemma Π(W_{0} ∧ W) = 0. This proves that ker b ⊂ ker Π. On the other hand, we have the obvious inclusion ker Π ⊂ ker (b ∘ e_{2} … e_{p′}) = ker b. Here the equation follows from the Spin(p′)- invariance of b, see Thm. 1, 3). □

Corollary 1. 𝔭(Π) is nondegenerate (see Def. 2) if and only if b(Π) is nondegenerate.

Theorem 2. Let 𝔭(Π) = 𝔭(V) + W be any extended Poincaré algebra of signature (p, q), p = p′ + p′′, p′ ≡ 3 (mod 4), and b the canonical symmetric bilinear form associated to these data. Then there exists a b-orthogonal decomposition W = ⊕_{i=0}^{l+m}W_{i} into Cℓ^{0}(V)-submodules with the following properties

1) [W_{0}, W] = 0, [W_{i}, W_{j}] = 0 if i ≠ j and [W_{i}, W_{i}] = V for all i = 1, 2, …, l.

2) W_{0} = ker b and W_{i} is b-nondegenerate for all i ≥ 1.

3) For i = 1, …, l the Cℓ^{0}(V)-submodule W_{i} is irreducible and for j = l + 1, …, l + m the Cℓ^{0}(V)-submodule W_{j} = X_{j} ⊕ X′_{j} is the direct sum of two irreducible b-isotropic Cℓ^{0}(V)-submodules.

4) The restriction of b to a bilinear form on any irreducible Cℓ^{0}(V)-submodule of X = ⊕_{j=l+1}^{l+m}W_{j} vanishes.

Proof: By Prop. 1, W_{0} := ker b = ker Π satisfies [W_{0}, W] = 0. As kernel of the 𝔬(V)-equivariant map Π the subspace W_{0} is 𝔬(V)-invariant and hence a Cℓ^{0}(V)- submodule. We denote by W′ a complementary Cℓ^{0}(V)-submodule. Every such submodule is b-nondegenerate. Let W_{i} ⊂ W′ be any irreducible Cℓ^{0}(V)-submodule. By Thm. 1, 4), ker(b|W_{i} × W_{i}) is 𝔬(V)-invariant and hence a Cℓ^{0}(V)-submodule. Now by Schur’s lemma we conclude that either b|W_{i} ×W_{i} = 0 or b is nondegenerate on W_{i}. In particular, we can decompose W′ = ⊕_{i=1}^{l}W_{i} ⊕ X as direct b-orthogonal sum of b-nondegenerate Cℓ^{0}(V)-submodules such that W_{i} is irreducible and the restriction of b to a bilinear form on any irreducible Cℓ^{0}(V)-submodule of X vanishes. Let Y, Z ⊂ X be two such submodules, Y ≠ Z. The bilinear form b induces a linear map Y → Z^{∗}. By Thm. 1, 4), the kernel of this map is 𝔬(V)-invariant and hence a Cℓ^{0}(V)-submodule. Now Schur’s lemma implies that either the kernel is Y and hence the restriction of b to a bilinear form on Y ⊕Z vanishes or the kernel is trivial and b is nondegenerate on Y ⊕Z. In the second case X splits as direct b-orthogonal sum: X = (Y ⊕ Z) ⊕ (Y ⊕ Z)^{⊥}. This shows that X = ⊕_{i=l+1}^{l+m}W_{i} is the direct orthogonal sum of b-nondegenerate Cℓ^{0}(V)-submodules W_{i} such that W_{i} = X_{i} ⊕ X′_{i} is the direct sum of two b-isotropic irreducible submodules. This proves 2), 3) and 4). Now 1) is established applying Schur’s lemma to the 𝔬(V)-equivariant map Π. In fact, b(W_{i}, W_{j}) = 0 (respectively, b(W_{i}, W_{i}) ≠ 0) implies Π(W_{i} ∧ W_{j}) ⊂ E′ (respectively, Π(∧^{2}W_{i}) ≠ 0) and thus [W_{i}, W_{j}] = Π(W_{i} ∧ W_{j}) = 0 (respectively, [W_{i}, W_{j}] = Π(∧^{2}W_{i}) = V). □

Next we will construct the subgroup K̂(p′, p′′) ⊂ Spin(p, q) which consists of all elements preserving the orthogonal decomposition V = E + E′ and the canonical symmetric bilinear form b on W. Its identity component is the group K(p′, p′′) ⊂ Spin_{0}(p, q) introduced above. We will see that if p′′ = 0 then K̂(p′, p′′) = K̂(p, 0) is a maximal compact subgroup of Spin(p, q), which together with the element 1 ∈ Cℓ_{p,q}^{0} generates the even Clifford algebra Cℓ_{p,q}^{0}. This property will be very useful in the next section.

We denote by x ↦ x′ the linear map ℝ^{q} = ℝ^{q,0} → ℝ^{0,q} which maps the standard orthonormal basis (e_{1}, …, e_{q}) of ℝ^{q} to the standard orthonormal basis (e′_{1}, …, e′_{q}) of ℝ^{0,q}. It is an antiisometry: ⟨x′, x′⟩ = −⟨x, x⟩. Let ω_{p′} = e_{1} … e_{p′} be the volume element of Cℓ_{p′} = Cℓ_{p′,0}, (e_{1}, …, e_{p′}) the standard orthonormal basis of ℝ^{p′} = ℝ^{p′,0} ⊂ ℝ^{p,q}. Note that, since p′ is odd, the volume element ω_{p′} commutes with ℝ^{p′} and anticommutes with ℝ^{p′′,q} ⊃ ℝ^{0,q}. Moreover, it satisfies ω_{p′}^{2} = 1, due to p′ ≡ 3 (mod 4).

Lemma 1. The map

ℝ^{q} ∋ x ↦ ω_{p′}x′ ∈ Cℓ_{p,q}^{0}

extends to an embedding ι : Cℓ_{q} ↪ Cℓ_{p,q}^{0} of algebras, which restricts to an embedding ι|Pin(q) : Pin(q) ↪ Spin(p, q) of groups.

Proof: It follows from (ω_{p′}x′)^{2} = −ω_{p′}^{2}x′^{2} = −x′^{2} = ⟨x′, x′⟩ = −⟨x, x⟩ that the map x ↦ ω_{p′}x′ extends to a homomorphism ι of Clifford algebras. Recall that ℂℓ_{q} is either simple or the sum of two simple ideals. In the first case, we can immediately conclude that ker ι is trivial and hence ι an embedding. In the second case, the two simple ideals of ℂℓ_{q} are ℂℓ_{q}^{±} := (1 ± ω_{q})ℂℓ_{q}, where ω_{q} = e_{1} … e_{q} is the volume element of ℂℓ_{q}. Now it is sufficient to check that ι(1 ± ω_{q}) ≠ 0. Using the fact that q is odd if Cℓ_{q} is not simple we compute

ι(ω_{q}) = ±ω_{p′}^{q}e′_{1}e′_{2} … e′_{q} = ±e_{1}e_{2} … e_{p′}e′_{1}e′_{2} … e′_{q} .

This shows that ι(1 ± ω_{q}) = 1 ± ι(ω_{q}) ≠ 0. □

We denote by K̂(p′, p′′) ⊂ Spin(p, q) the subgroup generated by the subgroups Spin(p′) · Spin_{0}(p′′, q) ⊂ Spin(p, q) and ι(Pin(q)) ⊂ Spin(p, q).

Theorem 3. The group K̂(p′, p′′) has the following properties:

1) K̂(p′, p′′) ⊂ Spin(p, q) consists of all elements preserving the orthogonal decomposition V = E + E′ and the canonical symmetric bilinear form b on W. Its identity component is the group K(p′, p′′) = Spin(p′) · Spin_{0}(p′′, q) = K̂(p′, p′′) ∩ Spin_{0}(p, q).

2) The homogeneous space Spin(p, q)/K̂(p′, p′′) is connected.

3) K̂(p′, p′′) is compact if and only if q = 0 or p′′ = 0 (and hence p = p′). In the latter case K̂(p′, p′′) = K̂(p, 0) = Spin(p) · ι(Pin(q)) ⊂ Spin(p, q) is a maximal compact subgroup and K̂(p, 0) ≅ (Spin(p)×Pin(q))/{±1}. Finally, in this case, the even Clifford algebra Cℓ_{p,q}^{0} is generated by 1 and K̂(p, 0).

Proof: The first part of 1) can be checked using Thm. 1, 4) and implies the second part of 1). To prove 2) it is sufficient to observe that ι(Pin(q)) ≅ Pin(q) has nontrivial intersection with all connected components of Spin(p, q) (due to our assumption p ≥ 3 there are two such components if q ≠ 0). The first part of 3) now follows simply from the fact that Spin_{0}(p′′, q) is compact if and only if p′′ = 0 or q = 0. The compact group K̂(p, 0) ⊂ Spin(p, q) is maximal compact, because it has the same number of connected components as Spin(p, q), and from Spin(p)∩ι(Pin(q)) = {±1} we obtain the isomorphism K̂(p, 0) = Spin(p)·ι(Pin(q)) ≅ (Spin(p)×Pin(q))/{±1}. Finally, to prove the last statement, one easily checks that K̂(p, 0) contains all quadratic monomials xy in unit vectors x, y ∈ ℝ^{p,0} ∪ ℝ^{0,q}. □

Corollary 2. The correspondence Π ↦ b(Π) defines an injective linear map (∧^{2}W^{∗} ⊗ V)^{𝔬(V)} → (∨^{2}W^{∗})^{K̂(p′,p′′)}.

Proof: The existence of the map follows from Thm. 3. We prove the injectivity. From b(Π) = 0 it follows that Π(∧^{2}W) ⊂ E′ and hence by Schur’s lemma Π = 0. □

1.3. The set of isomorphism classes of extended Poincaré algebras. Starting from the decomposition proven in Thm. 2 we will derive the classification of extended Poincaré algebras of signature (p, q), p ≡ 3 (mod 4), up to isomorphism. It will turn out that the space of isomorphism classes is naturally parametrized by a finite number of integers. We fix the decomposition p = p′ + p′′, p′ = p, p′′ = 0, and for any extended Poincaré algebra 𝔭(Π) of signature (p, q) as above we consider the canonical symmetric bilinear form b = b_{Π,(e_{1},...e_{p})}.

Theorem 4. Let 𝔭(Π) = 𝔭(V) + W be any extended Poincaré algebra of signature (p, q), p ≡ 3 (mod 4). Then there exists a b-orthogonal decomposition W = ⊕_{i=0}^{l}W_{i} into Cℓ^{0}(V)-submodules with the following properties 1) [W_{0}, W] = 0, [W_{i}, W_{j}] = 0 if i ≠ j and [W_{i}, W_{i}] = V for all i = 1, 2, …, l.

2) W_{0} = ker b and W_{i} is b-nondegenerate for all i ≥ 1.

3) W_{i}, i ≥ 1, is an irreducible Cℓ^{0}(V)-submodule on which b is (positive or negative) definite.

Proof: Let W = ⊕_{i=1}^{l+m}W_{i} be a decomposition as in Thm. 2. It only remains to prove that b is definite on W_{i} for i = 1, …, l and that X = ⊕_{i=l+1}^{l+m}W_{i} = 0. This follows from Lemma 3 and Lemma 4 below. □

Lemma 2. The restriction of an irreducible Cℓ_{p,q}^{0}-module Σ to a module of the maximal compact subgroup K̂ = K̂(p, 0) = Spin(p) · ι(Pin(q)) ⊂ Spin(p, q) is irreducible. Here ι : Pin(p) ↪ Spin(p, q) is the embedding of Lemma 1. Moreover, Σ is irreducible as module of the connected group K = K(p, 0) = Spin(p) · ι(Spin(q)) = Spin(p) · Spin(q) if and only if n = p + q ≡ 2, 4, 5 or 6 (mod 8). If n ≡ 0, 1, 3 or 7 (mod 8) then Σ is the sum of two irreducible K-submodules.

Proof: Recall that Cℓ_{p,q} = Cℓ_{p} ˆ⊗Cℓ_{0,q} is (identified with) the ℤ_{2}-graded tensor product of the Clifford algebras Cℓ_{p} = Cℓ_{p,0} and Cℓ_{0,q}. It is easily checked, using the classification of Clifford algebras and their modules, see [L-M], that any irreducible Cℓ_{p,q}^{0}-module Σ is irreducible as module of the subalgebra Cℓ_{p}^{0} ⊗ Cℓ_{0,q}^{0} ⊂ Cℓ_{p,q}^{0} = Cℓ_{p}^{0} ⊗Cℓ_{0,q}^{0} +Cℓ_{p}^{1} ⊗Cℓ_{0,q}^{1} if n ≡ 2, 4, 5 or 6 (mod 8) and is the sum of two irreducible submodules if n ≡ 0, 1, 3 or 7 (mod 8). Now Lemma 2 follows from the fact that Cℓ_{p}^{0} ⊗ Cℓ_{0,q}^{0} (respectively, Cℓ_{p,q}^{0}) is the subalgebra of Cℓ_{p,q} generated by 1 and K = Spin(p) · Spin(0, q) (respectively, by 1 and K̂ = Spin(p) · ι(Pin(q))). □

Lemma 3. A Cℓ_{p,q}^{0}-module W is irreducible if and only if it is irreducible as module of the maximal compact subgroup K̂ ⊂ Spin(p, q). In this case (∨^{2}W^{∗})K̂ is one-dimensional and is spanned by a positive definite scalar product on W. Let W be an irreducible Cℓ_{p,q}^{0}-module and Π : ∧^{2}W → V be any 𝔬(V)-equivariant linear map. Then either Π = 0 or b(Π) is a definite K̂-invariant symmetric bilinear form.

Proof: The first statement follows from Lemma 2. Since K̂ is compact there exists a positive definite K̂-invariant symmetric bilinear form on W. From the irreducibility of W we conclude by Schur’s lemma that (∨^{2}W^{∗})K̂ is spanned by this form. Now the last statement is an immediate consequence of Cor. 2. □

Lemma 4. Let W be a Cℓ^{0}(V)-module and Π : ∧^{2}W → V an 𝔬(V)-equivariant linear map such that b = b(Π) is nondegenerate. Suppose that W = Σ ⊕ Σ′ is the direct sum of two irreducible submodules Σ and Σ′. Then there exists a b-orthogonal decomposition W = Σ_{1} ⊕ Σ_{2} into two b-nondegenerate (and hence b-definite by Lemma 3) irreducible submodules Σ_{1} and Σ_{2}.

Proof: It is sufficient to show that W contains a b-nondegenerate irreducible Cℓ^{0}(V)-submodule Σ_{1}. Then Σ_{2} := Σ_{1}^{⊥} is a b-nondegenerate K̂-submodule. It is also a Cℓ^{0}(V)-submodule, because the algebra Cℓ^{0}(V) is generated by 1 and K̂, and it is irreducible since the Cℓ^{0}(V)-module W is the direct sum of only two irreducible submodules. If W does not contain any b-nondegenerate irreducible Cℓ^{0}(V)-submodule Σ_{1} then, by Schur’s lemma, the restriction of b to a bilinear form on any irreducible Cℓ^{0}(V)-submodule vanishes. In the following we derive a contradiction from this assumption. Since the bilinear form b is nondegenerate it defines a nondegenerate pairing between the b-isotropic subspaces Σ and Σ′. Due to the K̂-invariance of b (Thm. 3) b : Σ′ →^{∼} Σ^{∗} is a K̂-equivariant isomorphism. On the other hand Σ^{∗} ≅ Σ as irreducible modules of the compact group K̂. This shows that Σ and Σ′ are equivalent as K̂-modules and thus as Cℓ^{0}(V)-modules, because the algebra Cℓ^{0}(V) is generated by 1 and K̂. Hence there exists a Cℓ^{0}(V)- equivariant isomorphism φ : Σ →^{∼} Σ′. We define two K̂-invariant bilinear forms β_{±} on Σ by:

β_{±}(s, t) := b(φ(s), t) ± b(φ(t), s) , s, t ∈ Σ .

β_{+} is symmetric and β_{−} is skew symmetric. If β_{+} ≠ 0 then it is a definite K̂- invariant scalar product on Σ, since Σ is an irreducible module of the compact group K̂. So for s ∈ Σ − {0} we obtain

0 ≠ β_{+}(s, s) = b(φ(s), s) + b(s, φ(s)) = b(s + φ(s), s + φ(s)) .

This implies that the irreducible Cℓ^{0}(V)-submodule Σ_{φ} := {s+φ(s)|s ∈ Σ} ⊂ W is b-definite, which contradicts our assumption. We conclude that β_{+} = 0 and hence β_{−} = 2b ∘ φ is a K̂-invariant symplectic form on Σ. Let β be a K̂-invariant positive definite scalar product on Σ (such scalar products exist since K̂ is compact). We define a K̂-equivariant isomorphism χ : Σ → Σ by the equation

β(s, t) = β_{−}(χ(s), t) , s, t ∈ Σ .

Then ψ := φ ∘ χ : Σ → Σ′ is a K̂-equivariant and hence Cℓ^{0}(V)-equivariant isomorphism. Using β_{+} = 0, we compute for s ∈ Σ − {0}:

b(s + ψ(s), s + ψ(s)) = b(ψ(s), s) + b(s, ψ(s)) = b(ψ(s), s) − b(φ(s), χ(s))
= β_{−}(χ(s), s) = β(s, s) ≠ 0 .

As above, this implies that W contains a b-definite irreducible Cℓ^{0}(V)-submodule Σ_{ψ} ≅ Σ contradicting our assumption. □

Theorem 5. [A-C2] Let V = ℝ^{p,q}, p ≡ 3 (mod 4). If W is an irreducible Cℓ^{0}(V)-module then

dim(∧^{2}W^{∗} ⊗ V)^{𝔬(V)} = 1 .

Proof: Cor. 2 and Lemma. 3 show that dim(∧^{2}W^{∗} ⊗ V)^{𝔬(V)} ≤ 1. On the other hand, there exists a nontrivial 𝔬(V)-equivariant linear map ∧^{2}W → V, see [A-C2], and hence dim(∧^{2}W^{∗} ⊗ V)^{𝔬(V)} ≥ 1. This proves that dim(∧^{2}W^{∗} ⊗ V)^{𝔬(V)} = 1. □

Next, in order to parametrize the isomorphism classes of extended Poincaré algebras 𝔭(Π) = 𝔭(V) + W we associate a certain number of nonnegative integers to 𝔭(Π). Let us first consider the case when there is only one irreducible Cℓ^{0}(V)- module Σ up to equivalence. Then W is necessarily isotypical and, due to Thm. 4, there exists a b-orthogonal decomposition W = W_{0} ⊕ ⊕_{i=1}^{l}W_{i} with W_{0} = ker b = ker Π ≅ l_{0}Σ and irreducible Cℓ^{0}(V)-submodules W_{i} ≅ Σ for i = 1, …, l on which b is definite. We denote by l_{+} (repectively l_{−}) the number of summands W_{i} on which b is positive (respectively, negative) definite. Note that the triple (l_{0}, l_{+}, l_{−}) does not depend on the choice of decomposition.

Theorem 6. Let p ≡ 3 (mod 4) and q ≢ 3 (mod 4). Then the isomorphism class [𝔭(Π)] of an extended Poincaré algebra 𝔭(Π) of signature (p, q) is completely determined by the triple (l_{0}, l_{+}, l_{−}) introduced above. We put 𝔭(p, q, l_{0}, l_{+}, l_{−}) := [𝔭(Π)]. Then 𝔭(p, q, l_{0}, l_{+}, l_{−}) = 𝔭(p, q, l′_{0}, l′_{+}, l′_{−}) if and only if l_{0} = l′_{0} and {l_{+}, l_{−}} = {l′_{+}, l′_{−}}.

Proof: If p ≡ 3 (mod 4) then there is only one irreducible Cℓ_{p,q}^{0}-module Σ up to equivalence if and only if q ≢ 3 (mod 4). Let 𝔭(Π) = 𝔭(V) + W and 𝔭(Π′) = 𝔭(V) + W′ be two extended Poincaré algebras of signature (p, q) with the same integers l_{0} = l_{0}(Π) = l_{0}(Π′), l_{+} = l_{+}(Π) = l_{+}(Π′) and l_{−} = l_{−}(Π) = l_{−}(Π′). Then the modules W and W′ are equivalent and we can assume that W = W′ = W_{0} ⊕ ⊕_{i=1}^{l=l_{+}+l_{−}}W_{i} is a decomposition as above. In particular, it is b(Π)- and b(Π′)- orthogonal, b(Π) and b(Π′) are both positive definite or both negative definite on W_{i} for i ≥ 1, Π(W_{0} ∧ W) = Π′(W_{0} ∧ W) = 0, Π(W_{i} ∧ W_{j}) = Π′(W_{i} ∧ W_{j}) = 0 if i ≠ j and Π(∧^{2}W_{i}) = Π′(∧^{2}W_{i}) = V if i ≥ 1. So the maps Π and Π′ are completely determined by their restrictions Π_{i} := Π| ∧^{2} W_{i} ≠ 0 and Π′_{i} := Π′| ∧^{2} W_{i} ≠ 0 (i ≥ 1) respectively. By Thm. 5 Π′_{i} = λ_{i}Π_{i} (i ≥ 1) for some constant λ_{i} ∈ ℝ^{∗}. Now b = b(Π) and b′ = b(Π′) are both positive definite or both negative definite on W_{i} and hence λ_{i} = μ_{i}^{2} > 0. Now we can define an isomorphism φ : 𝔭(Π) → 𝔭(Π′) by φ|𝔭(V) + W_{0} = Id and φ|W_{i} = μ_{i}Id. This shows that the integers (l_{0}, l_{+}, l_{−}) determine the extended Poincaré algebra 𝔭(Π) of signature (p, q) up to isomorphism. The ℤ_{2}-graded Lie algebras 𝔭(Π) and 𝔭(−Π) are isomorphic via α : 𝔭(Π) → 𝔭(−Π) defined by: α|𝔬(V) + W = Id and α|V = −Id. This proves 𝔭(p, q, l_{0}, l_{+}, l_{−}) = 𝔭(p, q, l_{0}, l_{−}, l_{+}). It remains to show that 𝔭(p, q, l_{0}, l_{+}, l_{−}) = 𝔭(p, q, l′_{0}, l′_{+}, l′_{−}) implies l_{0} = l′_{0} and {l_{+}, l_{−}} = {l′_{+}, l′_{−}}. Let 𝔭(Π) = 𝔭(V) + W ∈ 𝔭(p, q, l_{0}, l_{+}, l_{−}) and 𝔭(Π′) = 𝔭(V) + W′ ∈ 𝔭(p, q, l′_{0}, l′_{+}, l′_{−}) be representative extended Poincaré algebras and W = W_{0} ⊕ ⊕_{i=1}^{l}W_{i} (l = l_{+} + l_{−}) a decomposition as above. We assume that there exists an isomorphism φ : 𝔭(Π) → 𝔭(Π′) of ℤ_{2}-graded Lie algebras, i.e. φ𝔭(V) = 𝔭(V) and φW = W. The automorphism φ|𝔭(V) preserves the radical V and maps the Levi subalgebra 𝔬(V) to an other Levi subalgebra of 𝔭(V). Now by Malcev’s theorem any two Levi subalgebras are conjugated by an inner automorphism, see [O-V]. So, using an inner automorphism of 𝔭(Π), we can assume that φ𝔬(V) = 𝔬(V). The subalgebra φ(𝔬(p)⊕𝔬(q)) ⊂ 𝔬(V) is maximal compact (i.e. the Lie algebra of a maximal compact subgroup of O(V) = O(p, q)) and hence conjugated by an inner automorphism to the maximal compact subalgebra 𝔬(p)⊕𝔬(q) ⊂ 𝔬(V). So, again, we can assume that φ preserves 𝔬(p) ⊕ 𝔬(q). Moreover, since p ≠ q any automorphism of 𝔬(p) ⊕ 𝔬(q) is inner and we can assume that φ|𝔬(p) ⊕ 𝔬(q) = Id. From the fact that φ is an automorphism of 𝔭(V) we obtain that ϕ := φ|V ∈ GL(V) normalizes 𝔬(V) and 𝔬(p)⊕𝔬(q). This implies ϕ ∈ O(p)×O(q), in particular, φℝ^{p} = ℝ^{p} and φℝ^{0,q} = ℝ^{0,q}. Using an inner automorphism of 𝔭(Π) we can further assume that φ|ℝ^{p} = ϵ(φ)Id, ϵ(φ) ∈ {+1, −1}, and hence φ|𝔬(p) = Ad_{ϕ}|𝔬(p) = Id. This implies that φ|W is Spin(p)-equivariant. Now we can compute

b′(φs, φt) = ⟨e_{1}, [e_{2} … e_{p}φs, φt]⟩ = ⟨e_{1}, [φe_{2} … e_{p}s, φt]⟩
= ⟨e_{1}, φ[e_{2} … e_{p}s, t]⟩ = ϵ(φ)⟨e_{1}, [e_{2} … e_{p}s, t]⟩
= ϵ(φ)b(s, t) , s, t ∈ W .

Put W′_{i} := φW_{i}. Then, since φ^{∗}b′ = ϵ(φ)b, we obtain a b′-orthogonal decomposition W′ = W′_{0} ⊕ ⊕_{i=1}^{l′}W′_{i} (l′ = l′_{+} + l′_{−}) as above; W′_{0} = ker Π′ ≅ l′_{0}Σ, W′_{i} ≅ Σ (i ≥ 1) etc. This shows that l′_{0} = l_{0}, l′_{±} = l_{±} if ϵ(φ) = +1 and l′_{±} = l_{∓} if ϵ(φ) = −1. □

Now we discuss the complementary case p ≡ q ≡ 3 (mod 4). In this case, the spinor module S of Cℓ^{0}(V) is the sum S^{+} ⊕ S^{−} of two irreducible inequivalent semispinor modules S^{+} and S^{−} and any irreducible Cℓ^{0}(V)-module is equivalent to S^{+} or S^{−}. As Spin_{0}(V)-modules, S^{+} and S^{−} are dual: S^{−} ≅ (S^{+})^{∗}. Let 𝔭(Π) = 𝔭(V) + W be an extended Poincaré algebra of signature (p, q) as above. Thanks to Thm. 4 there exists a b-orthogonal decomposition W = W_{0} ⊕ ⊕_{i=1}^{l}W_{i} as above with the following b-orthogonal refinements: W_{0} = W_{0}^{+} ⊕ W_{0}^{−}, W_{0}^{±} ≅ l_{0}^{±}S^{±}, and ⊕_{i=1}^{l}W_{i} = ⊕_{i=1}^{l^{+}}W_{i}^{+} ⊕ ⊕_{i=1}^{l^{−}}W_{i}^{−}, W_{i}^{±} ≅ S^{±}. We denote by l_{+}^{ϵ} (respectively, l_{−}^{ϵ}) the number of submodules W_{i}^{ϵ}, i = 1, …, l^{ϵ}, on which b is positive (respectively, negative) definite, ϵ ∈ {+, −}. So to 𝔭(Π) we have associated the nonnegative integers (l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}).

Theorem 7. Let p ≡ q ≡ 3 (mod 4). Then the isomorphism class [𝔭(Π)] of an extended Poincaré algebra 𝔭(Π) of signature (p, q) is completely determined by the tuple (l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) introduced above. We put 𝔭(p, q, l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) := [𝔭(Π)]. Then 𝔭(p, q, l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) = 𝔭(p, q, l̃_{0}^{+}, l̃_{+}^{+}, l̃_{−}^{+}, l̃_{0}^{−}, l̃_{+}^{−}, l̃_{−}^{−}) if and only if (l̃_{0}^{+}, l̃_{+}^{+}, l̃_{−}^{+}, l̃_{0}^{−}, l̃_{+}^{−}, l̃_{−}^{−}) ∈ Γ(l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}), where Γ ≅ ℤ_{2} × ℤ_{2} is the group generated by the following two involutions:

(l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) ↦ (l_{0}^{+}, l_{−}^{+}, l_{+}^{+}, l_{0}^{−}, l_{−}^{−}, l_{+}^{−})

and

(l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) ↦ (l_{0}^{−}, l_{+}^{−}, l_{−}^{−}, l_{0}^{+}, l_{+}^{+}, l_{−}^{+}) .

Proof: The proof uses again Thm. 4 and Thm. 5 and is similar to that of Thm. 6. Therefore, we explain only the reason for the appearance of the second involution. In terms of the standard basis (e_{1}, …, e_{p}, e′_{1}, …, e′_{q}) of V = ℝ^{p,q} we define an isometry ϕ ∈ SO(p)×O(q) by: ϕe_{1} := e_{1}, ϕe_{i} := −e_{i} (i ≥ 2) and ϕe′_{j} = −e′_{j} (j ≥ 1). Then Ad_{ϕ} ∈ Aut 𝔭(V) induces an (outer) automorphism of 𝔬(V) interchanging the two semispinor modules. Let (𝔭(V) + W, [·, ·]) ∈ 𝔭(p, q, l_{0}^{+}, l_{+}^{+}, l_{−}^{+}, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}) be an extended Poincaré algebra of signature (p, q). Then we define a new extended Poincaré algebra (𝔭(V) + W, [·, ·]′) ∈ 𝔭(p, q, l_{0}^{−}, l_{+}^{−}, l_{−}^{−}, l_{0}^{+}, l_{+}^{+}, l_{−}^{+}) by:

[·, ·]′ := [·, ·] on ∧^{2} 𝔭(V) ⊕ ∧^{2}W ⊕ V ∧ W

and

[A, s]′ := [Ad_{ϕ}(A), s] for A ∈ 𝔬(V) , s ∈ W .

The two ℤ_{2}-graded Lie algebras are isomorphic via φ : (𝔭(V) + W, [·, ·]) →^{∼} (𝔭(V) + W, [·, ·]′) defined by: φ|𝔭(V) = Ad_{ϕ} and φ|W = Id. □

2. The homogeneous quaternionic manifold (M, Q) associated to an extended Poincaré algebra

2.1. Homogeneous manifolds associated to extended Poincaré algebras. Any extended Poincaré algebra 𝔭 = 𝔭(Π) = 𝔭(V)+W has an even derivation D with eigenspace decomposition 𝔭 = 𝔬(V)+V +W and corresponding eigenvalues (0, 1, 1/2). Therefore, the ℤ_{2}-graded Lie algebra 𝔭 = 𝔭_{0} + 𝔭_{1} = 𝔭(V) + W is canonically extended to a ℤ_{2}-graded Lie algebra 𝔤 = 𝔤(Π) = ℝD + 𝔭 = 𝔤_{0} + 𝔤_{1}, where 𝔤_{0} = ℝD + 𝔭_{0} = ℝD + 𝔭(V) =: 𝔤(V) and 𝔤_{1} = 𝔭_{1} = W. The next proposition describes the basic structure of the Lie algebra 𝔤 = 𝔤(Π) = 𝔤(V) + W. To avoid trivial exceptions, in the following we assume that n = dim V > 2 and hence that 𝔬(V) is semisimple (for the construction of homogeneous quaternionic manifolds we will put V = ℝ^{p,q} and p ≥ 3).

Proposition 2. The Lie algebra 𝔤 = 𝔤(Π) has the Levi decomposition

𝔤 = 𝔬(V) + 𝔯 (6)

into the radical 𝔯 = ℝD + V + W and the complementary (maximal semisimple) Levi subalgebra 𝔬(V). The nilradical 𝔫 = [𝔯, 𝔯] = V + W is two-step nilpotent if Π ≠ 0 and Abelian otherwise.

For any Lie algebra 𝔩 we denote by der(𝔩) the Lie algebra of its derivations.

Proposition 3. The adjoint representation 𝔤 → der(𝔯) of 𝔤 = 𝔬(V) + 𝔯 on its ideal 𝔯 = ℝD + V + W is faithful.

Proof: Let x ∈ 𝔤. We show that [x, 𝔯] = 0 implies x = 0. First [x, D] = 0 implies x ∈ 𝔠𝔬(V) = ℝD + 𝔬(V). Then [x, V] = 0 implies x = 0, because the conformal Lie algebra 𝔠𝔬(V) acts faithfully on V. □

By Prop. 3 we can consider 𝔤(V) and 𝔤 = 𝔤(Π) = 𝔤(V)+W as linear Lie algebras via the embedding 𝔤(V) ⊂ 𝔤 ↪ der(𝔯) ⊂ 𝔤𝔩(𝔯). The corresponding connected Lie groups of Aut(𝔯) ⊂ GL(𝔯) will be denoted by G(V) and G = G(Π) respectively: Lie G(V) = 𝔤(V) ⊂ Lie G = 𝔤. Now let V = ℝ^{p,q} and fix a decomposition p = p′ + p′′. The subalgebra of 𝔬(V) preserving the corresponding orthogonal splitting V = E + E′ = ℝ^{p′,0} + ℝ^{p′′,q} is 𝔨 = 𝔨(p′, p′′) = 𝔬(p′) ⊕ 𝔬(p′′, q) ⊂ 𝔬(p, q) = 𝔬(V). We consider 𝔨 ⊂ 𝔬(V) ⊂ 𝔤 as a subalgebra of the linear Lie algebra 𝔤 ↪ der(𝔯) and denote by K = K(p′, p′′) ⊂ G(V) ⊂ G ⊂ Aut(𝔯) the corresponding connected linear Lie group. K is a closed Lie subgroup, see Cor. 3 below. We are interested in the homogeneous spaces

M(V) := G(V)/K ⊂ M = M(Π) := G/K = G(Π)/K .

Proposition 4. The Lie subalgebras 𝔨, 𝔫, 𝔯, 𝔬(V), 𝔭(V), 𝔭, 𝔤(V), 𝔤 ⊂ der(𝔯) are algebraic subalgebras of the (real) algebraic Lie algebra der(𝔯).

Proof: We use the following sufficient conditions for algebraicity, see [O-V] Ch. 3 §3 8^{o}:

a) A linear Lie algebra coinciding with its derived algebra is algebraic.

b) The radical of an algebraic linear Lie algebra is algebraic.

c) A linear Lie algebra generated by algebraic subalgebras is algebraic. The subalgebras 𝔬(V) ⊂ 𝔭(V) ⊂ 𝔭 ⊂ der(𝔯) are algebraic by a). By c), to prove that 𝔤 = ℝD + 𝔭 and 𝔤(V) are algebraic it is now sufficient to show that ℝD ⊂ der(𝔯) is algebraic. D preserves 𝔫 = V +W and acts trivially on the complement ℝD +𝔬(V). The Lie algebra ℝD ↪ der(𝔫) is the Lie algebra of the algebraic group

{λId_{V} ⊕ μId_{W} |λ = μ^{2} ≠ 0} ⊂ GL(V ⊕ W) .

This shows that ℝD, 𝔤(V), 𝔤 ⊂ der(𝔯) are algebraic. Now 𝔫, the radical of 𝔭, and 𝔯, the radical of 𝔤, are algebraic by b). Finally, 𝔨 ⊂ 𝔬(V) is the subalgebra which preserves the orthogonal splitting V = E + E′ and is hence algebraic. □

Corollary 3. The connected linear Lie groups K, S ≅ Spin_{0}(V), R = exp 𝔯, G(V), G ⊂ Aut(𝔯) with Lie algebras 𝔨, 𝔰 := 𝔬(V), 𝔯, 𝔤(V), 𝔤 ⊂ der(𝔯) are closed.

Proof: This follows from Prop. 4 and the fact that the identity component of a real algebraic linear group is a closed linear group. □

Proposition 5. The Lie group G(Π) has the following Levi decomposition:

G(Π) = S⋉R . (7)

Proof: This follows from the corresponding Levi decomposition (6) of Lie algebras since S ∩ R = {e}. □

Next we show that M(V) can be naturally endowed with a G(V)-invariant structure of pseudo-Riemannian locally symmetric space. With this in mind, we consider the pseudo-Euclidean vector space Ṽ = ℝ^{p+1,q+1} with scalar product ⟨·, ·⟩ and orthogonal decomposition Ṽ = ℝe_{0} + V + ℝe′_{0}, ⟨e_{0}, e_{0}⟩ = −⟨e′_{0}, e′_{0}⟩ = 1. Recall that 𝔬(Ṽ) is identified with ∧^{2}Ṽ via the pseudo-Euclidean scalar product ⟨·, ·⟩, see (3).

Proposition 6. The subspace

ℝe_{0} ∧ e′_{0} + ∧^{2}V + (e_{0} − e′_{0}) ∧ V ⊂ ∧^{2}Ṽ = 𝔬(Ṽ)

is a subalgebra isomorphic to 𝔤(V).

Proof: The canonical embedding 𝔬(V) = ∧^{2}V ↪ ∧^{2}Ṽ = 𝔬(Ṽ) is extended to an embedding 𝔤(V) ↪ 𝔬(Ṽ) via

D ↦ e_{0} ∧ e′_{0} , V ∋ v ↦ (e_{0} − e′_{0}) ∧ v .□

It is easy to see that the embedding 𝔤(V) ↪ 𝔬(Ṽ) lifts to a homomorphism of connected Lie groups G(V) → SO_{0}(Ṽ) with finite kernel. In particular, we have a natural isometric action of G(V) on the pseudo-Riemannian symmetric space M̃(V) := SO_{0}(p + 1, q + 1)/(SO(p′ + 1) × SO_{0}(p′′, q + 1)). We denote by [e] = eK̃ ∈ M̃(V) := SO_{0}(Ṽ)/K̃, K̃ := SO(p′ + 1) × SO_{0}(p′′, q + 1), the canonical base point and by G(V)[e] its G(V)-orbit.

Proposition 7. The action of G(V) on M̃(V) induces a G(V)-equivariant open embedding

M(V) = G(V)/K →^{∼} G(V)[e] ⊂ M̃(V) .

G(V) acts transitively on M̃(V) if and only if M̃(V) is Riemannian, i.e. if and only if p′′ = 0. In that case M(V) ≅ M̃(V) is the noncompact dual of the Grassmannian SO(p + q + 2)/(SO(p + 1) × SO(q + 1)) and admits a simply transitive splittable solvable subgroup I(S)⋉ exp(ℝD + V) ⊂ G(V). Here I(S) denotes the (solvable) Iwasawa subgroup of S ≅ Spin_{0}(V).

Proof: The stabilizer G(V)_{[e]} of [e] in G(V) coincides with K and hence G(V)[e] ≅ G(V)/G(V)_{[e]} = G(V)/K = M(V). Now a simple dimension count shows that the orbit G(V)[e] ⊂ M̃(V) is open. If M̃(V) is Riemannian then it is a Riemannian symmetric space of noncompact type and, by the Iwasawa decomposition theorem (see [H]) the Iwasawa subgroup I(SO_{0}(p + 1, q + 1)) ⊂ SO_{0}(p + 1, q + 1) is a splittable solvable subgroup which acts simply transitively on M̃(V). Now let G(V) = S⋉R(V) be the Levi decomposition associated to the Levi decomposition 𝔤(V) = 𝔬(V) + (ℝD + V), S ≅ Spin_{0}(V), R(V) = exp(ℝD + V) (cf. Prop. 2). We denote by I(S) ⊂ S the Iwasawa subgroup of S. Then I(S)⋉R(V) ⊂ G(V) is mapped isomorphically onto I(SO_{0}(p + 1, q + 1)) ⊂ SO_{0}(Ṽ) by the homomorphism G(V) → SO_{0}(Ṽ) introduced above. This shows that I(S)⋉R(V) and hence G(V) acts transitively on M̃(V) in the Riemannian case.

If M̃(V) is not Riemannian then the homogeneous spaces M̃(V) and M(V) are not homotopy equivalent and hence G(V) does not act transitively on M̃(V). In fact, M̃(V) (respectively, M(V)) has the homotopy type of the Grassmanian SO(p+ 1)/(SO(p′+1)×SO(p′′)) (respectively, SO(p)/(SO(p′)×SO(p′′))). Now it is sufficient to observe that the Stiefel manifolds SO(p + 1)/SO(p′ + 1) and SO(p)/SO(p′) are homotopy equivalent only if p = p′ and hence p′′ = 0, see [O3]. □

Before we can formulate the main result of the present paper in section 2.4 we need to recall the notions of quaternionic manifold and of (pseudo-) quaternionic Kähler manifold, cf. [A-M2]. The reader familiar with these concepts should skip the next section.

2.2. Quaternionic structures. It is instructive to introduce the basic concepts of quaternionic geometry as analogues of the more familiar concepts of complex geometry.

Definition 4. Let E be a (finite dimensional) real vector space. A complex structure on E is an endomorphism J ∈ End(E) such that J^{2} = −Id. A hypercomplex structure on E is a triple (J_{α}) = (J_{1}, J_{2}, J_{3}) of complex structures on E satisfying J_{1}J_{2} = J_{3}. A quaternionic structure on E is the three-dimensional subspace Q ⊂ End(E) spanned by a hypercomplex structure (J_{α}): Q = span{J_{1}, J_{2}, J_{3}}. In that case, we say that the hypercomplex structure (J_{α}) is subordinate to the quaternionic structure Q.

Note, first, that Q ⊂ 𝔤𝔩(E) is a Lie subalgebra isomorphic to 𝔰𝔭(1) ≅ Imℍ = span{i, j, k} the Lie algebra of the group Sp(1) = S^{3} ⊂ ℍ = span{1, i, j, k} of unit quaternions and, second, that the real associative subalgebra of End(E) generated by Id and a quaternionic structure Q on E is isomorphic to the algebra of quaternions ℍ. In both cases, the choice of such isomorphism is equivalent to the choice of a hypercomplex structure (J_{α}) = (J_{1}, J_{2}, J_{3}) subordinate to Q. In fact, given (J_{α}) we can define an isomorphism of associative algebras by (Id, J_{1}, J_{2}, J_{3}) ↦ (1, i, j, k) this induces also an isomorphism of Lie algebras Q →^{∼} 𝔰𝔭(1).

Definition 5. Let M be a (smooth) manifold. An almost complex structure J (respectively, almost hypercomplex structure (J_{1}, J_{2}, J_{3}), almost quaternionic structure Q) on M is a (smooth) field M ∋ m ↦ J_{m} ∈ End(T_{m}M) of complex structures (respectively, m ↦ (J_{1}, J_{2}, J_{3})_{m} of hypercomplex structures, m ↦ Q_{m} of quaternionic structures). The pair (M, J) (respectively, (M, (J_{α})), (M, Q)) is called an almost complex manifold (respectively, almost hypercomplex manifold, almost quaternionic manifold).

We recall that a connection on a manifold M (i.e. a covariant derivative ∇ on its tangent bundle T M) induces a covariant derivative ∇ on the full tensor algebra over T M and, in particular, on End(T M) ≅ T M ⊗ T^{∗}M. We will say that ∇ preserves a subbundle B ⊂ End(T M) if it maps (smooth) sections of B into sections of T^{∗}M ⊗ B.

Definition 6. A connection ∇ on an almost complex manifold (M, J) (respectively, almost hypercomplex manifold (M, (J_{α})), almost quaternionic manifold (M, Q)) is called almost complex (respectively, almost hypercomplex, almost quaternionic) connection if ∇J = 0 (respectively, if ∇J_{1} = ∇J_{2} = ∇J_{3} = 0, if ∇ preserves the rank 3 subbundle Q ⊂ End(T M)). A complex (respectively, hypercomplex, quaternionic) connection is a torsionfree almost complex (respectively, almost hypercomplex, almost quaternionic) connection.

Note that the equation ∇J = 0 is equivalent to the condition that ∇ preserves the rank 1 subbundle of End(T M) spanned by J, i.e. ∇J = θ ⊗ J for some 1-form θ on M. Therefore, the notion of almost quaternionic connection (as well as that of almost hypercomplex connection) is a direct quaternionic analogue of the notion of almost complex connection.

Definition 7. Let M be a manifold. An almost complex structure (respectively, almost hypercomplex structure, almost quaternionic structure) on M is called 1-integrable if there exists a complex (respectively, hypercomplex, quaternionic) connection on M. A complex structure (respectively, hypercomplex structure, quaternionic structure) on M is a 1-integrable almost complex structure (respectively, almost hypercomplex structure, almost quaternionic structure) on M.

Remark 3: It is well known, see [N-N] and [K-NII], that an almost complex manifold (M, J) is integrable, i.e. admits an atlas with holomorphic transition maps, if and only if it is 1-integrable. This justifies the following definition of complex manifold.

Definition 8. A complex manifold (respectively, hypercomplex manifold) is a manifold M together with a complex structure J (respectively, hypercomplex structure (J_{α})) on M. A quaternionic manifold of dimension d > 4 is a manifold M of dimension d together with a quaternionic structure Q on M. Finally, a quaternionic manifold of dimension d = 4 is a 4-dimensional manifold M together with an almost quaternionic structure Q which annihilates the Weyl tensor of the conformal structure defined by Q, see Remark 4 below.

For examples of hypercomplex and quaternionic manifolds (without metric condition) see [J1], [J2] and [B-D]. Remark 4: Notice that an almost quaternionic structure on a 4-manifold induces an oriented conformal structure. This follows from the fact that the normalizer in GL(4, ℝ) of the standard quaternionic structure(s) on ℝ^{4} = ℍ is the special conformal group CO_{0}(4). The definition of quaternionic 4-manifold (M, Q) implies that this conformal structure is half-flat. More precisely, if the orientation of M is chosen such that Q is locally generated by positively oriented almost complex structures, then the self-dual half of the Weyl tensor vanishes. The special treatment of the 4-dimensional case in Def. 8 and also in Def. 11 below has the advantage that with these definitions many important properties of quaternionic (and also of quaternionic Kähler) manifolds of dimension > 4 remain true in dimension 4, cf.

Remark 5. In particular, all future statements about quaternionic manifolds and quaternionic Kähler manifolds in the present paper, such as the integrability of the canonical almost complex structure on the twistor space (see section 3), are valid also in the 4-dimensional case.

Next we discuss (almost) complex, hypercomplex and quaternionic structures on a pseudo-Riemannian manifold (M, g).

Definition 9. A pseudo-Riemannian metric g on an almost complex manifold (M, J) (respectively, almost hypercomplex manifold (M, (J_{α})), almost quaternionic manifold (M, Q)) is called Hermitian if J is skew symmetric (respectively, the J_{α} are skew symmetric, Q consists of skew symmetric endomorphisms).

Note that, due to J^{2} = −Id, an almost complex structure J on a pseudo-Riemannian manifold (M, g) is skew symmetric if and only if J is orthogonal, i.e. if and only if g(JX, JY) = g(X, Y) for all vector fields X, Y on M. Similarly, an almost quaternionic structure Q on a pseudo-Riemannian manifold (M, g) consists of skew symmetric endomorphisms if and only if Z := {A ∈ Q|A^{2} = −Id} consists of orthogonal endomorphisms (here the equation A^{2} = −Id is on T_{πA}M, π : Q → M the bundle projection).

Definition 10. A complex (pseudo-) Hermitian manifold (M, J, g) (respectively, hypercomplex (pseudo-) Hermitian manifold (M, (J_{α}), g), quaternionic (pseudo-) Hermitian manifold (M, Q, g), almost complex (pseudo-) Hermitian manifold (M, J, g) etc.) is a complex manifold (M, J) (respectively, hypercomplex manifold (M, (J_{α})), quaternionic manifold (M, Q), almost complex manifold (M, J) etc.) with a Hermitian (pseudo-) Riemannian metric g.

Next we define the hypercomplex and quaternionic analogues of (pseudo-) Kähler manifolds.

Definition 11. A (pseudo-) Kähler manifold (respectively, (pseudo-) hyper-Kähler manifold, quaternionic (pseudo-) Kähler manifold of dimension d > 4) is an almost complex (pseudo-) Hermitian manifold (M, J, g) (respectively, almost hypercomplex (pseudo-) Hermitian manifold (M, (J_{α}), g), almost quaternionic (pseudo-) Hermitian manifold (M, Q, g) of dimension d > 4) with the property that the Levi-Civita connection ∇^{g} of the (pseudo-) Riemannian metric g is complex (respectively, hypercomplex, quaternionic). An almost quaternionic Hermitian 4manifold (M, Q, g) is called quaternionic Kähler manifold if Q annihilates the curvature tensor R of ∇^{g}.

See [Bes], [H-K-L-R], [C-F-G], [Sw1], [D-S1], [D-S2], [Bi1], [Bi2], [K-S2] and [C4] for examples of hyper-Kähler manifolds and [W1], [A3], [Bes], [G1], [G-L], [L2], [L3], [F-S], [dW-VP2], [A-G], [K-S1], [A-P], [D-S3] and [C2] for examples of quaternionic Kähler manifolds. Remark 5: As explained in Remark 4, an almost quaternionic structure Q on a 4-manifold M defines a conformal structure. It is clear that a pseudo-Riemannian metric g on M defines the same conformal structure as Q if and only if it is Q-Hermitian and that any such metric is definite. Moreover, the Levi-Civita connection ∇^{g} of an almost quaternionic Hermitian 4-manifold (M, Q, g) automatically preserves Q. In fact, its holonomy group at m ∈ M is a subgroup of SO(T_{m}M, g_{m}) because M is oriented (by Q) and the latter group normalizes Q because g is Q-Hermitian. Now let (M, Q, g) be a quaternionic Kähler 4-manifold. Since Q annihilates the curvature tensor R and the metric g it must also annihilate the the Ricci tensor Ric and the Weyl tensor of (M, g). This shows first that Ric is Q-Hermitian and hence proportional to g: Ric = cg. In other words, (M, g) is an Einstein manifold. Second, (M, Q) is a quaternionic manifold because Q annihilates the Weyl tensor of the conformal structure defined by g, which coincides with the conformal structure defined by Q. For any quaternionic pseudo-Kähler manifold (M, g) (of arbitrary dimension) it is known, see e.g. [A-M2], that (M, g) is Einstein and that Q annihilates R.

As next, we introduce the appropriate notions in order to discuss transitive group actions on manifolds with the special geometric structures defined above.

2.3. Invariant connections on homogeneous manifolds and 1-integra-bility of homogeneous almost quaternionic manifolds.

Definition 12. The automorphism group of an almost complex manifold (M, J) (respectively, almost hypercomplex manifold (M, (J_{α})), almost quaternionic manifold (M, Q), almost complex Hermitian manifold (M, J, g), etc.) is the group of diffeomorphisms of M which preserves J (respectively, (J_{α}), Q, (J, g), etc.). An almost complex manifold (respectively, almost hypercomplex manifold, almost quaternionic manifold, almost complex Hermitian manifold, etc.) is called homogeneous if it has a transitive automorphism group.

In the next section, we are going to construct an almost quaternionic structure Q on certain homogeneous manifolds M = G/K (G is a Lie group and K a closed subgroup). The almost quaternionic structure Q will be G-invariant by construction and hence (M, Q) will be a homogeneous almost quaternionic manifold. Similarly, we will construct homogeneous almost quaternionic (pseudo-) Hermitian manifolds (M = G/K, Q, g). In order to prove that (M = G/K, Q, g) is a quaternionic (pseudo-) Kähler manifold, or simply to establish the 1-integrability of the almost quaternionic structure Q it is useful to have an appropriate description of the affine space of G-invariant connections on the homogeneous manifold M = G/K. This is provided by the notion of Nomizu map, which we now recall, see [A-V-L], [V1]. Let ∇ be a connection on a manifold M. For any vector field X on M one defines the operator

L_{X} := L_{X} − ∇_{X} , (8)

where L_{X} is the Lie derivative (i.e. L_{X}Y = [X, Y] for any vector field Y on M). L_{X} is a C^{∞}(M)-linear map on the C^{∞}(M)-module Γ(T M) of vector fields on M, so it can be identified with a section L_{X} ∈ Γ(End(T M)). In particular, L_{X}|_{m} ∈ End(T_{m}M) for all m ∈ M.

Now let M = G/K be a homogeneous space and suppose that ∇ is G-invariant. The action of G on M defines an antihomomorphism of Lie algebras α : 𝔤 = Lie G → Γ(T M) from the Lie algebra 𝔤 of left-invariant vector fields on G to the Lie algebra Γ(T M) of vector fields on M. This antihomomorphism maps an element x ∈ 𝔤 to the fundamental vector field α(x) := X ∈ Γ(T M) defined by X(m) := \frac{d}{dt}|_{t=0} exp(tx)m. The statement that α is an antihomomorphism means that [α(x), α(y)] = −α([x, y]) for all x, y ∈ 𝔤. Note that α becomes a homomorphism if we replace 𝔤 by the Lie algebra of right-invariant vector fields on G. Without restriction of generality, we can assume that the action is almost effective, i.e. 𝔤 → Γ(T M) is injective. Then we can identify 𝔤 with its faithful image in Γ(T M). The isotropy subalgebra 𝔨 = Lie K is mapped (anti)isomorphically onto a subalgebra of vector fields vanishing at the base point [e] = eK ∈ M = G/K. In this situation we define the Nomizu map L = L(∇) : 𝔤 → End(T_{[e]}M), x ↦ L_{x}, by the equation

L_{x} := L_{X}|_{[e]} ,

where again X is the fundamental vector field on M associated to x ∈ 𝔤. The operators L_{x} ∈ End(T_{[e]}M) will be called Nomizu operators. They have the following properties:

L_{x} = dρ(x) for all x ∈ 𝔨 (9)

and

(10) L_{Ad_{k}x} = ρ(k)L_{x}ρ(k)^{−1} for all x ∈ 𝔤 , k ∈ K ,

where ρ : K → GL(T_{[e]}M) is the isotropy representation. The first equation follows directly from equation (8) since (∇_{X}Y)_{m} = 0 if X(m) = 0. The second equation expresses the G-invariance of ∇. Conversely, any linear map L : 𝔤 → End(T_{[e]}M) satisfying (9) and (10) is the Nomizu map of a uniquely defined G-invariant connection ∇ = ∇(L) on M. Its torsion tensor T and curvature tensor R are expressed at [e] by:

T (πx, πy) = −(L_{x}πy − L_{y}πx + π[x, y])

and

R(πx, πy) = [L_{x}, L_{y}] + L_{[x,y]} , x, y ∈ 𝔤

where π : 𝔤 → T_{[e]}M is the canonical projection x ↦ πx = X([e]) = \frac{d}{dt}|_{t=0} exp(tx)K. Remark 6: The difference between our formulas for torsion and curvature and those of [A-V-L] is due to the fact that in [A-V-L] everything is expressed in terms of right-invariant vector fields on the Lie group G whereas we use left-invariant vector fields. To obtain the corresponding expressions in terms of right-invariant vector fields from the expressions in terms of left-invariant vector fields and vice versa it is sufficient to replace [x, y] by −[x, y] in all formulas (x, y ∈ 𝔤). The same remark applies to formula (11) below, which expresses the Levi-Civita connection on a pseudo-Riemannian homogeneous manifold.

Suppose now that we are given a G-invariant geometric structure S on M (e.g. a G-invariant almost quaternionic structure Q) defined by a corresponding K-invariant geometric structure S_{[e]} on T_{[e]}M. Then a G-invariant connection ∇ preserves S if and only if the corresponding Nomizu operators L_{x}, x ∈ 𝔤, preserve S_{[e]}. So to construct a G-invariant connection preserving S it is sufficient to find a Nomizu map L : 𝔤 → End(T_{[e]}M) such that L_{x} preserves S_{[e]} for all x ∈ 𝔤. We observe that, due to the K-invariance of S_{[e]}, the Nomizu operators L_{x} preserve S_{[e]} already for x ∈ 𝔨. The above considerations can be specialized as follows:

Proposition 8. Let Q be a G-invariant almost quaternionic structure on a homogeneous manifold M = G/K. There is a natural one-to-one correspondence between G-invariant almost quaternionic connections on (M, Q) and Nomizu maps L : 𝔤 → End(T_{[e]}M), whose image normalizes Q_{[e]}, i.e. whose Nomizu operators L_{x}, x ∈ 𝔤, belong to the normalizer n(Q) ≅ 𝔰𝔭(1) ⊕ 𝔤𝔩(d, ℍ) (d = dim M/4) of the quaternionic structure Q_{[e]} in the Lie algebra 𝔤𝔩(T_{[e]}M).

Corollary 4. Let (M = G/K, Q) be a homogeneous almost quaternionic manifold and L : 𝔤 → End(T_{[e]}M) a Nomizu map such that

(1) L_{x}πy − L_{y}πx = −π[x, y] for all x, y ∈ 𝔤 (i.e. T = 0) and

(2) L_{x} normalizes Q_{[e]} ⊂ End(T_{[e]}M). Then the connection ∇(L) associated to the Nomizu map L is a G-invariant quaternionic connection on (M, Q) and hence Q is 1-integrable.

For future use we give the well known formula for the Nomizu map L^{g} associated to the Levi-Civita connection ∇^{g} of a G-invariant pseudo-Riemannian metric g on a homogeneous space M = G/K. Let ⟨·, ·⟩ = g_{[e]} be the K-invariant scalar product on T_{[e]}M induced by g. Then L_{x}^{g} ∈ End(T_{[e]}M), x ∈ 𝔤, is given by the following Koszul type formula:

−2⟨L_{x}^{g}πy, πz⟩ = ⟨π[x, y], πz⟩ − ⟨πx, π[y, z]⟩ − ⟨πy, π[x, z]⟩ , x, y, z ∈ 𝔤 . (11)

Corollary 5. Let (M = G/K, Q, g) be a homogeneous almost quaternionic (pseudo-) Hermitian manifold and assume that L_{x}^{g} normalizes Q_{[e]} for all x ∈ 𝔤. Then the Levi-Civita connection ∇^{g} = ∇(L^{g}) is a G invariant quaternionic connection on (M, Q, g) and hence (M, Q, g) is a quaternionic (pseudo-) Kähler manifold if dim M > 4.

2.4. The main theorem. Let 𝔭(Π) = 𝔭(V) + W be an extended Poincaré algebra of signature (p, q) and p ≥ 3. We fix the decomposition p = p′ + p′′, where p′ = 3. For notational convenience we put r := p′′ = p − 3. Then we consider the linear groups K = K(p′, p′′) = K(3, r) ⊂ G(V) ⊂ G = G(Π) ⊂ Aut(𝔯) introduced in section 2.1. As before 𝔯 denotes the radical of 𝔤 = Lie G.

Theorem 8. 1) There exists a G-invariant quaternionic structure Q on

M = M(Π) = G(Π)/K.

2) If Π is nondegenerate (see Def. 2) then there exists a G-invariant pseudo-Riemannian metric g on M such that (M, Q, g) is a quaternionic pseudo-Kähler manifold.

Proof: The main idea of the proof, which we will carry out in detail, is first to observe that the submanifold M(V) = G(V)/K ⊂ G/K = M has a natural G(V)- invariant structure of quaternionic pseudo-Kähler manifold and then to study the problem of extending this structure to the manifold M. As shown in section 2.1, we can embed M(V) as open G(V)-orbit into the pseudo-Riemannian symmetric space M̃(V) = SO_{0}(Ṽ)/K̃, Ṽ = ℝ^{p+1,q+1}, K̃ = SO(p′ + 1) × SO_{0}(p′′, q + 1) = SO(4) × SO_{0}(r, q + 1). Now we claim that M̃(V) carries an SO_{0}(Ṽ)-invariant (and hence G(V)-invariant) almost quaternionic structure Q. It is sufficient to specify the corresponding K̃-invariant quaternionic structure Q_{[e]} ⊂ End(T_{[e]}M̃(V)) at the canonical base point [e] = eK̃ ∈ M̃(V). We first define a hypercomplex structure (J_{α}) on the isotropy module T_{[e]}M̃(V) ≅ ℝ^{4} ⊗ ℝ^{r,q+1}:

J_{1} := μ_{l}(i) ⊗ Id , J_{2} := μ_{l}(j) ⊗ Id , J_{3} := μ_{l}(k) ⊗ Id . (12)

Here μ_{l}(x) ∈ End(ℝ^{4}) stands for left-multiplication by the quaternion x ∈ ℍ = span{1, i, j, k}: μ_{l}(x)y = xy for all x, y ∈ ℍ = ℝ^{4}. Right-multiplication by x will be denoted by μ_{r}(x). Now it is easy to check that the quaternionic structure Q_{[e]} := span{J_{1}, J_{2}, J_{3}} is normalized under the isotropy representation of K̃ and hence extends to an SO_{0}(Ṽ)-invariant almost quaternionic structure Q on M̃(V). Note that the complex structures J_{α} are not invariant under the isotropy representation and hence do not extend to SO_{0}(Ṽ)-invariant almost complex structures on M̃(V). Next we observe that the tensor product of the canonical pseudo-Euclidean scalar products on ℝ^{4} and ℝ^{r,q+1} defines a pseudo-Euclidean scalar product −g_{[e]} on T_{[e]}M̃(V) ≅ ℝ^{4} ⊗ ℝ^{r,q+1}. Notice that g_{[e]} is positive definite if r = 0 and indefinite otherwise. It is invariant under the isotropy representation and hence extends to a (Q-Hermitian) SO_{0}(Ṽ)-invariant pseudo-Riemannian metric on M̃(V), which is unique, up to scaling, and defines on M̃(V) the well known structure of pseudo-Riemannian symmetric space. In fact, it is the symmetric space associated to the symmetric pair (𝔬(Ṽ), 𝔨̃ := Lie K̃). Let 𝔪̃ ⊂ 𝔬(Ṽ) be the 𝔨̃-invariant complement to the isotropy algebra 𝔨̃. Then [𝔪̃, 𝔪̃] ⊂ 𝔨̃ and hence the Nomizu operators L_{x}^{g} ∈ End(T_{[e]}M̃(V)) associated to the Levi-Civita connection ∇^{g} of g vanish for x ∈ 𝔪̃, see (11). On the other hand if x ∈ 𝔨̃ then the Nomizu operator L_{x}^{g} on T_{[e]}M̃(V) ≅ 𝔪̃ coincides with the image of x under the isotropy representation: L_{x}^{g} = ad_{x}|𝔪̃. This shows that L_{x}^{g} normalizes the quaternionic structure Q_{[e]} for all x ∈ 𝔬(Ṽ) and by Cor. 5 we conclude that (M̃(V), Q, g) is a homogeneous quaternionic pseudo-Kähler manifold if dim M̃(V) > 4. The manifold M̃(V) is 4-dimensional only if r = q = 0 and in this case M̃(V) = SO_{0}(4, 1)/SO(4) reduces to real hyperbolic 4-space Hℝ^{4}, i.e. to the quaternionic hyperbolic line Hℍ^{1} = Sp(1, 1)/Sp(1) · Sp(1), which is a standard example of (conformally half-flat Einstein) quaternionic Kähler 4-manifold. Of course, the pair (Q, g) defined on the manifold M̃(V) (for all r and q) restricts to a G(V)-invariant quaternionic pseudo-Kähler structure on the open G(V)-orbit M(V) ↪ M̃(V). So we have proven that (M(V) = G(V)/K, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold.

Our strategy is now to extend the geometric structures Q and g from M(V) to G-invariant structures on M = G/K ⊃ G(V)/K = M(V). First we will extend Q to a G-invariant almost quaternionic structure on M. Using the infinitesimal action of 𝔤 = 𝔤(Π) = 𝔤(V) + W on M we can identify T_{[e]}M = (𝔤(V)/𝔨) ⊕ W = T_{[e]}M(V) ⊕ W. Note that the isotropy representation of K on T_{[e]}M preserves this decomposition and acts on T_{[e]}M(V) = T_{eK}M(V) ≅ T_{eK̃} M̃(V) as restriction of the isotropy representation of K̃ on T_{eK̃} M̃(V) to the subgroup K ⊂ K̃. We extend the hypercomplex structure (J_{α}) defined above on T_{eK̃} M̃(V) ≅ T_{eK}M(V) to a hypercomplex structure on T_{eK}M = T_{eK}M(V) ⊕ W as follows:

J_{α}s := e_{β}e_{γ}s , s ∈ W , (13)

where (e_{α}, e_{β}, e_{γ}) is a cyclic permutation of the standard orthonormal basis of E = R^{3,0} ⊂ V = E + E′ (the product e_{β}e_{γ} is in the Clifford algebra). Now we can extend the quaternionic structure Q_{[e]} on T_{[e]}M(V) to a quaternionic structure on T_{[e]}M such that Q_{[e]} = span{J_{1}, J_{2}, J_{3}}. To prove that Q_{[e]} ⊂ End(T_{[e]}M) extends to a G-invariant almost quaternionic structure on M it is sufficient to check that the isotropy algebra 𝔨 normalizes Q_{[e]}. It is obvious that the subalgebra 𝔬(E′) = 𝔬(r, q) ⊂ 𝔨 = 𝔬(E) ⊕ 𝔬(E′) centralizes Q_{[e]}. We have to show that 𝔬(E) = 𝔬(3) normalizes Q_{[e]}. In terms of the standard basis (e_{1}, e_{2}, e_{3}) = (i, j, k) of E = ℝ^{3} = Imℍ a basis of 𝔬(3) = ∧^{2}E is given by (e_{2} ∧ e_{3}, e_{3} ∧ e_{1}, e_{1} ∧ e_{2}). For any cyclic permutation (α, β, γ) of (1, 2, 3) we have the following easy formulas, cf. (4):

−2dρ(e_{β} ∧ e_{γ}) = (μ_{l}(e_{α}) − μ_{r}(e_{α})) ⊗ Id (14) on T_{eK}M(V) ≅ T_{eK̃} M̃(V) ≅ ℝ^{4} ⊗ ℝ^{r,q+1} and

−2dρ(e_{β} ∧ e_{γ}) = e_{β}e_{γ} = J_{α} (15)

on W. As before dρ : 𝔨 → 𝔤𝔩(T_{[e]}M) denotes the isotropy representation. From the equations (12)-(15) we immediately obtain that

[dρ(e_{β} ∧ e_{γ}), J_{α}] = 0 and [dρ(e_{β} ∧ e_{γ}), J_{β}] = −J_{γ} .

This shows that Q_{[e]} ⊂ End(T_{[e]}M) is invariant under 𝔨 and hence extends to a G-invariant almost quaternionic structure Q on M. The 1-integrability of Q will be proven in the sequel.

Let us first treat the case where Π is nondegenerate. First of all, we extend the G(V)-invariant Q-Hermitian pseudo-Riemannian metric g on M(V) to a Ginvariant Q-Hermitian pseudo-Riemannian metric on M. It is sufficient to extend the K-invariant pseudo-Euclidean scalar product g_{[e]} on T_{[e]}M(V) to a K-invariant and J_{α}-invariant (α = 1, 2, 3) scalar product on T_{[e]} = T_{[e]}M(V) ⊕ W. We do this in such a way that the above decomposition is orthogonal for the extended scalar product g_{[e]} on T_{[e]}M and define

g_{[e]}|W × W := −b ,

where b = b_{Π,(e_{1},e_{2},e_{3})} is the canonical symmetric bilinear form associated to Π and to the decomposition p = p′ + p′′ = 3 + r. By Prop. 1 the nondegeneracy of Π implies the nondegeneracy of b. This shows that the symmetric bilinear form g_{[e]} on T_{[e]}M defined above is indeed a pseudo-Euclidean scalar product. This scalar product is invariant under the isotropy group K = K(p′, p′′) = K(3, r), by virtue of Thm. 1, 3), and hence extends to a G-invariant pseudo-Riemannian metric g on M. Moreover, the metric g is Q-Hermitian. In fact, it is sufficient to observe that b is J_{α}-invariant. This property follows from the K-invariance of b, since J_{α}|W ∈ dρ(𝔨)|W, see (15). Summarizing, we obtain: (M, Q, g) is a homogeneous pseudo-Hermitian almost quaternionic manifold. The next step is to compute the Nomizu map L^{g} : 𝔤 → End(T_{[e]}M) associated to the Levi-Civita connection ∇^{g} of g. It is convenient to identify T_{[e]}M with a 𝔨-invariant complement 𝔪 to 𝔨 in 𝔤. Such complement is easily described in terms of the K-invariant orthogonal decomposition V = E +E′ = ℝ^{3,0} +ℝ^{r,q}. Indeed, using the canonical identification 𝔬(V) = ∧^{2}V we have the following 𝔨-invariant decompositions:

𝔬(V) = 𝔨 + E ∧ E′ ,

𝔤(V) = 𝔬(V) + ℝD + V = 𝔨 + 𝔪(V) , 𝔪(V) = E ∧ E′ + ℝD + V ,

𝔤 = 𝔤(V) + W = 𝔨 + 𝔪 , 𝔪 = 𝔪(Π) := 𝔪(V) + W .

In the following, we will make the identifications T_{[e]}M(V) = 𝔤(V)/𝔨 = 𝔪(V) ⊂ T_{[e]}M = 𝔤/𝔨 = 𝔪. Let (e′_{a}), a = 1, …, r + q, be the standard orthonormal basis of E′ = ℝ^{r,q}. Then we have the following orthonormal basis of (𝔪(V), g_{[e]}):

(e_{α} ∧ e′_{a}, D, e_{α}, e′_{a}) ,

where α = 1, 2, 3 and a = 1, …, r + q. For notational convenience we denote this basis by (e_{ιi}), ι = 0, … 3, i = 0, …, r + q, where

e_{00} := e′_{0} := e_{0} := D
e_{0a} := e′_{a} (a = 1, …, r + q)
e_{α0} := e_{α} (α = 1, 2, 3)
e_{αa} := e_{α} ∧ e′_{a} .

In terms of this basis the hypercomplex structure J_{α} is expressed on 𝔪(V) simply by:

J_{α}e_{0i} = J_{α}e′_{i} = e_{αi} and J_{α}e_{βi} = e_{γi} ,

where i = 0, …, r + q and (α, β, γ) is a cyclic permutation of (1, 2, 3). The scalar product g_{[e]} is completely determined on 𝔪(V) by the condition that (e_{ιi}) is orthonormal and that

ϵ_{i} := g(e_{ιi}, e_{ιi}) = \{ \begin{matrix}−1 if & i = 1, …, r \\ +1 if & i = 0, r + 1, …, r + q . \end{matrix}

The scalar product g_{[e]} on 𝔪 induces an identification x ∧ y ↦ x ∧_{g} y of the exterior square ∧^{2}𝔪 with the vector space of g_{[e]}-skew symmetric endomorphisms on 𝔪, where x∧_{g} y(z) := g(y, z)x−g(x, z)y, x, y, z ∈ 𝔪. We denote by n(Q) (respectively, z(Q)) the normalizer (respectively, centralizer) of the quaternionic structure Q_{[e]} in the Lie algebra 𝔤𝔩(𝔪).

Lemma 5. The Nomizu map L^{g} = L(∇^{g}) associated to the Levi-Civita connection ∇^{g} of the homogeneous pseudo-Riemannian manifold (M = G/K, g) is given by the following formulas:

L_{e_{0}}^{g} = L_{e_{αa}}^{g} = 0 ,
L_{e_{α}}^{g} = \frac{1}{2}J_{α} + L̄_{e_{α}}^{g} ,

where

L̄_{e_{α}}^{g} = \frac{1}{2} ∑_{i=0}^{r+q} ϵ_{i}e_{αi} ∧_{g} e′_{i} − \frac{1}{2} ∑_{i=0}^{r+q} ϵ_{i}e_{γi} ∧_{g} e_{βi} ∈ z(Q) ,

L_{e′_{a}}^{g} ∈ z(Q) is given by:

L_{e′_{a}}^{g}|𝔪(V) = ∑_{ι=0}^{3} e_{ιa} ∧_{g} e_{ι} ,
L_{e′_{a}}^{g}|W = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a} .

For all s ∈ W the Nomizu operator L_{s}^{g} ∈ z(Q) maps the subspace 𝔪(V) ⊂ 𝔪 = 𝔪(V) + W into W and W into 𝔪(V). The restriction L_{s}^{g}|W (s ∈ W) is completely determined by L_{s}^{g}|𝔪(V) (and vice versa) according to the relation

g(L_{s}^{g}t, x) = −g(t, L_{s}^{g}x) , s, t ∈ W , x ∈ 𝔪(V) .

Finally, L_{s}^{g}|𝔪(V) (s ∈ W) is completely determined by its values on the quaternionic basis (e′_{i}), i = 0, …, r + q, which are as follows:

L_{s}^{g}e′_{0} = \frac{1}{2}s ,
L_{s}^{g}e′_{a} = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a}s .

(It is understood that L_{x}^{g} = dρ(x) = ad_{x}|𝔪 for all x ∈ 𝔨, cf. equation (9).) In the above formulas a = 1, …, r + q, α = 1, 2, 3 and (α, β, γ) is a cyclic permutation of (1, 2, 3).

Proof: This follows from equation (11) by a straightforward computation. □

Corollary 6. The Levi-Civita connection ∇^{g} of the homogeneous almost quaternionic pseudo-Hermitian manifold (M, Q, g) preserves Q and hence (M, Q, g) is a quaternionic pseudo-Kähler manifold.

Proof: From the G-invariance of Q, we know already that L_{x}^{g} ∈ n(Q) for all x ∈ 𝔨 and the formulas of Lemma 5 show that L_{x}^{g} ∈ n(Q) for all x ∈ 𝔪. Now the corollary follows from Cor. 5. □

By Cor. 6 we have already established part 2) of Thm. 8. Part 1) is a consequence of 2) provided that Π is nondegenerate. It remains to discuss the case of degenerate Π.

By Thm. 2, we have a direct decomposition W = W_{0} + W′ of Cℓ^{0}(V)-modules, where W_{0} = ker Π. We put Π′ := Π| ∧^{2} W′ and denote by L′ : 𝔤(Π′) → End(𝔪(Π′)) the Nomizu map associated to the Levi-Civita connection of the quaternionic pseudo-Kähler manifold (M′ := M(Π′), Q, g). Note that the quaternionic structure Q of M′ coincides with the almost quaternionic structure induced by the obvious G(Π′)- equivariant embedding M′ = G(Π′)/K ⊂ G(Π)/K = M. By the next lemma, we can extend the map L′ to a torsionfree Nomizu map L : 𝔤(Π) → End(𝔪(Π)), whose image normalizes Q_{[e]}. This proves the 1-integrability of Q (by Cor. 4), completing the proof of Thm. 8. □

By Cor. 6 we can decompose L′_{x} = ∑_{α=1}^{3} ω′_{α}(x)J_{α} + L̄′_{x}, where the ω′_{α} are 1-forms on 𝔪(Π′) and L̄′_{x} ∈ z(Q) belongs to the centralizer of the quaternionic structure on 𝔪(Π′).

Lemma 6. The Nomizu map L′ : 𝔤(Π′) → End(𝔪(Π′)) associated to the Levi-Civita connection of (M(Π′) = G(Π′)/K, g) can be extended to the Nomizu map L : 𝔤(Π) → End(𝔪(Π)) of a G(Π)-invariant quaternionic connection ∇ on the homogeneous almost quaternionic manifold (M(Π) = G(Π)/K, Q). The extension is defined as follows:

L_{x} := ∑_{α=1}^{3} ω_{α}(x)J_{α} + L̄_{x} , x ∈ 𝔪(Π) ,

where L̄_{x} ∈ z(Q), with centralizer taken in 𝔤𝔩(𝔪(Π)), is defined below and the 1forms ω_{α} on 𝔪(Π) := 𝔪(Π′) + W_{0} satisfy ω_{α}|𝔪(Π′) := ω′_{α} and ω_{α}|W_{0} := 0. The operators L̄_{x} are given by:

L̄_{x}|𝔪(Π′) := L̄′_{x} if x ∈ 𝔪(Π′) ,
L̄_{e_{0}}|W_{0} := L̄_{e_{αa}}|W_{0} := L̄_{e_{α}}|W_{0} := 0 ,

L̄_{e′_{a}}|W_{0} := \frac{1}{2}e_{1}e_{2}e_{3}e′_{a} ,

L̄_{s}|W_{0} := 0 if s ∈ W ,

L̄_{s}x := L_{x}s − [s, x] if s ∈ W_{0} , x ∈ 𝔪(Π′) .

(It is understood that L_{x} = ad_{x}|𝔪(Π) for all x ∈ 𝔨. In the above formulas, as usual, a = 1, …, r + q and α = 1, 2, 3.)

Proof: To check that L : 𝔤(Π) → End(𝔪(Π)) is a Nomizu map it is sufficient to check (10); (9) is satisfied by definition. The equation (10) expresses the Kinvariance of L ∈ 𝔤(Π)^{∗} ⊗ End(𝔪(Π)) which is equivalent to the invariance of L under the Lie algebra 𝔨, since K is a connected Lie group. So (10) is equivalent to the following equation:

L_{[x,y]}z = [ad_{x}, L_{y}]z x ∈ 𝔨 , y, z ∈ 𝔪(Π) . (16)

We first check this equation. Let always x ∈ 𝔨. Due to L_{W} W_{0} = 0 and [𝔨, W_{0}] ⊂ W_{0} we have

L_{[x,y]}z = [ad_{x}, L_{y}]z = 0

if y ∈ W and z ∈ W_{0}. Also (16) is satisfied if y, z ∈ 𝔪(Π′) because L_{x}|𝔪(Π′) = L′_{x} is the Nomizu operator associated to the Nomizu map L′. Now let y ∈ W_{0} and z ∈ 𝔪(Π′). Then we compute:

L_{[x,y]}z − [ad_{x}, L_{y}]z = (L_{z}[x, y] − [[x, y], z]) − [x, L_{y}z] + L_{y}[x, z]
= L_{z}[x, y] − [[x, y], z] − [x, L_{z}y] + [x, [y, z]] + L_{[x,z]}y − [y, [x, z]]
= L_{[x,z]}y − [x, L_{z}y] + L_{z}[x, y] = L_{[x,z]}y − [ad_{x}, L_{z}]y .

This shows that it is sufficient to check (16) for y ∈ 𝔪(Π′) and z ∈ W_{0}; the case y ∈ W_{0} and z ∈ 𝔪(Π′) then follows from the above computation. In the following let x ∈ 𝔨 and z ∈ W_{0}. We check (16) for all y ∈ 𝔪(Π′). From L_{e_{0}} = 0 and [𝔨, e_{0}] = 0 it follows that

L_{[x,e_{0}]}z − [x, L_{e_{0}}z] + L_{e_{0}}[x, z] = 0 .

Next we check (16) for y = e_{α}:

L_{[x,e_{α}]}z − [x, L_{e_{α}}z] + L_{e_{α}}[x, z] = L_{[x,e_{α}]}z − \frac{1}{2}[x, J_{α}z] + \frac{1}{2}J_{α}[x, z] .

It is clear that the first summand and the sum of the second and third summands vanish if x ∈ 𝔬(E′) ⊂ 𝔨 = 𝔬(E) ⊕ 𝔬(E′). For x = e_{α} ∧ e_{β} ∈ 𝔬(E) = 𝔬(3) ((α, β, γ) cyclic) we compute:

L_{[e_{α}∧e_{β},e_{α}]}z − \frac{1}{2}[e_{α} ∧ e_{β}, J_{α}z] + \frac{1}{2}J_{α}[e_{α} ∧ e_{β}, z]

= −L_{e_{β}}z + \frac{1}{4}e_{α}e_{β}e_{β}e_{γ}z − \frac{1}{4}e_{β}e_{γ}e_{α}e_{β}z

= −\frac{1}{2}e_{γ}e_{α}z + \frac{1}{2}e_{γ}e_{α}z = 0

and for x = e_{β} ∧ e_{γ} we obtain:

L_{[e_{β}∧e_{γ},e_{α}]}z − \frac{1}{2}[e_{β} ∧ e_{γ}, J_{α}z] + \frac{1}{2}J_{α}[e_{β} ∧ e_{γ}, z]

= 0 + \frac{1}{4}e_{β}e_{γ}e_{β}e_{γ}z − \frac{1}{4}e_{β}e_{γ}e_{β}e_{γ}z = 0 .

Next we check (16) for y = e′_{a}:

L_{[x,e′_{a}]}z − [x, L_{e′_{a}}z] + L_{e′_{a}}[x, z] =

\frac{1}{2}e_{1}e_{2}e_{3}[x, e′_{a}]z − \frac{1}{2}[x, e_{1}e_{2}e_{3}e′_{a}z] + \frac{1}{2}e_{1}e_{2}e_{3}e′_{a}[x, z] .

It is easy to see that this is zero if [x, e′_{a}] = 0. So we can put x = e′_{b} ∧ e′_{a} obtaining:

\frac{1}{2}⟨e′_{a}, e′_{a}⟩e_{1}e_{2}e_{3}e′_{b}z + \frac{1}{4}e′_{b}e′_{a}e_{1}e_{2}e_{3}e′_{a}z − \frac{1}{4}e_{1}e_{2}e_{3}e′_{a}e′_{b}e′_{a}z = 0 .

Finally, for y ∈ E ∧ E′ we immediately obtain:

L_{[x,y]}z − [x, L_{y}z] + L_{y}[x, z] = L_{[x,y]}z = 0 ,

since L_{y} = 0 for y ∈ E ∧ E′ and [𝔨, E ∧ E′] ⊂ E ∧ E′. So we have proven that L is a Nomizu map. It is easily checked that L_{x}y − L_{y}x = −π[x, y] for all x, y ∈ 𝔪(Π), where π : 𝔤(Π) → 𝔪(Π) ≅ T_{[e]}M(Π) is the projection along 𝔨. This shows that ∇ = ∇(L) has zero torsion. It only remains to check that L_{x} ∈ n(Q) for all x ∈ 𝔤(Π). This is easy to see for x ∈ 𝔤(Π′) from the definition of the map L as extension of the Nomizu map L′. We present the calculation only for L_{s}, s ∈ W_{0}:

L_{s}e_{0} = L_{e_{0}}s − [s, e_{0}] = 0 + \frac{1}{2}s = \frac{1}{2}s ,

L_{s}J_{α}e_{0} = L_{s}e_{α} = L_{e_{α}}s − [s, e_{α}] = \frac{1}{2}J_{α}s − 0 = J_{α}L_{s}e_{0} ,

L_{s}e′_{a} = L_{e′_{a}}s − [s, e′_{a}] = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a}s − 0 = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a}s ,

L_{s}J_{α}e′_{a} = L_{s}e_{αa} = L_{e_{αa}}s − [s, e_{αa}] = 0 − \frac{1}{2}e_{α}e′_{a}s

= e_{β}e_{γ}(\frac{1}{2}e_{α}e_{β}e_{γ}e′_{a}s) = J_{α}L_{s}e′_{a} ,

(α, β, γ) cyclic,

L_{s}t = L_{t}s − [s, t] = 0 ,
L_{s}J_{α}t = L_{J_{α}t}s − [s, J_{α}t] = 0 = J_{α}L_{t}s

if t ∈ W. We have used that L_{W} W_{0} = 0 and [W_{0}, W] = 0. This shows that L_{s} ∈ z(Q) for all s ∈ W_{0} finishing the proof of the lemma. □

2.5. The Riemannian case.

Proposition 9. Let 𝔭(Π) = 𝔭(V) + W be a nondegenerate extended Poincaré algebra of signature (p, q), p ≥ 3, and (M(Π), Q, g) the corresponding homogeneous quaternionic pseudo-Kähler manifold, see Thm. 8. Then the pseudo-Riemannian metric g is positive definite, and hence a Riemannian metric, if and only if −b is positive definite and p = 3. In all other cases g is indefinite. Here b = b_{Π,(e_{1},e_{2},e_{3})} is the canonical symmetric bilinear form associated to Π and to the decomposition p = 3 + r.

Proof: By construction, the restriction of g to the submanifold M(V) ⊂ M(Π) is a (positive definite) Riemannian metric if and only if p = 3 and is indefinite otherwise. Now the proposition follows from the fact that −b is precisely the restriction of the scalar product g_{[e]} on T_{[e]}M(Π) ≅ T_{[e]}M(V) ⊕ W to the subspace W. □

Next we will use the classification of extended Poincaré algebras of signature (3, q) up to isomorphism (see 1.3) to derive the classification of the quaternionic Kähler manifolds (M(Π), Q, g) up to isometry. We recall that 𝔭(3, q, 0, 0, l) = 𝔭(p = 3, q, l_{0} = 0, l_{+} = 0, l_{−} = l) (respectively, 𝔭(3, q, 0, 0, l^{+}, 0, 0, l^{−}) = 𝔭(p = 3, q, l_{0}^{+} = 0, l_{+}^{+} = l^{+}, l_{0}^{−} = 0, l_{+}^{−} = 0, l_{−}^{−} = l^{−})) is the set of isomorphism classes of extended Poincaré algebras for which −b is positive definite if q ≢ 3 (mod 4) (respectively, if q ≡ 3 (mod 4)). We denote by M(q, l) (respectively, M(q, l^{+}, l^{−})) the homogeneous quaternionic Kähler manifold (M(Π), Q, g) associated to Π ∈ 𝔭(3, q, 0, 0, l) (respectively, Π ∈ 𝔭(3, q, 0, 0, l^{+}, 0, 0, l^{−})).

Theorem 9. Every homogeneous quaternionic pseudo-Kähler manifold of the form (M(Π), Q, g) for which g is positive definite is isometric to one of the homogeneous quaternionic Kähler manifolds M(q, l) (q ≢ 3 (mod 4)) or M(q, l^{+}, l^{−}) ≅ M(q, l^{−}, l^{+}) (q ≡ 3 (mod 4)). In particular, there are only countably many such Riemannian manifolds up to isometry.

Proof: This is a direct consequence of Prop. 9, Thm. 6 and Thm. 7. □

Any real vector space E admitting a quaternionic structure has dim E ≡ 0 (mod 4). Therefore, we can define its quaternionic dimension dimℍ E := dim E/4. Similarly, the quaternionic dimension of a quaternionic manifold (M, Q) is dimℍ M := dim M/4. We denote by N(q) the quaternionic dimension of an irreducible Cℓ_{3,q}^{0}-module.

Proposition 10. 1) N(0) = N(1) = N(2) = N(3) = 1, N(4) = 2, N(5) = 4, N(6) = N(7) = 8 and N(q + 8) = 16N(q) for all q ≥ 0. In particular, N(q) coincides with the dimension of an irreducible ℤ_{2}-graded Cℓ_{q−3}-module if q ≥ 3.

2) The quaternionic dimension of the homogeneous quaternionic Kähler manifolds M(q, l) and M(q, l^{+}, l^{−}) is given by:

dimℍ M(q, l) = q + 1 + lN(q)

and

dimℍ M(q, l^{+}, l^{−}) = q + 1 + (l^{+} + l^{−})N(q) .

Proof: The first part follows from the classification of Clifford algebras. The second part follows from dim M = dim M(V) + dim W. □

The next theorem identifies the spaces M(q, l) and M(q, l^{+}, l^{−}) with Alekseevsky’s quaternionic Kähler manifolds. We recall that an Alekseevsky space is a quaternionic Kähler manifold which admits a simply transitive non Abelian splittable solvable group of isometries, see [A3] and [C2]. Due to Iwasawa’s decomposition theorem any symmetric quaternionic Kähler manifold of noncompact type is an Alekseevsky space. These are precisely the noncompact duals of the Wolf spaces. We recall that a Wolf space is a symmetric quaternionic Kähler manifold of compact type and that such manifolds are in 1-1-correspondence with the complex simple Lie algebras [W1]. The nonsymmetric Alekseevsky spaces are grouped into 3 series: V-spaces, W-spaces and T -spaces, see [A3], [dW-VP2] and [C2]. These 3 series contain also all symmetric Alekseevsky spaces of rank > 2 and no symmetric spaces of smaller rank. By definition an Alekseevsky space can be presented as metric Lie group, i.e. as homogeneous Riemannian manifold of the form (L, g), where L is a Lie group and g a left-invariant Riemannian metric on L.

Theorem 10. Let (M = M(Π) = G(Π)/K, Q, g) be a homogeneous quaternionic Kähler manifold as in Prop. 9,

G(Π) = S⋉R ≅ Spin_{0}(V)⋉R

the Levi decomposition (7) and I(S) the Iwasawa subgroup of S. Then L := L(Π) := I(S)⋉R ⊂ G(Π) is a (non Abelian) splittable solvable Lie subgroup which acts simply transitively on M. In particular, (M, Q, g) is an Alekseevsky space. More precisely, we have the following identifications with the V-spaces, W-spaces, T - spaces and symmetric Alekseevsky spaces:

1) M(q, l) = V(l, q − 3) and M(q, l^{+}, l^{−}) = V(l^{+}, l^{−}, q − 3) if q ≥ 4,

2) M(3, l^{+}, l^{−}) = W(l^{+}, l^{−}),
3) M(2, l) = T (l),
4) M(1, l) = SU(l + 2, 2)/S(U(l + 2) × U(2)),

5) M(0, l) = Sp(l + 1, 1)/Sp(l + 1)Sp(1) = ℍH^{l+1} (quaternionic hyperbolic

(l + 1)-space).

The above Riemannian manifolds M(q, l) and M(q, l^{+}, l^{−}) ≅ M(q, l^{−}, l^{+}) are pairwise nonisometric and exhaust all Alekseevsky spaces with the following two symmetric exceptions: the complex hyperbolic plane ℂH^{2} = SU(1, 2)/U(2) =: M(1, −1) and G_{2}^{(2)}/SO(4). (Note that these two symmetric spaces have rank ≤ 2 and so do not belong to any of the 3 series V, W and T of Alekseevsky spaces.)

Proof: The fact that L acts simply transitively on M follows from Prop. 7. This shows that (M, Q, g) is an Alekseevsky space. The Riemannian metric g induces a left-invariant metric g_{L} on the Lie group L. To establish the identifications given in the theorem it is sufficient to check that (L, g_{L}) is isomorphic (as metric Lie group) to one of the metric Lie groups which occur in the classification of Alekseevsky spaces, see [A3] and [C2]. (The quaternionic structure can be reconstructed from the holonomy of the Levi-Civita connection, up to an automorphism of the full isometry group which preserves the isotropy group.) Finally, to prove that M(q, l) and M(q, l^{+}, l^{−}) ≅ M(q, l^{−}, l^{+}) are pairwise nonisometric it is, by [A4], sufficient to check that the corresponding metric Lie groups (which occur in the classification of Alekseevsky spaces) are pairwise nonisomorphic. This was done in [C2]. □

2.6. A class of noncompact homogeneous quaternionic Hermitian manifolds with no transitive solvable group of isometries. There is a widely known conjecture by D.V. Alekseevsky which says that any noncompact homogeneous quaternionic Kähler manifold admits a transitive solvable group of isometries [A3]. The next theorem shows that this conjecture becomes false if we replace “Kähler” by “Hermitian”.

Theorem 11. Let 𝔭(Π) be any extended Poincaré algebra of signature (p, q) = (3 + r, 0), r ≥ 0, and (M = G/K, Q) the corresponding homogeneous quaternionic manifold, see Thm. 8, 1). Then there exists a G-invariant Q-Hermitian Riemannian metric h on M. Moreover, the noncompact homogeneous quaternionic Hermitian manifold (M, Q, h) does not admit any transitive solvable Lie group of isometries if r > 0.

Proof: Since K = K(3, r) ≅ Spin(3)·Spin(r) is compact, one can easily construct a K-invariant Q_{[e]}-Hermitian Euclidean scalar product h_{[e]} on T_{[e]}M by the standard averaging procedure and extend it to a G-invariant Q-Hermitian Riemannian metric h on M. More explicitly, we can construct such a scalar product h_{[e]} on T_{[e]}M ≅ T_{[e]}M(V) ⊕ W as orthogonal sum of K-invariant Euclidean scalar products on T_{[e]}M(V) and W as follows. Using the open embedding M(V) ↪ M̃(V) we can identify T_{eK}M(V) ≅ T_{eK̃} M̃(V) ≅ ℝ^{4} ⊗ ℝ^{r,1} ≅ ℝ^{4} ⊗ ℝ^{r+1} and choose the standard O(4) × O(r + 1)-invariant Euclidean scalar product on ℝ^{4} ⊗ ℝ^{r+1}. This scalar product is automatically K-invariant and Q_{[e]}-Hermitian. On W we choose any K-invariant Euclidean scalar product (which exists by compactness of K). It is automatically Q_{[e]}-Hermitian because Q_{[e]} ⊂ End(W) is precisely the image of 𝔬(3) ⊂ 𝔨 = Lie K under the isotropy representation of 𝔨 on the 𝔨-invariant subspace W ⊂ T_{[e]}M ≅ T_{[e]}M(V)⊕W. It remains to show that Isom(M, h) does not contain any transitive solvable Lie subgroup if r > 0. In fact, M is homotopy equivalent to the simply connected real Grassmannian SO(3 + r)/SO(3) × SO(r) of oriented 3-planes in ℝ^{3+r} (r > 0). On the other hand, if (M, h) admits a transitive solvable group of isometries then M must be homotopy equivalent to a (possibly trivial) torus, which contradicts the fact that M is simply connected (and not contractible). □ The last argument proves, in fact, the following theorem.

Theorem 12. Let 𝔭(Π) be any extended Poincaré algebra of signature (p, q), p > 3 and M = M(Π) the manifold constructed in Thm. 8. Then M does not admit any transitive (topological) action by a solvable Lie group.

3. Bundles associated to the quaternionic manifold (M, Q)

To any almost quaternionic manifold (M, Q) one can canonically associate the following bundles over M: the twistor bundle Z(M), the canonical SO(3)-principal bundle S(M) and the Swann bundle U(M). The twistor bundle (or twistor space) Z(M) → M is the subbundle of Q whose fibre Z(M)_{m} at m ∈ M consists of all complex structures subordinate to the quaternionic structure Q_{m}, i.e.

Z(M)_{m} = {A ∈ Q_{m}|A^{2} = −Id} .

So Z(M) is a bundle of 2-spheres. The fibre S(M)_{m} of the SO(3)-principal bundle S(M) at m ∈ M consists of all hypercomplex structures (J_{1}, J_{2}, J_{3}) subordinate to Q_{m}. Finally,

U(M) = S(M) ×_{SO(3)} (ℍ^{∗}/{±1})

is associated to the action of SO(3) ≅ Sp(1)/{±1} on ℍ^{∗}/{±1} induced by leftmultiplication of unit quaternions on ℍ. The total space Z(M) carries a canonical almost complex structure 𝕁, which is integrable if Q is quaternionic, see [A-M-P]. Similarly, one can define an almost hypercomplex structure (𝕁_{1}, 𝕁_{2}, 𝕁_{3}) on U(M), which is integrable if Q is quaternionic, cf. [P-P-S]. We recall the definition of the complex structure 𝕁 on the twistor space Z = Z(M) of a quaternionic manifold (M, Q). Since Q is 1-integrable, there exists a quaternionic connection ∇ on M, see Def. 6 and Def. 7. The holonomy of ∇ preserves not only Q ⊂ End(T M) but also its sphere subbundle Z ⊂ Q, simply because Id is a parallel section of End(T M). Let

T Z = T Z^{ver} ⊕ T Z^{hor} (17)

be the corresponding decomposition into the vertical space T Z^{ver} tangent to the fibres of the twistor bundle Z → M and its ∇-horizontal complement T Z^{hor}. The complex structure 𝕁 preserves the decomposition (17). Let m ∈ M be a point in M and z := J ∈ Z_{m} ⊂ Q_{m} a complex structure on T_{m}M subordinate to Q_{m}. Then 𝕁_{z} ∈ End(T_{z}Z) is defined by:

𝕁A := JA and 𝕁X̃ = J̃X

for all A ∈ T_{z}Z^{ver} = T_{z}Z_{m} = {A ∈ Q_{m}|AJ = −JA} and all X ∈ T_{m}M, where X̃ ∈ T_{z}Z^{hor} denotes the ∇-horizontal lift of X. It was proven in [A-M-P] that 𝕁 does not depend on the choice of quaternionic connection ∇.

If (M, Q) admits a quaternionic pseudo-Kähler metric g (of nonzero scalar curvature) then it is known that (Z(M), 𝕁) admits a complex contact structure and a pseudo-Kähler-Einstein metric [S1], that S(M) admits a pseudo-3-Sasakian structure [Ko] and that (U(M), 𝕁_{1}, 𝕁_{2}, 𝕁_{3}) admits a pseudo-hyper-Kähler metric [Sw1]. Moreover, all these special geometric structures are canonically associated to the data (M, Q, g). We recall that a complex contact structure on a complex manifold Z is a holomorphic distribution D of codimension one whose Frobenius form [·, ·] : ∧^{2}D → T Z/D is (pointwise) nondegenerate; for the definition of 3-Sasakian structure see [I-K] and [T]. If a Lie group G acts (smoothly) on an almost quaternionic manifold (M, Q) preserving Q then there is an induced 𝕁-holomorphic action on Z. Similarly, if a Lie group G acts on a quaternionic pseudo-Kähler manifold (M, Q, g) preserving the data (Q, g) then it acts on any of the bundles Z(M), S(M) and U(M) preserving all the special geometric structures mentioned above.

Theorem 13. Let (M(Π) = G(Π)/K, Q) be the homogeneous quaternionic manifold associated to an extended Poincaré algebra of signature (p, q), p ≥ 3, Z(Π) := Z(M(Π)) its twistor space, S(Π) := S(M(Π)) its canonical SO(3)-principal bundle and U(Π) := U(M(Π)) its Swann bundle. Then G(Π) acts transitively on the manifolds Z(Π) and S(Π) and acts on U(Π) with an orbit of codimension one.

Proof: Since G = G(Π) acts transitively on the base M = M(Π) of any of the bundles Z(Π) → M, S(Π) → M and U(Π) → M, it is sufficient to consider the action of the stabilizer K = Spin(3) · Spin_{0}(r, q) on the fibres Z(Π)_{[e]}, S(Π)_{[e]} and U(Π)_{[e]}, [e] = eK ∈ M = G/K. The subgroup Spin_{0}(r, q) ⊂ K acts trivially on S(Π)_{[e]} and hence also on Z(Π)_{[e]} and U(Π)_{[e]}, whereas Spin(3) acts transitively on the set S(Π)_{[e]} of hypercomplex structures subordinate to Q_{[e]} and hence also on the set Z(Π)_{[e]} of complex structures subordinate to Q_{[e]}. From this it follows that G acts transitively on Z(Π) and S(Π) and with an orbit of codimension one on U(Π). □

From now on we denote by m_{0} := [e] = eK ∈ M = G/K the canonical base point of M and fix the complex structure J_{1} ∈ Q_{m_{0}} as base point z_{0} := J_{1} ∈ Z_{m_{0}} in Z = Z(Π).

Corollary 7. (Z = Gz_{0} ≅ G/G_{z_{0}}, 𝕁) is a homogeneous complex manifold of the group G. The stabilizer of the point z_{0} ∈ Z in G is the centralizer of J_{1} in K: G_{z_{0}} = Z_{K}(J_{1}) = Z_{Spin(3)}(J_{1}) · Spin_{0}(r, q), Z_{Spin(3)}(J_{1}) ≅ U(1).

Now we are going to construct a natural holomorphic immersion Z → Z̄ = Z̄(Π) = Gℂ/H of Z into a homogeneous complex manifold of the complexified linear group Gℂ ⊂ Aut(𝔯ℂ), where (G_{z_{0}})ℂ ⊂ H ⊂ Gℂ are closed complex Lie subgroups.

First of all, we give an explicit description of the complex structure 𝕁 on the twistor space Z. The choice of base point z_{0} ∈ Z determines a G-equivariant diffeomorphism Z = Gz_{0} →^{∼} G/G_{z_{0}}, which maps z_{0} to the canonical base point eG_{z_{0}} ∈ G/G_{z_{0}}. From now on we will identify Z and G/G_{z_{0}} via this map. The complex structure 𝕁 being G-invariant, it is completely determined by the G_{z_{0}}- invariant complex structure 𝕁_{z_{0}} on T_{z_{0}}Z. In order to describe 𝕁_{z_{0}} we introduce the following G_{z_{0}}-invariant complement 𝔷 = 𝔷(Π) to 𝔤_{z_{0}} = Lie G_{z_{0}} = ℝe_{2} ∧ e_{3} ⊕ 𝔬(r, q) in 𝔤:

𝔷 = ℝe_{1} ∧ e_{2} + ℝe_{1} ∧ e_{3} + 𝔪 .

The G_{z_{0}}-invariant decomposition 𝔤 = 𝔤_{z_{0}} + 𝔷 determines a G_{z_{0}}-equivariant isomorphism T_{z_{0}}Z ≅ 𝔤/𝔤_{z_{0}} →^{∼} 𝔷. Using it we can consider the G_{z_{0}}-invariant complex structure 𝕁_{z_{0}} as a G_{z_{0}}-invariant complex structure on 𝔷.

Proposition 11. The G-invariant complex structure 𝕁 on the twistor space Z = G/G_{z_{0}} is given on T_{z_{0}}Z ≅ 𝔷 = ℝe_{1} ∧ e_{2} + ℝe_{1} ∧ e_{3} + 𝔪 by:

𝕁_{z_{0}}e_{1} ∧ e_{2} = e_{1} ∧ e_{3} , 𝕁_{z_{0}}|𝔪 = J_{1} .

Proof: Let ∇ be the G-invariant quaternionic connection on (M, Q) constructed in Lemma 6 and L_{x} = ∑_{α=1}^{3} ω_{α}(x)J_{α} + L̄_{x}, x ∈ 𝔪, its Nomizu operators, where L̄_{x} ∈ z(Q) ≅ 𝔤𝔩(d, ℍ) (d = dimℍ M). The connection ∇ induces the decomposition T_{z_{0}}Z = T_{z_{0}}Z^{ver} ⊕ T_{z_{0}}Z^{hor} ≅ 𝔷 = 𝔷^{ver} ⊕ 𝔷^{hor} into vertical space and horizontal space. The vertical space is T_{z_{0}}Z^{ver} = T_{z_{0}}Z_{z_{0}} = ℝJ_{2} ⊕ ℝJ_{3} ⊂ Q_{z_{0}} and 𝔷^{ver} = ℝe_{1}∧e_{2}⊕ℝe_{1}∧e_{3} respectively, the identification being J_{2} ↦ −e_{1}∧e_{2}, J_{3} ↦ −e_{1}∧e_{3}.

For any vector x ∈ 𝔪 we consider the curve c(t) = exp txK ∈ G/K = M (t ∈ ℝ) and define a lift s(t) ∈ Z_{c(t)} by the differential equation L_{X}s = 0 with initial condition s(0) = z_{0} = J_{1}. Here X = α(x) is the fundamental vector field on M associated to x (as defined on p. 22) and L_{X} is the Lie derivative with respect to X. Then s(t) = (exp tx)z_{0} is precisely the orbit of z_{0} ∈ Z = G/G_{z_{0}} under the 1-parameter subgroup of G generated by x. The vector

\frac{d}{dt}|_{t=0}(s − t∇_{X}s) = s′(0) − ∇_{X}s|_{t=0}
= s′(0) + [L_{x}, J_{1}] = s′(0) − 2ω_{2}(x)J_{3} + 2ω_{3}(x)J_{2} ∈ T_{z_{0}}Z

is horizontal. It is precisely the horizontal lift of X(m_{0}) ∈ T_{m_{0}}M and corresponds to

x̃ := x + 2ω_{2}(x)e_{1} ∧ e_{3} − 2ω_{3}(x)e_{1} ∧ e_{2} ∈ 𝔷 (18)

under the identification T_{z_{0}}Z ≅ 𝔷. This shows that

𝔷^{hor} = {x̃|x ∈ 𝔪}
= ℝ(e_{2} + e_{1} ∧ e_{3}) + ℝ(e_{3} − e_{1} ∧ e_{2}) +
ℝe_{0} + ℝe_{1} + span{e_{ιa}|ι = 0, …, 3, a = 1, …, r + q} + W ,

see Lemma 6. Now the formulas for 𝕁_{z_{0}} follow easily. In fact, it is clear that 𝕁_{z_{0}} coincides with J_{1} on the J_{1}-invariant subspace 𝔪 ∩ 𝔷^{hor} = ℝe_{0} + ℝe_{1} + span{e_{ιa}|ι = 0, …, 3, a = 1, …, r + q} + W = e_{2}^{⊥} ∩ e_{3}^{⊥} ⊂ 𝔪. The equation 𝕁_{z_{0}}e_{1} ∧ e_{2} = e_{1} ∧ e_{3} follows immediately from J_{1}J_{2} = J_{3}, since e_{1} ∧ e_{2}, e_{1} ∧ e_{3} ∈ 𝔷^{ver} are identified with the vertical vectors −J_{2}, −J_{3} ∈ T_{z_{0}}Z^{ver}. It is now sufficient to check that 𝕁_{z_{0}}e_{2} = e_{3}. This is done in the next computation:

𝕁_{z_{0}}e_{2} = 𝕁_{z_{0}}(e_{2} + e_{1} ∧ e_{3}) − 𝕁_{z_{0}}(e_{1} ∧ e_{3}) = 𝕁_{z_{0}} ẽ_{2} + e_{1} ∧ e_{2}
= J̃_{1}e_{2} + e_{1} ∧ e_{2} = ẽ_{3} + e_{1} ∧ e_{2} = e_{3} .□

We denote by 𝔷^{1,0} (respectively, 𝔷^{0,1}) the eigenspace of 𝕁_{z_{0}} ∈ End(𝔷) for the eigenvalue i (respectively, −i). The integrability of the complex structure 𝕁 implies that 𝔥 := 𝔥(Π) := (𝔤_{z_{0}})ℂ + 𝔷^{0,1} ⊂ 𝔤ℂ is a (complex) Lie subalgebra. Let H = H(Π) ⊂ Gℂ ⊂ Aut(𝔯ℂ) be the corresponding connected linear Lie group. Let us also consider the Lie algebra 𝔥(V) := 𝔥∩𝔤(V)ℂ = (𝔤_{z_{0}})ℂ +𝔷(V)^{0,1} ⊂ 𝔤(V)ℂ, where 𝔷(V) = 𝔷 ∩ 𝔤(V) and 𝔷(V)^{0,1} := 𝔷^{0,1} ∩ 𝔤(V)ℂ.

Proposition 12. H = H(Π) ⊂ Gℂ ⊂ Aut(𝔯ℂ) are complex algebraic subgroups.

Proof: It follows from Prop. 4 that 𝔤ℂ ⊂ der(𝔯ℂ) is a complex algebraic subalgebra and hence Gℂ ⊂ Aut(𝔯ℂ) a complex algebraic subgroup. It only remains to show that 𝔥 is a complex algebraic subalgebra. Let us consider the decomposition 𝔥 = (𝔤_{z_{0}})ℂ + 𝔷^{0,1}. The subalgebra (𝔤_{z_{0}})ℂ is algebraic. In fact, 𝔤_{z_{0}} = z_{𝔨}(e_{2} ∧ e_{3}) is a centralizer in the real algebraic subalgebra 𝔨 ⊂ 𝔤. If the subalgebra ⟨𝔷^{0,1}⟩ ⊂ 𝔥 generated by the subspace 𝔷^{0,1} is an algebraic subalgebra of der(𝔯ℂ), then 𝔥 is generated by algebraic linear Lie algebras and hence is itself algebraic, see [O-V]. The algebraicity of ⟨𝔷^{0,1}⟩ is proven in the next lemma. □

Lemma 7. 𝔷^{0,1} generates the algebraic subalgebra ⟨𝔷^{0,1}⟩ = 𝔬(E′)ℂ + 𝔷^{0,1} ⊂ 𝔤ℂ.

Proof: First we compute the subalgebra ⟨𝔷^{0,1}⟩ of 𝔤ℂ generated by 𝔷^{0,1} = 𝔷^{0,1}(V) + W^{0,1}. Note that

𝔷^{0,1}(V) = span{e_{1} ∧ e_{2} + ie_{1} ∧ e_{3}, e_{0} + ie_{1}, e_{2} + ie_{3}}
+ span{e_{0a} + ie_{1a}, e_{2a} + ie_{3a}|a = 1, …, r + q} .

It is easy to check that

[𝔷^{0,1}(V), 𝔷^{0,1}(V)] = 𝔬(E′)ℂ + span{e_{1} ∧ e_{2} + ie_{1} ∧ e_{3}, e_{2} + ie_{3}}
+ span{e_{2a} + ie_{3a}|a = 1, …, r + q}

and

[𝔷^{0,1}(V), W^{0,1}] = W^{0,1} .

We show that [W^{0,1}, W^{0,1}] ⊂ ℂ(e_{2} + ie_{3}). This shows that ⟨𝔷^{0,1}⟩ ⊃ 𝔬(E′)ℂ + 𝔷^{0,1}. Since the right-hand side is closed under Lie brackets, we conclude that ⟨𝔷^{0,1}⟩ = 𝔬(E′)ℂ + 𝔷^{0,1}. It is easy to check that the subalgebra

𝔬(E′)ℂ + span{e_{1} ∧ e_{2} + ie_{1} ∧ e_{3}, e_{2} + ie_{3}}
+ span{e_{0a} + ie_{1a}, e_{2a} + ie_{3a}|a = 1, …, r + q} + W^{0,1}

coincides with its derived Lie algebra. Therefore, it is an algebraic subalgebra of 𝔤ℂ, see [O-V]. Now, to prove that ⟨𝔷^{0,1}⟩ is algebraic, it is sufficient to check that ℂ(e_{0} + ie_{1}) is algebraic. The element e_{0} + ie_{1} ∈ 𝔤ℂ is conjugated to the algebraic element e_{0} ∈ 𝔤ℂ via exp(−ie_{1}) ∈ Gℂ. Consequently, it is algebraic. □

Theorem 14. For the twistor space Z of any of the homogeneous quaternionic manifolds (M = G/K, Q) constructed in Thm. 8 there is a natural open G-equivariant holomorphic immersion

Z ≅ G/G_{z_{0}} → Z̄ = Z̄(Π) := Gℂ/H

into a homogeneous complex (Hausdorff) manifold of the complex algebraic group Gℂ. This immersion is a (universal) finite covering over its image, which is an open G-orbit. (The group H was defined on p. 36.)

Proof: It follows from Prop. 12 that H ⊂ Gℂ is closed. This shows that Gℂ/H is a homogeneous complex Hausdorff manifold. The inclusions G ⊂ Gℂ and G_{z_{0}} ⊂ H define a G-equivariant map G/G_{z_{0}} → Gℂ/H, which is an immersion since 𝔤 ∩ 𝔥 = 𝔤_{z_{0}}. The differential of this immersion at eG_{z_{0}} is canonically identified with the restriction ϕ : 𝔷 → 𝔤ℂ/𝔥 of the canonical projection 𝔤ℂ → 𝔤ℂ/𝔥 to 𝔷 ⊂ 𝔤 = 𝔤_{z_{0}} + 𝔷. Obviously, the complex linear extension ϕℂ maps 𝔷^{1,0} isomorphically to 𝔤ℂ/𝔥 and 𝔷^{0,1} to zero. This shows that Z ≅ G/G_{z_{0}} → Gℂ/H is open and holomorphic with respect to the G-invariant complex structure 𝕁 on Z and the canonical complex structure on Gℂ/H. □

Theorem 15. The homogeneous complex manifold Z̄ = Gℂ/H carries a Gℂinvariant holomorphic hyperplane distribution D ⊂ T Z̄. The hyperplane D_{z_{0}} = T_{z_{0}}Z^{hor} ⊂ T_{z_{0}}Z = T_{z_{0}} Z̄ is the horizontal space associated to the G-invariant quaternionic connection ∇ on M constructed in Lemma 6. Moreover, D defines a complex contact structure on Z̄ = Z̄(Π) if and only if Π is nondegenerate. In this case the restriction D|Z coincides with the canonical complex contact structure on the twistor space Z of the quaternionic pseudo-Kähler manifold M.

Proof: Recall that we identify T_{z_{0}}Z with the 𝔤_{z_{0}}-invariant subspace 𝔷 ⊂ 𝔤 = 𝔤_{z_{0}} +𝔷 complementary to 𝔤_{z_{0}}. Hereby the subspace D_{z_{0}} = T_{z_{0}}Z^{hor} is identified with 𝔷^{hor} = {x̃|x ∈ 𝔪} ⊂ 𝔷, where x̃ = x + 2ω_{2}(x)e_{1} ∧ e_{3} − 2ω_{3}(x)e_{1} ∧ e_{2} is the ∇- horizontal lift of x, see equation (18). The subspace 𝔷^{hor} ⊂ 𝔷 is 𝕁_{z_{0}}-invariant by the very definition of the complex structure 𝕁_{z_{0}} and 𝕁_{z_{0}}|𝔷^{hor} is given by 𝕁_{z_{0}} x̃ = J̃_{1}x for all x ∈ 𝔪. The subspace (𝔷^{hor})^{1,0} = (𝔷^{hor})ℂ ∩ 𝔷^{1,0} ⊂ 𝔷^{1,0} is identified with the i-eigenspace D_{z_{0}}^{1,0} ⊂ T_{z_{0}}^{1,0}Z = T_{z_{0}}^{1,0}Z̄ of 𝕁_{z_{0}} on D_{z_{0}}. In order to prove that D_{z_{0}}^{1,0} extends to a Gℂ-invariant holomorphic distribution D^{1,0} ⊂ T^{1,0}Z̄ it is sufficient to check the following lemma.

Lemma 8. The complex Lie algebra 𝔥 = 𝔤ℂ_{z_{0}} + 𝔷^{0,1} preserves the projection of 𝔷^{hor})^{1,0} = {x̃ − iJ̃_{1}x|x ∈ 𝔪} = spanℂ{e_{2} + e_{1} ∧ e_{3} − i(e_{3} − e_{1} ∧ e_{2}), e_{0} − ie_{1}, e_{0a} − ie_{1a}, e_{2a} − ie_{3a}|a = 1, …, r + q} + W^{1,0} into 𝔤ℂ/𝔥, i.e.

[𝔥, (𝔷^{hor})^{1,0}] ⊂ 𝔥 + (𝔷^{hor})^{1,0} .

Now D^{1,0} defines a complex contact structure if and only if the Frobenius form

∧^{2}D^{1,0} →^{[·,·]} T^{1,0}Z̄/D^{1,0} is nondegenerate, which is equivalent to the nondegeneracy of the skew symmetric complex bilinear form

ω : ∧^{2}(𝔷^{hor})^{1,0} →^{[·,·]} 𝔤ℂ → 𝔤ℂ/(𝔥 + (𝔷^{hor})^{1,0}) ≅ ℂ .

Lemma 9. Let (𝔷^{hor})^{1,0} = (𝔷(V)^{hor})^{1,0} + W^{1,0} be the decomposition of (𝔷^{hor})^{1,0} induced by the decomposition 𝔷 = 𝔷(V) + W. Then ω((𝔷(V)^{hor})^{1,0}, W^{1,0}) = 0, ω| ∧^{2} (𝔷(V)^{hor})^{1,0} is nondegenerate and ω| ∧^{2} W^{1,0} is given by

ω(s − iJ_{1}s, t − iJ_{1}t) = 2(⟨e_{2}, [s, t]⟩ + i⟨e_{3}, [s, t]⟩)(e_{2} − ie_{3}) (mod 𝔥 + (𝔷^{hor})^{1,0}) . (19)

From the lemma it follows that ω is nondegenerate if and only if ω| ∧^{2} W^{1,0} is nondegenerate. The explicit formula for ω|∧^{2}W^{1,0} given in equation (19) now shows that ω is nondegenerate if and only if b(Π) = b_{Π,(e_{1},e_{2},e_{3})} is nondegenerate, which is in turn equivalent to the nondegeneracy of Π by Cor. 1. This proves that D is a complex contact structure if and only if Π is nondegenerate. If Π is nondegenerate then D_{z_{0}} = T_{z_{0}}Z^{hor} is precisely the horizontal space associated to the Levi-Civita connection ∇^{g} of the quaternionic pseudo-Kähler manifold (M, Q, g) and D is the canonical complex contact structure on its twistor space Z, as defined by Salamon [S1]. □

4. Homogeneous quaternionic supermanifolds associated to superextended Poincaré algebras

In this section we will show that our main result, Thm. 8, has a natural supergeometric analogue, Thm. 17. The fundamental idea is to replace the map Π : ∧^{2}W → V defined on the exterior square ∧^{2}W by an 𝔬(V)-equivariant linear map Π : ∨^{2}W → V defined on the symmetric square ∨^{2}W = Sym^{2}W. We will freely use the language of supergeometry. The necessary background is outlined in the appendix.

4.1. Superextended Poincaré algebras. Let (V, ⟨·, ·⟩) be a pseudo-Euclid-ean vector space, W a Cℓ^{0}(V)-module and Π : ∨^{2}W → V an 𝔬(V)-equivariant linear map. We recall that 𝔬(V) acts on W via ad^{−1} : 𝔬(V) → 𝔰𝔭𝔦𝔫(V) ⊂ Cℓ^{0}(V), see equation (2).

Given these data we extend the Lie bracket on 𝔭_{0} := 𝔭(V) to a super Lie bracket (see Def. 16) [·, ·] on the ℤ_{2}-graded vector space 𝔭_{0} + 𝔭_{1}, 𝔭_{1} = W, by the following requirements:

1) The adjoint representation (see Def. 28) of 𝔬(V) on 𝔭_{1} concides with the natural representation of 𝔬(V) ≅ 𝔰𝔭𝔦𝔫(V) on W = 𝔭_{1} and [V, 𝔭_{1}] = 0.

2) [s, t] = Π(s ∨ t) for all s, t ∈ W. The super Jacobi identity follows from 1) and 2). The resulting super Lie algebra will be denoted by 𝔭(Π).

Definition 13. Any super Lie algebra 𝔭(Π) as above is called a superextended Poincaré algebra (of signature (p, q) if V ≅ ℝ^{p,q}). 𝔭(Π) is called nondegenerate if Π is nondegenerate, i.e. if the map W ∋ s ↦ Π(s ∨ ·) ∈ W^{∗} ⊗ V is injective.

The structure of superextended Poincaré algebra on the vector space 𝔭(V)+W is completely determined by the map Π : ∨^{2}W → V. An explicit basis for the vector space (∨^{2}W^{∗} ⊗ V)^{𝔬(V)} of such 𝔬(V)-equivariant linear maps was constructed in [A-C2] for all V and W.

4.2. The canonical supersymmetric bilinear form b. Let V = ℝ^{p,q} be the standard pseudo-Euclidean vector space with scalar product ⟨·, ·⟩ of signature (p, q). From now on we fix a decomposition p = p′ + p′′ and assume that p′ ≡ 3 (mod 4), see Remark 7 below. We denote by (e_{i}) = (e_{1}, …, e_{p′}) the first p′ basis vectors of the standard basis of V and by (e′_{i}) = (e′_{1}, …, e′_{p′′+q}) the remaining ones. The two complementary orthogonal subspaces of V spanned by these bases are denoted by E = ℝ^{p′} = ℝ^{p′,0} and E′ = ℝ^{p′′,q} respectively. The vector spaces V, E and E′ are oriented by their standard orthonormal bases. E.g. the orientation of Euclidean p′-space E defined by the basis (e_{i}) is e_{1}^{∗} ∧ ⋯ ∧ e_{p′}^{∗} ∈ ∧^{p′}E^{∗}. Here (e_{i}^{∗}) denotes the basis of E^{∗} dual to (e_{i}). Now let 𝔭(Π) = 𝔭(V) + W be a superextended Poincaré algebra of signature (p, q) and (ẽ_{i}) any orhonormal basis of E. Then we define a ℝ-bilinear form b_{Π,(ẽ_{i})} on the Cℓ^{0}(V)-module W by:

b_{Π,(ẽ_{i})}(s, t) = ⟨ẽ_{1}, [ẽ_{2} … ẽ_{p′}s, t]⟩ = ⟨ẽ_{1}, Π(ẽ_{2} … ẽ_{p′}s ∨ t)⟩ , s, t ∈ W . (20)

We put b = b(Π) := b_{Π,(e_{i})} for the standard basis (e_{i}) of E. As in 4.1 we consider W = 𝔭_{1} as ℤ_{2}-graded vector space of purely odd degree and recall that, by Def. 22, an even supersymmetric (respectively, super skew symmetric) bilinear form on W is simply an ordinary skew symmetric (respectively, symmetric) bilinear form on W. Remark 7: Equation (20) defines an even super skew symmetric bilinear form on W if p′ ≡ 1 (mod 4). For even p′ the above formula does not make sense, unless one assumes that W is a Cℓ(V)-module rather than a Cℓ^{0}(V)-module. Here we are only interested in the case p′ ≡ 3 (mod 4). Moreover, later on, for the construction of homogeneous quaternionic supermanifolds we will put p′ = 3.

Theorem 16. The bilinear form b has the following properties:

1) b_{Π,(ẽ_{i})} = ±b if ẽ_{1} ∧ ⋯ ∧ ẽ_{p′} = ±e_{1} ∧ ⋯ ∧ e_{p′}. In particular, our definition of b does not depend on the choice of positively oriented orthonormal basis of E.

2) b is an even supersymmetric bilinear form.

3) b is invariant under the connected subgroup K(p′, p′′) = Spin(p′)·Spin_{0}(p′′, q) ⊂ Spin_{0}(p, q) (and is not Spin_{0}(p, q)-invariant, unless p′′ + q = 0).

4) Under the identification 𝔬(V) = ∧^{2}V = ∧^{2}E + ∧^{2}E′ + E ∧ E′, see equation (3), the subspace E ∧ E′ acts on W by b-symmetric endomorphisms and the subalgebra ∧^{2}E ⊕ ∧^{2}E′ ≅ 𝔬(p′) ⊕ 𝔬(p′′, q) acts on W by b-skew symmetric endomorphisms.

Proof: The proof is the same as for Thm. 1, up to the modifications caused by the fact that Π is now symmetric instead of skew symmetric. Part 2) e.g. follows from the next computation, in which we use that p′ ≡ 3 (mod 4):

b(t, s) = ⟨e_{1}, [e_{2} … e_{p′}t, s]⟩
= −⟨e_{1}, [e_{4} … e_{p′}t, e_{2}e_{3}s]⟩ + ⟨e_{1}, ad(e_{2}e_{3})[e_{4} … e_{p′}t, s]⟩
= −⟨e_{1}, [e_{4} … e_{p′}t, e_{2}e_{3}s]⟩ = ⋯ = −⟨e_{1}, [t, e_{2} … e_{p′}s]⟩
= −⟨e_{1}, [e_{2} … e_{p′}s, t]⟩ = −b(s, t) .□

Definition 14. The bilinear form b = b(Π) = b_{Π,(e_{1},... ,e_{p′})} defined above is called the canonical supersymmetric bilinear form on W associated to the 𝔬(V)-equivariant map Π : ∨^{2}W → V = ℝ^{p,q} and the decomposition p = p′ + p′′.

Proposition 13. The kernels of the linear maps Π : W → W^{∗} ⊗ V and b = b(Π) : W → W^{∗} coincide: ker Π = ker b.

Proof: See the proof of Prop. 1. □

Corollary 8. 𝔭(Π) is nondegenerate (see Def. 13) if and only if b(Π) is nondegenerate.

4.3. The main theorem in the super case. Any superextended Poincaré algebra 𝔭 = 𝔭(Π) = 𝔭(V) + W has an even derivation D with eigenspace decomposition 𝔭 = 𝔬(V) + V + W and corresponding eigenvalues (0, 1, 1/2). Therefore, the super Lie algebra 𝔭 = 𝔭_{0} + 𝔭_{1} = 𝔭(V) + W is canonically extended to a super Lie algebra 𝔤 = 𝔤(Π) = ℝD + 𝔭 = 𝔤_{0} + 𝔤_{1}, where 𝔤_{0} = ℝD + 𝔭_{0} = ℝD + 𝔭(V) = 𝔤(V) and 𝔤_{1} = 𝔭_{1} = W.

Proposition 14. The adjoint representation (see Def. 28) of 𝔤 is faithful and moreover it induces a faithful representation on its ideal 𝔯 = ℝD + V + W ⊂ 𝔤 = 𝔬(V) + 𝔯.

By Prop. 14 we can consider 𝔤 as subalgebra of the super Lie algebra 𝔤𝔩(𝔯) (defined in 4.6), i.e. 𝔤 is a linear super Lie algebra (see Def. 28). We denote by G = G(Π) the corresponding linear Lie supergroup, see 4.6. Its underlying Lie group is G_{0} = G(V) and has 𝔤_{0} = 𝔤(V) as Lie algebra. For the construction of homogeneous quaternionic supermanifolds we will assume that V = ℝ^{p,q} with p ≥ 3. Then we fix the decomposition p = p′ + p′′, where now p′ = 3 and p′′ = p − 3 =: r. As before, we have a corresponding orthogonal decomposition V = E + E′, the subalgebra 𝔨 = 𝔨(p′, p′′) = 𝔨(3, r) = 𝔬(3) ⊕ 𝔬(r, q) ⊂ 𝔬(p, q) = 𝔬(V) preserving this decomposition and the corresponding linear Lie subgroup K = K(3, r) ⊂ G_{0}. We are interested in the homogeneous supermanifold (see 4.7):

M = M(Π) := G/K = G(Π)/K .

Its underlying manifold is the homogeneous manifold

M_{0} := G_{0}/K = G(V)/K = M(V) .

Theorem 17. 1) There exists a G-invariant quaternionic structure Q on

M = G/K.

2) If Π is nondegenerate (see Def. 13) then there exists a G-invariant pseudo-Riemannian metric g on M such that (M, Q, g) is a quaternionic pseudo-Kähler supermanifold.

Proof: First let (Q_{0}, g_{0}) denote the G_{0}-invariant quaternionic pseudo-Kähler structure on M_{0} = G_{0}/K which was introduced in the proof of Thm. 8 (previously it was denoted simply by (Q, g)). As in that proof, see equation (13), the corresponding K-invariant quaternionic structure on T_{eK}M_{0} ≅ 𝔤_{0}/𝔨 is extended to a K-invariant quaternionic structure Q_{eK} on T_{eK}M ≅ 𝔤/𝔨 defining a G-invariant almost quaternionic structure Q on G/K (see 4.5 and 4.7). Similarly, the G_{0}-invariant pseudo-Riemannian metric g_{0} on M_{0} corresponds to a K-invariant pseudo-Euclidean scalar product on 𝔤_{0}/𝔨. This scalar product is extended by b = b_{Π,(e_{1},e_{2},e_{3})}, in the obvious way, to a K-invariant supersymmetric bilinear form g_{eK} on 𝔤/𝔨, which is nondegenerate if Π is nondegenerate. Let us first treat the case where Π is nondegenerate. In this case g_{eK} defines a G-invariant Q-Hermitian pseudo-Riemannian metric g on G/K. The Levi-Civita connection of the homogeneous pseudo-Riemannian supermanifold (M, g) is computed in Lemma 10 below. As on p. 26, 𝔤/𝔨 is identified with the complement 𝔪 = 𝔪(V) + W to 𝔨 in 𝔤. We use the g_{0}-orthonormal basis (e_{ιi}) of 𝔪(V) introduced on p. 27 and recall that ϵ_{i} = g_{0}(e_{ιi}, e_{ιi}). Also we will continue to write x ∧_{g} y for the g_{eK}-skew symmetric endomorphism of 𝔪 defined for x, y ∈ 𝔪(V) by: x ∧_{g} y(z) := g_{eK}(y, z)x − g_{eK}(x, z)y, z ∈ 𝔪.

Lemma 10. The Nomizu map L^{g} = L(∇^{g}) associated to the Levi-Civita connection ∇^{g} of the homogeneous pseudo-Riemannian supermanifold (M = G/K, g) is given by the following formulas:

L_{e_{0}}^{g} = L_{e_{αa}}^{g} = 0 ,
L_{e_{α}}^{g} = \frac{1}{2}J_{α} + L̄_{e_{α}}^{g} ,

where

L̄_{e_{α}}^{g} = \frac{1}{2} ∑_{i=0}^{r+q} ϵ_{i}e_{αi} ∧_{g} e′_{i} − \frac{1}{2} ∑_{i=0}^{r+q} ϵ_{i}e_{γi} ∧_{g} e_{βi} ∈ z(Q_{eK}) ,

L_{e′_{a}}^{g} ∈ z(Q_{eK}) is given by:

L_{e′_{a}}^{g}|𝔪(V) = ∑_{ι=0}^{3} e_{ιa} ∧_{g} e_{ι} ,
L_{e′_{a}}^{g}|W = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a} .

For all s ∈ W the Nomizu operator L_{s}^{g} ∈ z(Q_{eK}) maps the subspace 𝔪(V) ⊂ 𝔪 = 𝔪(V) + W into W and W into 𝔪(V). The restriction L_{s}^{g}|W (s ∈ W) is completely determined by L_{s}^{g}|𝔪(V) (and vice versa) according to the relation

g_{eK}(L_{s}^{g}t, x) = g_{eK}(t, L_{s}^{g}x) , s, t ∈ W , x ∈ 𝔪(V) .

Finally, L_{s}^{g}|𝔪(V) (s ∈ W) is completely determined by its values on the quaternionic basis (e′_{i}), i = 0, …, r + q, which are as follows:

L_{s}^{g}e′_{0} = \frac{1}{2}s ,
L_{s}^{g}e′_{a} = \frac{1}{2}e_{1}e_{2}e_{3}e′_{a}s .

(It is understood that L_{x}^{g} = ad_{x}|𝔪 for all x ∈ 𝔨, cf. equation (24).) In the above formulas a = 1, …, r+q, α = 1, 2, 3 and (α, β, γ) is a cyclic permutation of (1, 2, 3).

Proof: This follows from equation (26) by a straightforward computation. □

Corollary 9. The Levi-Civita connection ∇^{g} of the homogeneous almost quaternionic pseudo-Hermitian supermanifold (M, Q, g) preserves Q and hence (M, Q, g) is a quaternionic pseudo-Kähler supermanifold.

By Cor. 9 we have already established part 2) of Thm. 17. Part 1) is a consequence of 2) provided that Π is nondegenerate. It remains to discuss the case of degenerate Π. From the 𝔬(V)-equivariance of Π it follows that W_{0} = ker Π ⊂ W is an 𝔬(V)-submodule. Let W′ be a complementary 𝔬(V)-submodule. Then W_{0} and W′ are Cℓ^{0}(V)-submodules; we put Π′ := Π| ∨^{2} W′. We denote by (M′ := M(Π′), Q′, g′) the corresponding quaternionic pseudo-Kähler supermanifold and by L′ := L^{g′} : 𝔤(Π′) → End(𝔪(Π′)) the Nomizu map associated to its Levi-Civita connection. By the next lemma, we can extend the map L′ to a torsionfree Nomizu map L : 𝔤(Π) → End(𝔪(Π)), whose image normalizes Q_{eK}. This proves the 1integrability of Q (by Cor. 10), completing the proof of Thm. 17. □

By Cor. 9 we can decompose L′_{x} = ∑_{α=1}^{3} ω′_{α}(x)J_{α} + L̄′_{x}, where the ω′_{α} are 1-forms on 𝔪(Π′) and L̄′_{x} ∈ z(Q′_{eK}) belongs to the centralizer of the quaternionic structure Q′_{eK} = Q_{eK}|𝔪(Π′) on 𝔪(Π′).

Lemma 11. The Nomizu map L′ : 𝔤(Π′) → End(𝔪(Π′)) associated to the Levi-Civita connection of (M(Π′) = G(Π′)/K, g′) can be extended to the Nomizu map L : 𝔤(Π) → End(𝔪(Π)) of a G(Π)-invariant quaternionic connection ∇ on the homogeneous almost quaternionic supermanifold (M(Π) = G(Π)/K, Q). The extension is defined as follows:

L_{x} := ∑_{α=1}^{3} ω_{α}(x)J_{α} + L̄_{x} , x ∈ 𝔪(Π) ,

where L̄_{x} ∈ z(Q) (the centralizer is taken in 𝔤𝔩(𝔪(Π))) is defined below and the 1-forms ω_{α} on 𝔪(Π) := 𝔪(Π′) + W_{0} satisfy ω_{α}|𝔪(Π′) := ω′_{α} and ω_{α}|W_{0} := 0. The operators L̄_{x} are given by:

L̄_{x}|𝔪(Π′) := L̄′_{x} if x ∈ 𝔪(Π′) ,
L̄_{e_{0}}|W_{0} := L̄_{e_{αa}}|W_{0} := L̄_{e_{α}}|W_{0} := 0 ,

L̄_{e′_{a}}|W_{0} := \frac{1}{2}e_{1}e_{2}e_{3}e′_{a} ,

L̄_{s}|W_{0} := 0 if s ∈ W ,

L̄_{s}x := (−1)x̃L_{x}s − [s, x] if s ∈ W_{0} , x ∈ 𝔪(Π′) .

(It is understood that L_{x} = ad_{x}|𝔪(Π) for all x ∈ 𝔨.) In the above formulas, as usual, a = 1, …, r + q, α = 1, 2, 3 and x̃ ∈ ℤ_{2} = {0, 1} stands for the ℤ_{2}-degree of x ∈ 𝔪(Π′) = 𝔪(Π′)_{0} + 𝔪(Π′)_{1} = 𝔪(V) + W′.

Proof: The proof is similar to that of Lemma 6. □

Appendix. Supergeometry

In this appendix we summarize the supergeometric material needed in 4. Standard references on supergeometry are [M], [L] and [K], see also [B-B-H], [Ba], [Be], [Bern], [B-O], [DW], [F], [O1], [O2], [Sch] and [S-W]. (D.A. Leites has informed us that he will soon publish a monograph on supergeometry.)

4.4. Supermanifolds. Let V = V_{0} + V_{1} be a ℤ_{2}-graded vector space. We recall that an element x ∈ V is called homogeneous (or of pure degree) if x ∈ V_{0} ∪ V_{1}. The degree of a homogeneous element x ∈ V is the number x̃ ∈ ℤ_{2} = {0, 1} such that x ∈ Vx̃. The element x ∈ V said to be even if x̃ = 0 and odd if x̃ = 1. For V_{0} and V_{1} of finite dimension, the dimension of V is defined as dim V := dim V_{0}| dim V_{1} = m|n and a basis of V is by definition a tuple (x_{1}, …, x_{m}, ξ_{1}, …, ξ_{n}) such that (x_{1}, …, x_{m}) is a basis of V_{0} and (ξ_{1}, …, ξ_{n}) is a basis of V_{1}.

Definition 15. Let A be a ℤ_{2}-graded algebra. The supercommutator is the bilinear map [·, ·] : A × A → A defined by:

[a, b] := ab − (−1)^{ãb̃}ba

for all homogeneous elements a, b ∈ A. The algebra A is called supercommutative if [a, b] = 0 for all a, b ∈ A. A ℤ_{2}-graded supercommutative associative (real) algebra A = A_{0} + A_{1} will simply be called a superalgebra.

Example 1: The exterior algebra ∧E = ∧^{even}E + ∧^{odd}E over a finite dimensional vector space E is a superalgebra.

Definition 16. A super Lie bracket on a ℤ_{2}-graded vector space V = V_{0}+V_{1} is a bilinear map [·, ·] : V × V → V such that for all x, y, z ∈ V_{0} ∪ V_{1} we have:

i) { \atop [x, y]} = x̃ + ỹ,

ii) [x, y] = −(−1)^{x̃ỹ}[y, x] and iii) [x, [y, z]] = [[x, y], z] + (−1)^{x̃ỹ}[y, [x, z]] (“super Jacobi identity”).

The ℤ_{2}-graded algebra with underlying ℤ_{2}-graded vector space V and product defined by the super Lie bracket [·, ·] is called a super Lie algebra.

Example 2: The supercommutator of any associative ℤ_{2}-graded algebra A is a super Lie bracket and hence defines on it the structure of super Lie algebra. For example, we may take A = End(V) with the obvious structure of ℤ_{2}-graded associative algebra (with unit). The corresponding super Lie algebra is denoted by 𝔤𝔩(V) and is called the general linear super Lie algebra.

Let M_{0} be a (differentiable) manifold of dimension m. We denote by C_{M_{0}}^{∞} its sheaf of functions. Sections of the sheaf C_{M_{0}}^{∞} over an open set U ⊂ M_{0} are simply smooth functions on U: C_{M_{0}}^{∞}(U) = C^{∞}(U). Now let A = A_{0} + A_{1} be a sheaf of superalgebras over M_{0}.

Definition 17. The pair M = (M_{0}, A) is called a (differentiable) supermanifold of dimension dim M = m|n over M_{0} if for all p ∈ M_{0} there exists an open neighborhood U ∋ p and a rank n free sheaf E_{U} of C_{U}^{∞}-modules over U such that A|_{U} ≅ ∧E_{U} (as sheaves of superalgebras). A function on M (over an open set U ⊂ M_{0}) is by definition a section of A (over U). The sheaf A = A_{M} is called the sheaf of functions on M and M_{0} is called the manifold underlying the supermanifold M. Let M = (M_{0}, A_{M}) and N = (N_{0}, A_{N}) be supermanifolds. A morphism φ : M → N is a pair φ = (φ_{0}, φ^{∗}), where φ_{0} : M_{0} → N_{0} is a smooth map and φ^{∗} : A_{N} → (φ_{0})_{∗}A_{M} is a morphism of sheaves of superalgebras. It is called an isomorphism if φ_{0} is a diffeomorphism and φ^{∗} an isomorphism. An isomorphism φ : M → M is called an automorphism of M. The set of all morphisms φ : M → N (respectively, automorphisms φ : M → M) is denoted by Mor(M, N) (respectively, Aut(M)).

From Def. 17 it follows that there exists a canonical epimorphism of sheaves ϵ^{∗} : A → C_{M_{0}}^{∞}, which is called the evaluation map. Its kernel is the ideal generated by A_{1}: ker ϵ^{∗} = ⟨A_{1}⟩ = A_{1} + A_{1}^{2}.

Given supermanifolds L, M, N and morphisms ψ ∈ Mor(L, M) and φ ∈ Mor(M, N), there is a composition φ ∘ ψ ∈ Mor(L, N) defined by:

(φ ∘ ψ)_{0} = φ_{0} ∘ ψ_{0} and (φ ∘ ψ)^{∗} = ψ^{∗} ∘ φ^{∗} .

Here we have used the same symbol ψ^{∗} for the map (φ_{0})_{∗}A_{M} → (φ_{0})_{∗}(ψ_{0})_{∗}A_{L} = (φ_{0} ∘ ψ_{0})_{∗}A_{L} induced by ψ^{∗} : A_{M} → (ψ_{0})_{∗}A_{L}. Similarly, if φ : M → N is an isomorphism, then we can define it inverse isomorphism by:

φ^{−1} := (φ_{0}^{−1}, (φ^{∗})^{−1}) : N → M .

Here, again, we have used the same notation (φ^{∗})^{−1} for the map A_{M} → (φ_{0}^{−1})_{∗}A_{N} induced by (φ^{∗})^{−1} : (φ_{0})_{∗}A_{M} → A_{N}. Finally, for every supermanifold M = (M_{0}, A), there is the identity automorphism Id_{M} := (Id_{M_{0}}, Id_{A}). The above operations turn the set Aut(M) into a group. Example 3: Let E → M_{0} be a (smooth) vector bundle of rank n over the mdimensional manifold M_{0} and E its sheaf of sections. It is a locally free sheaf of C_{M_{0}}^{∞}- modules and SM(E) := (M_{0}, ∧E) is a supermanifold of dimension m|n. Its evaluation map is the canonical projection onto 0-forms ∧E = C_{M_{0}}^{∞} + ∑_{j=1}^{n} ∧^{j}E → C_{M_{0}}^{∞}. It is well known, see [Ba], that any supermanifold is isomorphic to a supermanifold of the form SM(E). However, the isomorphism is not canonical, unless n = 0. Example 4: Any manifold (M_{0}, C_{M_{0}}^{∞}) of dimension m can be considered as a supermanifold of dimension m|0. In fact, it is associated to the vector bundle of rank 0 over M_{0} via the construction of Example 3. For any supermanifold M = (M_{0}, A) the pair (Id_{M_{0}}, ϵ^{∗}) defines a canonical morphism ϵ : M_{0} → M. The composition of ϵ with the canonical constant map p : {p} → M_{0} (p ∈ M_{0}) defines a morphism ϵ_{p} = (p, ϵ_{p}^{∗}) : {p} → M. The epimorphism ϵ_{p}^{∗} : A → ℝ onto the constant sheaf is called the evaluation at p:

ϵ_{p}^{∗}f = (ϵ^{∗}f)(p) .

f(p) := ϵ_{p}^{∗}f ∈ ℝ is called the value of f at the point p. Example 5: Let V = V_{0} + V_{1} be a ℤ_{2}-graded vector space of dimension m|n and E_{V} = V_{1} × V_{0} → V_{0} the trivial vector bundle over V_{0} with fibre V_{1}. Then to V we can canonically associate the supermanifold SM(V) := SM(E_{V}), see Example 3.

Let (x_{0}^{i}) = (x_{0}^{1}, …, x_{0}^{m}) be local coordinates for M_{0} defined on an open set U ⊂ M_{0}, E_{U} a rank n free sheaf of C_{U}^{∞}-modules over U and ϕ : ∧E_{U} → A|_{U} an isomorphism. We can choose sections (ξ_{0}^{j}) = (ξ_{0}^{1}, …, ξ_{0}^{n}) of E_{U} which generate E_{U} freely over C_{U}^{∞}. Then any section of ∧E_{U} is of the form:

f = ∑_{α∈ℤ_{2}^{n}} f_{α}(x_{0}^{1}, …, x_{0}^{m})ξ_{0}^{α} , f_{α}(x_{0}^{1}, …, x_{0}^{m}) ∈ C^{∞}(U) , (21)

where ξ_{0}^{α} := (ξ_{0}^{1})^{α_{1}} ∧ … ∧ (ξ_{0}^{n})^{α_{n}} for α = (α_{1}, …, α_{n}). The tuple (ϕ, x_{0}^{i}, ξ_{0}^{j}) is called a local coordinate system for M over U. The open set U ⊂ M_{0} is called a coordinate neighborhood for M. Any function on M over U is of the form ϕ(f) for some section f as in (21). The functions x^{i} := ϕ(x_{0}^{i}) ∈ A(U)_{0}, ξ^{j} := ϕ(ξ_{0}^{j}) ∈ A_{1} are called local coordinates for M over U. The evaluation map ϵ^{∗} : A → C_{M_{0}}^{∞} is expressed in a local coordinate system simply by putting ξ_{0}^{1} = ⋯ = ξ_{0}^{n} = 0 in (21):

ϵ^{∗}(ϕ(f)) = f(x_{0}^{1}, …, x_{0}^{m}, 0, …, 0) := f_{(0,... ,0)}(x_{0}^{1}, …, x_{0}^{m}) .

In particular, we have ϵ^{∗}x^{i} = x_{0}^{i} and ϵ^{∗}ξ^{j} = 0. Example 6: Let V = V_{0}+V_{1} be a ℤ_{2}-graded vector space and (x^{i}, ξ^{j}) = (x^{i}, …, x^{m}, ξ^{1}, …, ξ^{n}) a basis of the ℤ_{2}-graded vector space V^{∗} = Hom(V, ℝ). Then (x^{i}, ξ^{j}) can be considered as global coordinates on the supermanifold SM(V), i.e. as local coordinates for SM(V) over V_{0} = SM(V)_{0}, see Example 5.

Let M and N be supermanifolds of dimension m|n and p|q respectively. In local coordinates (x^{i}, ξ^{j}) for M and (y^{k}, η^{l}) for N a morphism φ is expressed by p even functions y^{k}(x^{1}, …, x^{m}, ξ^{1}, …, ξ^{n}) := φ^{∗}y^{k} and q odd functions η^{l}(x^{1}, …, x^{m}, ξ^{1}, …, ξ^{n}) := φ^{∗}η^{l}.

There exists a supermanifold M ×N = (M_{0}×N_{0}, A_{M×N}) called the product of the supermanifolds M and N and morphisms π_{M} : M ×N → M, π_{N} : M ×N → N such that (π_{M}^{∗}x^{i}, π_{N}^{∗}y^{k}, π_{M}^{∗}ξ^{j}, π_{N}^{∗}η^{l}) are local coordinates for M × N over U × V if (x^{i}, ξ^{j}) are local coordinates for M over U and (y^{k}, η^{l}) are local coordinates for N over V. The morphism π_{1} = π_{M} (respectively, π_{2} = π_{N}) is called the projection of M × N onto the first (respectively, second) factor. Given morphisms φ_{i} : M_{i} → N_{i}, i = 1, 2, there is a corresponding morphism φ_{1} × φ_{2} : M_{1} × M_{2} → N_{1} × N_{2} such that π_{N_{1}} ∘ (φ_{1} × φ_{2}) = φ_{1} ∘ π_{M_{1}} and π_{N_{2}} ∘ (φ_{1} × φ_{2}) = φ_{2} ∘ π_{M_{2}}.

As next, we will discuss the notion for tangency on supermanifolds. For this purpose, we recall the following definition.

Definition 18. An endomorphism X = X_{0} +X_{1} ∈ End(A) = Endℝ(A) (here X̃_{α} = α, α = 0, 1) of a ℤ_{2}-graded algebra A is called a derivation if it satisfies the Leibniz-rule

X_{α}(ab) = X_{α}(a)b + (−1)^{αã}aX_{α}(b)

for all homogeneous a, b ∈ A and α ∈ ℤ_{2}. The ℤ_{2}-graded vector space of all derivations of A is denoted by Der A = (Der A)_{0} + (Der A)_{1}.

Notice that the supercommutator on End(A) restricts to a super Lie bracket on Der A.

Definition 19. Let M = (M_{0}, A) be a supermanifold. The tangent sheaf of M is the sheaf of derivations of A and is denoted by T_{M} = (T_{M})_{0} + (T_{M})_{1}. A vector field on M is a section of T_{M}. The cotangent sheaf is the sheaf T_{M}^{∗} = Hom_{A}(T_{M}, A). The full tensor superalgebra over T_{M} is the sheaf of superalgebras generated by tensor products (ℤ_{2}-graded over A) of T_{M} and T_{M}^{∗}. It is denoted by ⊗_{A}⟨T_{M}, T_{M}^{∗}⟩. A tensor field on M is a section of ⊗_{A}⟨T_{M}, T_{M}^{∗}⟩.

Explicitly, a section X ∈ T_{M}(U) (U ⊂ M_{0} open) associates to any open subset V ⊂ U a derivation X|_{V} ∈ Der A(V) such that

X|_{V} (f|_{V}) = X|_{U}(f)|_{V} for all f ∈ A(U) .

Given local coordinates (x^{i}, ξ^{j}) on M over U there exist unique vector fields

\frac{∂}{∂x^{i}} ∈ T_{M}(U)_{0} and \frac{∂}{∂ξ^{j}} ∈ T_{M}(U)_{1}

such that

\frac{∂x^{k}}{∂x^{i}} = δ_{i}^{k} , \frac{∂ξ^{l}}{∂x^{i}} = 0 , \frac{∂x^{k}}{∂ξ^{j}} = 0 , \frac{∂ξ^{l}}{∂ξ^{j}} = δ_{j}^{l} .

(Here, of course, δ_{i}^{k} ∈ A(U) is the unit element in the algebra A(U) if i = k and is zero otherwise.) Moreover, the vector fields (∂/∂x^{i}, ∂/∂ξ^{j}) freely generate T_{M}(U) ≅ Der A(U) over A(U). This shows that T_{M} is a locally free sheaf of rank m|n = dim M over A. It is also a sheaf of super Lie algebras; simply because Der A(U) is a subalgebra of the super Lie algebra Endℝ A(U) for all open U ⊂ M_{0}. As in the case of ordinary manifolds, given a vector field X there exists a unique derivation L_{X} of the full tensor superalgebra ⊗_{A}⟨T_{M}, T_{M}^{∗}⟩ over T_{M} compatible with contractions such that

L_{X}f = X(f) and L_{X}Y = [X, Y]

for all functions f and vector fields Y on M ([X, Y] is the supercommutator of vector fields).

Definition 20. Let M = (M_{0}, A) be a supermanifold. A tangent vector to M at p ∈ M_{0} is an ℝ-linear map v = v_{0} + v_{1} : A_{p} → ℝ such that

v_{α}(fg) = v_{α}(f)ϵ_{p}^{∗}(g) + (−1)^{αf̃}ϵ_{p}^{∗}(f)v_{α}(g) , α = 0, 1 ,

for all germs of functions f, g ∈ A_{p} of pure degree. (A_{p} denotes the stalk of A at p.) The ℤ_{2}-graded vector space of all tangent vectors to M at p is denoted by T_{p}M = (T_{p}M)_{0} + (T_{p}M)_{1} and is called the tangent space to M at p.

Let X be a vector field defined on some open set U ⊂ M_{0} and p ∈ U. Then we can define the value X(p) ∈ T_{p}M of X at p:

X(p)(f) := ϵ_{p}^{∗}(X(f)) , f ∈ A_{p} .

However, unless dim M = m|n = m|0, a vector field is not determined by its values at all p ∈ M_{0}. The above definition of value at a point p is naturally extended to arbitrary tensor fields S; the value of S at p is denoted by S(p). Again, unless dim M = m|n = m|0, a tensor field on M is not determined by its values at all p ∈ M_{0}.

Given a morphism φ : M → N, to any local vector field X ∈ T_{M}(U) on M we can associate a vector field dφX ∈ (φ^{∗}T_{N})(U) on N with values in A_{M} which is defined by:

(dφX)(f) = X(φ^{∗}f) , f ∈ A_{N}(V) ,

where V ⊂ N_{0} is an open set such that φ_{0}^{−1}(V) ⊃ U. We recall that φ^{∗}T_{N} is the sheaf of A_{M}-modules over M_{0} defined by:

φ^{∗}T_{N} := A_{M} ⊗_{φ_{0}^{−1}A_{N}} φ_{0}^{−1}T_{N} .

Here the action of φ_{0}^{−1}A_{N} on A_{M} is defined by the map

φ_{0}^{−1}A_{N} → φ_{0}^{−1}φ_{∗}A_{M} → A_{M}

induced by φ^{∗} : A_{N} → φ_{∗}A_{M}. By the above construction, we obtain a section dφ of the sheaf Hom_{A}(T_{M}, φ^{∗}T_{N}), which is expressed with respect to local coordinates (u^{1}, …, u^{m+n}) = (x^{1}, …, x^{m}, ξ^{1}, …, ξ^{n}) on M and (v^{1}, …, v^{p+q}) = (y^{1}, …, y^{p}, η^{1}, …, η^{q}) on N by the Jacobian matrix (\frac{∂φ^{∗}v^{i}}{∂u^{j}}). The value dφ(p) ∈ Hom(T_{p}M, T_{φ_{0}(p)}N) of the differential at a point p ∈ M_{0} is defined by:

dφ(p)X(p) = (dφX)(p)

for all vector fields X on M. The differential dϵ of the canonical morphism ϵ : M_{0} → M provides the canonical isomorphism dϵ(p) : T_{p}M_{0} →^{∼} (T_{p}M)_{0} for all p ∈ M.

Definition 21. A morphism φ : M → N is called an immersion (respectively, a submersion) if dφ has constant rank m|n = dim M (respectively, p|q = dim N), i.e. if for all local coordinates as above the matrix (ϵ^{∗}(∂φ^{∗}y^{i}/∂x^{j})) has constant rank m (respectively, p) and (ϵ^{∗}(∂φ^{∗}η^{i}/∂ξ^{j})) has constant rank n (respectively, q). An immersion φ : M → N is called injective (respectively, an embedding, a closed embedding) and is denoted by φ : M ↪ N if φ_{0} : M_{0} → N_{0} is injective (respectively, an embedding, a closed embedding). Two immersions φ : M → N and φ′ : M′ → N are called equivalent if there exists an isomorphism ψ : M → M′ such that φ = φ′ ∘ ψ. A submanifold (respectively, an embedded submanifold, a closed submanifold) is an equivalence class of injective immersions (respectively, embeddings, closed embeddings).

Notice that for any supermanifold the canonical morphism ϵ = (Id_{M_{0}}, ϵ^{∗}) : M_{0} ↪ M is a closed embedding.

As for ordinary manifolds, immersions and submersions admit adapted coordinates:

Proposition 15. (see [L], [K]) Let M and N be supermanifolds of dimension dim M = m|n and dim N = p|q. A morphism φ : M → N is an immersion (respectively, submersion) if and only if for all (x, y = φ_{0}(x)) ∈ M_{0} × φ_{0}(M_{0}) ⊂ M_{0} × N_{0} there exists local coordinates (x^{1}, …, x^{m}, ξ^{1}, …, ξ^{n}) for M defined near x and (y^{1}, …, y^{p}, η^{1}, …, η^{q}) for N defined near y such that

φ^{∗}y^{i} = \{ \begin{matrix}x^{i} for & i = 1, …, m ≤ p \\ 0 for & m < i ≤ p \end{matrix} and φ^{∗}η^{j} = \{ \begin{matrix}ξ^{j} for & j = 1, …, n ≤ q \\ 0 for & n < j ≤ q \end{matrix}

(respectively,

φ^{∗}y^{i} = x^{i} for i = 1, …, p ≤ m and φ^{∗}η^{j} = ξ^{j} for j = 1, …, q ≤ n) .

For any submanifold φ : M ↪ N we define its vanishing ideal J_{y} ⊂ (A_{N})_{y} at y ∈ N_{0} as follows: J_{y} := (A_{N})_{y} if y ∉ φ_{0}(M_{0}) and J_{y} := ker(r∘φ^{∗}) if y = φ_{0}(x), x ∈ M_{0}, where r : ((φ_{0})_{∗}A_{M})_{y} → (A_{M})_{x} is the natural restriction homomorphism. The union J := ∪_{y∈N_{0}}J_{y} ⊂ A_{N} is called the vanishing ideal of the submanifold φ : M ↪ N. By Prop. 15, it has the following property (P): If y ∈ φ_{0}(M_{0}) then there exists p−m even functions (y^{i}) and q−n odd functions η^{j} on N vanishing at y whose germs at y generate J_{y} and which can be complemented to local coordinates for N defined near y. (Here m|n = dim M and p|q = dim N.) We denote by Z_{U} ⊂ N_{0} the submanifold defined by the equations ϵ^{∗}y^{i} = 0 on some sufficiently small open neighborhood U ⊂ N_{0} of y. Then U can be chosen such that the germs of the functions (y^{i}, η^{j}) at z generate J_{z} for all z ∈ Z_{U}.

Conversely, let φ_{0} : M_{0} ↪ N_{0} be any submanifold and J_{y} ⊂ (A_{N})_{y}, y ∈ N, a collection of ideals with the above property (P) and such that J_{y} = (A_{N})_{y} for y ∉ φ_{0}(M_{0}), then there exists a supermanifold M = (M_{0}, A_{M}) and an injective immersion φ = (φ_{0}, φ^{∗}) : M ↪ N with J = ∪_{y∈N}J_{y} as its vanishing ideal.

Notice that if M ↪ N is a closed submanifold then its vanishing ideal can be defined directly as the sheaf of ideals J := ker φ^{∗}.

4.5. Pseudo-Riemannian metrics, connections and quaternionic structures on supermanifolds. Let A be a superalgebra and T a free A-module of rank m|n.

Definition 22. An even (respectively, odd) bilinear form on T is a biadditive map g : T × T → A such that

g(aX, bY) = (−1)^{b̃X̃}abg(X, Y)
(respectively, g(aX, bY) = (−1)^{b̃X̃+ã+b̃}abg(X, Y)) ,

for all homogeneous a, b ∈ A and X, Y ∈ T. A bilinear form g on T is called supersymmetric (respectively, super skew symmetric) if

g(X, Y) = (−1)^{X̃Ỹ} g(Y, X)
(respectively, g(X, Y) = −(−1)^{X̃Ỹ} g(Y, X)) ,

for all homogeneous X, Y ∈ T. It is called nondegenerate if

T ∋ X ↦ g(X, ·) ∈ T^{∗} = Hom_{A}(T, A)

is an isomorphism of A-modules. The A-module of bilinear forms on T is denoted by Bil_{A}(T) = Bil_{A}(T)_{0} + Bil_{A}(T)_{1}.

Notice that Bil_{A}(T) ≅ Hom_{A}(T, T^{∗}) ≅ T^{∗} ⊗_{A} T^{∗}, where T^{∗} = Hom_{A}(T, A). For a supermanifold M = (M_{0}, A) the sheaf Bil_{A} T_{M} of bilinear forms on T_{M}

is defined in the obvious way such that (Bil_{A} T_{M})(U) = Bil_{A(U)}(T_{M}(U)) for every open subset U ⊂ M_{0} with the property that T_{M}(U) is a free A(U)-module. We have obvious isomorphisms of sheaves of A-modules: Bil_{A} T_{M} ≅ Hom_{A}(T_{M}, T_{M}^{∗}) ≅ T_{M}^{∗} ⊗_{A} T_{M}^{∗}. Since any section g of Bil_{A} T_{M} can be considered as a tensor field on M, it has a well defined value g(p) ∈ Bilℝ(T_{p}M) for all p ∈ M_{0}. The restriction g(p)|(T_{p}M)_{0} × (T_{p}M)_{0} defines a section g_{0} of Bil_{C_{M_{0}}^{∞}} T_{M_{0}} via the canonical

identification dϵ(p) : T_{p}M_{0} →^{∼} (T_{p}M)_{0}.

Definition 23. A pseudo-Riemannian metric on a supermanifold M = (M_{0}, A), M_{0} connected, is an even nondegenerate supersymmetric section g of Bil_{A} T_{M}. The signature (k, l) of g is the signature of the pseudo-Riemannian metric g_{0} on M_{0}. The pseudo-Riemannian metric g is said to be a Riemannian metric if g_{0} is Riemannian. Let M = (M_{,}A) be a supermanifold and E a locally free sheaf of A-modules. A connection on E is an even section ∇ of the sheaf Hom_{A}(T_{M}, Endℝ E), which to any vector field X on M associates a section ∇_{X} of Endℝ E such that

∇_{X}fs = X(f)s + (−1)^{X̃f̃}f∇_{X}s

for all vector fields X on M, functions f on M and sections s of E of pure degree. The curvature R of ∇ is the even super skew symmetric section of End_{A} E ⊗_{A} Bil_{A} T_{M} defined by:

R(X, Y) := ∇_{X}∇_{Y} − (−1)^{X̃Ỹ} ∇_{Y} ∇_{X} − ∇_{[X,Y]}

for all vector fields X, Y on M of pure degree. A connection on a supermanifold M is by definition a connection on its tangent sheaf T_{M}. Its torsion is the even super skew symmetric section of T_{M} ⊗_{A} Bil_{A} T_{M} defined by:

T (X, Y) := ∇_{X}Y − (−1)^{X̃Ỹ} ∇_{Y} X − [X, Y]

for all vector fields X, Y on M of pure degree.

As for ordinary manifolds, a connection ∇ on a supermanifold M induces a connection on the full tensor superalgebra ⊗_{A}⟨T_{M}, T_{M}^{∗}⟩ . In particular, if g is e.g. an even section of Bil_{A} T_{M} ≅ T_{M}^{∗} ⊗ T_{M}^{∗} then we have

(∇_{X}g)(Y, Z) = Xg(Y, Z) − g(∇_{X}Y, Z) − (−1)^{X̃Ỹ} g(Y, ∇_{X}Z)

for all vector fields X, Y and Z on M of pure degree. As in the case of ordinary manifolds, see e.g. [O’N], one can prove that a pseudo-Riemannian supermanifold (M, g) has a unique torsionfree connection ∇ = ∇^{g} such that ∇g = 0. We will call this connection the Levi-Civita connection of (M, g). It is computable from the following superversion of the Koszul-formula:

2g(∇_{X}Y, Z) = Xg(Y, Z) + (−1)^{X̃(Ỹ +Z̃)}Y g(Z, X)
− (−1)^{Z̃(X̃+Ỹ)}Zg(X, Y) − g(X, [Y, Z])

+ (−1)^{X̃(Ỹ +Z̃)}g(Y, [Z, X]) + (−1)^{Z̃(X̃+Ỹ)}g(Z, [X, Y]) (22)

for all vector fields X, Y and Z on M of pure degree.

Next we are going to define quaternionic supermanifolds and quaternionic Kähler supermanifolds. First we define the notion of almost quaternionic structure.

Definition 24. Let M = (M_{0}, A) be a supermanifold. An almost complex structure on M is an even global section J ∈ (End_{A} T_{M})(M_{0}) such that J^{2} = −Id. An almost hypercomplex structure on M is a triple (J_{α}) = (J_{1}, J_{2}, J_{3}) of almost complex structures on M satisfying J_{1}J_{2} = J_{3}. An almost quaternionic structure on M is a subsheaf Q ⊂ End_{A} T_{M} with the following property: for every p ∈ M_{0} there exists an open neighborhood U ⊂ M_{0} and an almost hypercomplex structure (J_{α}) on M|_{U} such that Q(U) is a free A(U)-module of rank 3|0 with basis (J_{1}, J_{2}, J_{3}). A pair (M, J) (respectively, (M, (J_{α})), (M, Q)) as above is called an almost complex supermanifold (respectively, almost hypercomplex supermanifold, almost quaternionic supermanifold).

Second we introduce the basic compatibility conditions between almost quaternionic structures, connections and pseudo-Riemannian metrics.

Definition 25. Let (M, Q) be an almost quaternionic supermanifold of dimension dim M = m|n. A connection ∇ on (M, Q) is called an almost quaternionic connection if ∇ preserves Q, i.e. if ∇_{X}S is a section of Q for any vector field X on M and any section S of Q. A quaternionic connection on (M, Q) is a torsionfree almost quaternionic connection. If the almost quaternionic structure Q on M admits a quaternionic connection, then it is called 1-integrable or quaternionic structure. In this case the pair (M, Q) is called a quaternionic supermanifold, provided that m = dim M_{0} > 4.

Definition 26. A (pseudo-) Riemannian metric g on an almost quaternionic supermanifold (M, Q) is called Hermitian if sections of Q are g-skew symmetric, i.e. if g(SX, Y) = −g(X, SY) for all sections S of Q and vector fields X, Y on M. In this case the triple (M, Q, g) is called an almost quaternionic (pseudo-) Hermitian supermanifold. If, moreover, the Levi-Civita connection ∇^{g} of the Q-Hermitian metric g is quaternionic, then (M, Q, g) is called a quaternionic (pseudo-) Kähler supermanifold, provided that m = dim M_{0} > 4.

We recall that to any (pseudo-) Riemannian metric g on a supermanifold M we have associated the (pseudo-) Riemannian metric g_{0} on the manifold M_{0}, see Def. 23. Now we will associate an almost quaternionic structure Q_{0} on M_{0} to any almost quaternionic structure Q on M. To any even section S of End_{A} T_{M} we associate a section S_{0} of End_{C_{M_{0}}^{∞}} T_{M_{0}} defined by S_{0}(p) = S(p)|T_{p}M_{0}, where S(p) ∈ (Endℝ(T_{p}M))_{0} is the value of the tensor field S at p ∈ M_{0} and, as usual, T_{p}M_{0} is canonically identified with (T_{p}M)_{0}. Let Q_{0} ⊂ End_{C_{M_{0}}^{∞}} T_{M_{0}} be the subsheaf of C_{M_{0}}^{∞}-modules spanned by the local sections of the form S_{0}, where S is a local section of Q. Then Q_{0} is a locally free sheaf of rank 3 and hence it defines a rank 3 subbundle of the vectorbundle End T M_{0}, more precisely, an almost quaternionic structure on M_{0} in the usual sense, see Def. 5.

Finally we give the definition of quaternionic supermanifolds and of quaternionic Kähler supermanifolds of dimension 4|n.

Definition 27. An almost quaternionic supermanifold (M, Q) of dimension dim M = 4|n is called a quaternionic supermanifold if

i) there exists a quaternionic connection ∇ on (M, Q), ii) (M_{0}, Q_{0}) is a quaternionic manifold in the sense of Def. 8.

An almost quaternionic Hermitian supermanifold (M, Q, g) of dimension dim M = 4|n is called a quaternionic Kähler supermanifold if

i) the Levi-Civita connection ∇^{g} preserves g and ii) (M_{0}, Q_{0}) is a quaternionic Kähler manifold in the sense of Def. 11.

4.6. Supergroups. Let V = V_{0} + V_{1} be a ℤ_{2}-graded vector space. We recall that 𝔤𝔩(V) denotes the general linear super Lie algebra. Its super Lie bracket is the supercommutator on Endℝ V, see Example 2.

Definition 28. A representation of a super Lie algebra 𝔤 on a ℤ_{2}-graded vector space V is a homomorphism of super Lie algebras 𝔤 → 𝔤𝔩(V). The adjoint representation of 𝔤 is the representation ad : 𝔤 ∋ x ↦ ad_{x} ∈ 𝔤𝔩(𝔤) defined by:

ad_{x}y = [x, y] , x, y ∈ 𝔤 .

Here [·, ·] denotes the super Lie bracket in 𝔤. A linear super Lie algebra is a subalgebra 𝔤 ⊂ 𝔤𝔩(V).

Notice that, by the super Jacobi identity, see Def. 16, ad_{x} is a derivation of 𝔤 for all x ∈ 𝔤.

Now let E be a module over a superalgebra A; say E = V ⊗ A, where V is a ℤ_{2}-graded vector space (and ⊗ stands for the ℤ_{2}-graded tensor product over ℝ). The invertible elements of (End_{A} E)_{0} form a group, which is denoted by GL_{A}(E). Given a ℤ_{2}-graded vector space V, the correspondence

A ↦ GL_{A}(V ⊗ A)

defines a covariant functor from the category of superalgebras into the category of groups. We can compose this functor with the contravariant functor

M = (M_{0}, A) ↦ A(M_{0})

from the category of supermanifolds into that of superalgebras. The resulting contravariant functor

M ↦ GL(V)[M] := GL_{A(M_{0})}(V ⊗ A(M_{0}))

from the category of supermanifolds into that of groups is denoted by GL(V)[·].

Definition 29. A supergroup G[·] is a contravariant functor from the category of supermanifolds into that of groups. The supergroup GL(V)[·], defined above, is called the general linear supergroup over the ℤ_{2}-graded vector space V. A subgroup of a supergroup G[·] is a supergroup H[·] such that H[M] is a subgroup of G[M] for all supermanifolds M and H[φ] = G[φ]|H[N] for all morphisms φ : M → N. We will write H[·] ⊂ G[·] if H[·] is a subgroup of the supergroup G[·]. A linear supergroup is a subgroup H[·] ⊂ GL(V)[·].

Example 7: To any linear super Lie algebra 𝔤 ⊂ 𝔤𝔩(V) we can associate the linear supergroup G[·] ⊂ GL(V)[·] defined by:

G[M] := ⟨exp(𝔤 ⊗ A(M_{0}))_{0}⟩

for any supermanifold M = (M_{0}, A). The right-hand side is the subgroup of GL_{A(M_{0})} (V ⊗ A(M_{0})) generated by the exponential image of the Lie algebra (𝔤 ⊗ A(M_{0}))_{0} ⊂ (𝔤𝔩(V) ⊗ A(M_{0}))_{0} ≅ (End_{A(M_{0})}(V ⊗ A(M_{0})))_{0}. The convergence of the exponential series, which is locally uniform in all derivatives of arbitrary order, follows from the analyticity of the exponential map by a (finite) Taylor expansion with respect to the odd coordinates. G[·] is called the linear supergroup associated to the linear super Lie algebra 𝔤.

For any supermanifold M we denote by Δ_{M} = (Δ_{M_{0}}, Δ_{M}^{∗}) the diagonal embedding defined by: Δ_{M_{0}}(x) = (x, x) (x ∈ M_{0}) and Δ_{M}^{∗}π_{i}^{∗}f = f for all functions f on M. Here π_{i} : M × M → M denotes the projection onto the i-th factor of M × M (i = 1, 2).

Definition 30. A Lie supergroup is a supermanifold G = (G_{0}, A_{G}) whose underlying manifold is a Lie group G_{0} (which, for convenience, we will always assume to be connected) with neutral element e ∈ G_{0}, multiplication μ_{0} : G_{0}×G_{0} → G_{0} and inversion ι_{0} : G_{0} → G_{0}, together with morphisms μ = (μ_{0}, μ^{∗}) : G×G → G and ι = (ι_{0}, ι^{∗}) : G → G such that

i) μ ∘ (Id_{G} × μ) = μ ∘ (μ × Id_{G}) ∈ Mor(G × G × G, G),

ii) μ ∘ (Id_{G} × ϵ_{e}) = π_{1} : G × {e} →^{∼} G, μ ∘ (ϵ_{e} × Id_{G}) = π_{2} : {e} × G →^{∼} G and

iii) μ ∘ (Id_{G} × ι) ∘ Δ_{G} = μ ∘ (ι × Id_{G}) ∘ Δ_{G} = Id_{G}.

Here ϵ_{e} = (e, ϵ_{e}^{∗}) : {e} ↪ G_{0} ↪^{ϵ} G is the canonical embedding. The morphisms μ and ι are called, respectively, multiplication and inversion in the Lie supergroup G. A Lie subgroup of a Lie supergroup G = (G_{0}, A_{G}) is a submanifold φ : H = (H_{0}, A_{H}) ↪ G such that the immersion φ_{0} : H_{0} ↪ G_{0} induces on H_{0} the structure of Lie subgroup of G_{0} and φ induces on H the structure of Lie supergroup with underlying Lie group H_{0}. We will write H ⊂ G if H is a Lie subgroup of G. An action of a Lie supergroup G on a supermanifold M is a morphism α : G×M → M such that

i) α ∘ (Id_{G} × α) = α ∘ (μ × Id_{M}) and

ii) α ∘ (ϵ_{e} × Id_{M}) = π_{2} : {e} × M →^{∼} M.

Given an action α : G × M → M we have the notion of fundamental vector field X associated to x ∈ T_{e}G. It is defined by X(f) := x(α^{∗}f). The correspondence x ↦ X defines an even ℝ-linear map T_{e}G → T_{M}(M_{0}) from the ℤ_{2}-graded vector space T_{e}G to the free A_{M}(M_{0})-module of global vector fields on M.

Definition 31. Let an action α of a Lie supergroup G on a supermanifold M be given. A tensor field S on M is called G-invariant if the Lie derivative L_{X}S = 0 for all fundamental vector fields X. (Notice that, in particular, this defines the notion of G-invariant pseudo-Riemannian metric on M.) An almost quaternionic structure Q on M is called G-invariant if L_{X}S is a section of Q for all sections S of Q and fundamental vector fields X on M.

We can specialize the above definition to the (left-) action μ : G×G → G given by the multiplication in the Lie supergroup G. The G-invariant tensor fields with respect to that action are called left-invariant. A tensor field S is called rightinvariant if L_{X}S = 0 for all left-invariant vector fields X on G. The right-invariant vector fields are precisely the fundamental vector fields for the action μ.

For any Lie supergroup G, we can define a group homomorphism μ_{l} : G_{0} → Aut(G) by:

μ_{l}(g) := μ ∘ (ϵ_{g} × Id_{G}) : {g} × G ≅ G → G , g ∈ G_{0} .

Here ϵ_{g} : {g} ↪ G is the canonical embedding and {g} × G is canonically identified with G via the projection π_{2} : {g} × G →^{∼} G onto the second factor. Similarly, we can define a group antihomomorphism μ_{r} : G_{0} → Aut(G) by:

μ_{r}(g) := μ ∘ (Id_{G} × ϵ_{g}) : G × {g} ≅ G → G , g ∈ G_{0} .

Notice that the canonical embedding ϵ : G_{0} ↪ G is G_{0}-equivariant with respect to the usual left- (respectively, right-) action on G_{0} and the action on G defined by μ_{l} (respectively, μ_{r}).

Given a Lie supergroup G and a supermanifold M there is a natural group structure on Mor(M, G) with multiplication defined by:

φ · ψ := μ ∘ (φ × ψ) ∘ Δ_{M} , φ, ψ ∈ Mor(M, G) .

The correspondence M ↦ Mor(M, G) defines a supergroup.

Definition 32. The supergroup G[·] := Mor(·, G) is called the supergroup subordinate to the Lie supergroup G.

Example 8: Let V be a ℤ_{2}-graded vector space. Then GLℝ(V) is, by definition, the group of invertible elements of (Endℝ V)_{0}. The Lie group GLℝ(V) ≅ GLℝ(V_{0}) × GLℝ(V_{1}) is an open submanifold of the vector space (Endℝ V)_{0} = SM(Endℝ V)_{0}, see Example 5. We define the submanifold

GL(V) := SM(Endℝ V)|_{GLℝ(V)} ↪ SM(Endℝ V) .

The manifold underlying the supermanifold GL(V) = (GL(V)_{0}, A_{GL(V)}) is the above Lie group: GL(V)_{0} = GLℝ(V). From the definition of the supermanifold GL(V) it is clear that Mor(M, GL(V)) is canonically identified with the set GL_{A(M_{0})}(V ⊗ A(M_{0})) = GL(V)[M] (cf. Def. 29) for any supermanifold M = (M_{0}, A). Moreover, GL(V) has a unique structure of Lie supergroup inducing the canonical group structure on GL(V)[M] for any supermanifold M. In other words, the general linear supergroup GL(V)[·] is the supergroup subordinate to the Lie supergroup GL(V), see Def. 32.

Definition 33. The Lie supergroup GL(V) is called the general linear Lie supergroup. A linear Lie supergroup is a Lie subgroup of GL(V).

There exists a unique morphism

Exp : SM(𝔤𝔩(V)) → GL(V)

such that

Exp ∘ φ = exp(φ)

for all supermanifolds M = (M_{0}, A) and all morphisms

φ ∈ Mor(M, SM(𝔤𝔩(V))) = (𝔤𝔩(V) ⊗ A(M_{0}))_{0} = (End_{A(M_{0})} V ⊗ A(M_{0}))_{0} ,

where, on the right-hand side,

exp : (End_{A(M_{0})} V ⊗ A(M_{0}))_{0} → GL_{A(M_{0})}(V ⊗ A(M_{0})) = Mor(M, GL(V))

is the exponential map for even endomorphisms of V ⊗ A(M_{0}) (for the definition of the supermanifold SM(𝔤𝔩(V)) see Example 5). The underlying map Exp_{0} : 𝔤𝔩(V)_{0} = SM(𝔤𝔩(V))_{0} → GL(V)_{0} is the ordinary exponential map for the Lie group GL(V)_{0}: Exp_{0} = exp. The morphism Exp is called the exponential morphism of 𝔤𝔩(V). Example 9: Let 𝔤 ⊂ 𝔤𝔩(V) be a linear super Lie algebra and G[·] ⊂ GL(V)[·] the correponding linear supergroup, see Example 7. We denote by G_{0} ⊂ GL(V)_{0} the connected linear Lie group with Lie algebra 𝔤_{0}. It is the immersed Lie subgroup generated by the exponential image of 𝔤_{0} = 𝔤𝔩(V)_{0}. We define an ideal J_{g} ⊂ (A_{GL(V)})_{g} as follows: A germ of function f ∈ (A_{GL(V)})_{g} (g ∈ G_{0}) belongs to J_{g} if and only if r(φ^{∗}f) = 0 for all injective immersions φ ∈ G[M] ⊂ Mor(M, GL(V)), where r : ((φ_{0})_{∗}A_{M})_{g} → (A_{M})_{φ_{0}^{−1}(g)} is the natural restriction map. We claim that J_{G} := ∪_{g∈G_{0}}J_{g} is the vanishing ideal of a submanifold G ↪ GL(V). Due to the invariance of J_{G} under the group μ_{l}(G_{0}) ⊂ Aut(G) it is sufficient to prove the claim locally over an open set of the form exp U ⊂ G_{0}, U ⊂ 𝔤_{0} an open neighborhood of 0 ∈ 𝔤_{0}. The local statement follows from the fact that the morphism SM(𝔤) ↪ SM(𝔤𝔩(V)) →^{Exp} GL(V) has maximal rank at 0 ∈ 𝔤_{0} = SM(𝔤)_{0} and hence defines a submanifold SM(𝔤)|_{U} ↪ GL(V) for some open neighborhood of U of 0 ∈ 𝔤_{0}. The vanishing ideal of this submanifold coincides with J_{G} over exp U by the definition of the exponential morphism. Next we claim that multiplication μ : GL(V)×GL(V) → GL(V) and inversion ι : GL(V) → GL(V) have the property that μ^{∗}J_{G} ⊂ J_{G×G} and ι^{∗}J_{G} = J_{G}. Here J_{G×G} is the vanishing ideal of the submanifold G × G ↪ GL(V)×GL(V). The claim follows from the fact that G[·] is a subgroup of GL(V)[·] and implies that the morphisms G × G ↪ GL(V) × GL(V) →^{μ} GL(V) and G ↪ GL(V) →^{ι} GL(V) induce morphisms G × G → G and G → G, which induce on G the structure of Lie supergroup. In other words, G is a Lie subgroup of the general linear Lie supergroup GL(V). It is called the linear Lie supergroup associated to the linear Lie superalgebra 𝔤 ⊂ 𝔤𝔩(V). Notice that Exp^{∗} maps J_{G} into the vanishing ideal of SM(𝔤) ↪ SM(𝔤𝔩(V)) and hence the restriction SM(𝔤) ↪ SM(𝔤𝔩(V)) →^{Exp} GL(V) of the exponential morphism induces a morphism SM(𝔤) → G, which is called the exponential morphism of 𝔤 and is again denoted by Exp. Its differential at 0 ∈ 𝔤_{0} yields an isomorphism 𝔤 ≅ T_{0}SM(𝔤) →^{∼} T_{e}G.

4.7. Homogeneous supermanifolds. Let G = (G_{0}, A_{G}) be a Lie supergroup, K ⊂ G_{0} a closed subgroup and π_{0} : G_{0} → G_{0}/K the canonical projection. Then the subsheaf A_{G}^{K} := A_{G}^{μ_{r}(K)} ⊂ A_{G} of functions on G invariant under the subgroup μ_{r}(K) ⊂ Aut(G) is again a sheaf of superalgebras on G_{0}. Explicitly, a function f ∈ A_{G}(U) (U ⊂ G_{0} open) belongs to A_{G}^{K}(U) if it can be extended to a μ_{r}(K)- invariant function over UK ⊂ G_{0}. Its pushed forward sheaf A_{G/K} := (π_{0})_{∗}A_{G}^{K} is a sheaf of superalgebras on the homogeneous manifold G_{0}/K.

Theorem 18. (cf. [K]) Let G ↪ GL(V) be a closed linear Lie supergroup and K ⊂ G_{0} a closed subgroup. Then G/K = (G_{0}/K, A_{G/K}) is a supermanifold with a canonical submersion π : G → G/K and a canonical action α : G × G/K → G/K.

Proof: Since μ_{l}(g) ∈ Aut(G) induces isomorphisms

A_{G/K}(gU) →^{∼} A_{G/K}(U)

for all g ∈ G_{0} and U ⊂ G_{0}/K open, it is sufficient to check that G/K|_{U} = (U, A_{G/K}|_{U}) is a supermanifold for some open neighborhood U of eK ∈ G_{0}/K.

Lemma 12. Under the assumptions of Thm. 18, there exists local coordinates (x, y, ξ) for GL(V) over some neighborhood U = UK ⊂ GL(V)_{0} of e ∈ GL(V)_{0} such that

1) x = (x^{i}) and y = (y^{j}) consist of even functions and ξ = (ξ^{k}) of odd functions,

2) A_{GL(V)}^{K}(U) is the subalgebra of A_{GL(V)}(U) which consists of functions f(x, ξ) only of (x, ξ), more precisely,

f(x, ξ) = ∑_{α} f_{α}(x)ξ^{α} ,

where the f_{α}(x) ∈ A_{GL(V)}(U) are functions only of x, i.e. if f_{α}(x) ≠ 0 then f_{α}(x) does not belong to the ideal generated by ξ and ϵ^{∗}f_{α}(x) ∈ C^{∞}(U) are functions only of ϵ^{∗}x (independent of ϵ^{∗}y). Here we used the multiindex notation α = (α_{1}, …, α_{2mn}) ∈ ℤ_{2}^{2mn} (m|n = dim V) and ξ^{α} = ∏_{j=1}^{2mn}(ξ^{j})^{α_{j}}.

Proof: The natural (global) coordinates on the supermanifold GL(V) are the matrix coefficients with respect to some basis of V. We will denote them simply by (z^{i}, ζ^{j}) instead of using matrix notation. For these coordinates it is clear that μ_{r}(g)^{∗}z^{k} is a linear combination (over the real numbers) of the even coordinates z := (z^{i}) and μ_{r}(g)^{∗}ζ^{l} is a linear combination (over the real numbers) of the odd coordinates ζ := (ζ^{j}) for all g ∈ GL(V)_{0}. In particular, we obtain a representation ρ of K ⊂ GL(V)_{0} on the vector space spanned by the odd coordinates. Let E_{ρ} → G_{0}/K denote the vector bundle associated to this representation. Any μ_{r}(K)-invariant function on GL(V) linear in ζ defines a section of the dual vector bundle E_{ρ}^{∗} and vice versa. Now, since E_{ρ}^{∗} is locally trivial (like any vector bundle), we can find μ_{r}(K)-invariant local functions ξ = (ξ^{j}) linear in ζ such that (z, ξ) are local coordinates for GL(V) over some open neighborhood U = UK ⊂ GL(V)_{0} of e. Next, by way of a local diffeomorphism in the even coordinates z, we can arrange that z = (x, y), where the x are μ_{r}(K)-invariant functions on GL(V) such that ϵ^{∗}x ∈ C^{∞}(U) induce local coordinates on G_{0}/K. Now any function f ∈ A_{GL(V)}(U) has a unique expression of the form

∑_{α} f_{α}(x, y)ξ^{α} , (23)

where the f_{α}(x, y) are functions only of (x, y). From the μ_{r}(K)-invariance of the ξ^{α} it follows that f is μ_{r}(K)-invariant if and only if the f_{α}(x, y) are μ_{r}(K)-invariant. The function f_{α}(x, y) is μ_{r}(K)-invariant if and only if ϵ^{∗}f_{α}(x, y) ∈ C^{∞}(U) is invariant under the right-action of K on GL_{0}, i.e. if and only if ϵ^{∗}f_{α}(x, y) is a function only of ϵ^{∗}x. This shows that f ∈ A_{GL(V)}(U)^{K} if and only if the coefficients f_{α}(x, y) in the expansion (23) are functions only of x □

We continue the proof of Thm. 18. From Lemma 12 it follows that GL(V)/K is a supermanifold. In fact, the K-invariant local functions (x, ξ) on GL(V) constructed in that lemma induce local coordinates on GL(V)/K. Next we restrict the coordinates (x, y, ξ) to the submanifold G ↪ GL(V). We can decompose x = (x′, x′′), ξ = (ξ′, ξ′′) such that (x′, y, ξ′) restrict to local coordinates on G over some open neighborhood (again denoted by) U of e ∈ G_{0} (notice that, by construction, y restrict to local coordinates on K). Now, as in the proof of the corresponding statement for GL(V) (see Lemma 12), it follows that A_{G}^{K}(U) consists precisely of all functions of the form f(x′, ξ′) = ∑ f_{α}(x′)(ξ′)^{α}. This proves that G/K is a supermanifold with local coordinates (x′, ξ′) over U.

The inclusion A_{G}^{K} ⊂ A_{G} defines the canonical submersion π : G → G/K. Finally, if f is a μ_{r}(K)-invariant (local) function on G then μ^{∗}f is a (local) function on G×G invariant under the group Id_{G}×μ_{r}(K) ⊂ Aut(G×G). This shows that the composition G × G →^{μ} G →^{π} G/K factorizes to a morphism α : G × G/K → G/K, which defines an action of G on G/K. □

Definition 34. The supermanifold M = G/K is called the homogeneous supermanifold associated to the pair (G, K).

For the rest of the paper let 𝔤 ⊂ 𝔤𝔩(V) be a linear super Lie algebra, 𝔨 ⊂ 𝔤_{0} a subalgebra and 𝔤 = 𝔨 + 𝔪 a 𝔨-invariant direct decomposition compatible with the ℤ_{2}-grading. We denote by K ⊂ G_{0} ⊂ G ⊂ GL(V) the corresponding linear Lie supergroups (see Example 9) and assume that the (connected) Lie subgroups K ⊂ G_{0} ⊂ GL(V)_{0} are closed. Then, by Thm. 18, M = G/K is a supermanifold with a canonical action of G. We have the canonical identification 𝔪 ≅ 𝔤/𝔨 ≅ T_{eK}M given by x ↦ X(eK), where X(eK) is the value of the fundamental vector field X on M associated to x ∈ 𝔪 at the base point eK ∈ G_{0}/K = M_{0}. We claim that any ad_{𝔨}-invariant tensor S_{eK} over 𝔪 defines a correponding G-invariant (see Def.

31) tensor field on M such that S(eK) = S_{eK}. Here by a tensor over 𝔪 we mean an element of the full tensor superalgebra ⊗⟨𝔪, 𝔪^{∗}⟩ over 𝔪. In fact, for any tensor S_{eK} over 𝔪 there exists a corresponding left-invariant tensor field S on G such that S(eK) = S_{eK}. In order for S to define a tensor field on G/K it is necessary and sufficient that S is μ_{r}(K)-invariant, or, equivalently, that S_{eK} is ad_{𝔨}-invariant. In particular, we have the following proposition:

Proposition 16. Let g_{eK} be an ad_{𝔨}-invariant nondegenerate supersymmetric bilinear form on 𝔪. Then there exists a unique G-invariant pseudo-Riemannian metric g on M = G/K (see Def. 23 and Def. 31) such that g(eK) = g_{eK}. Let Q_{eK} be an ad_{𝔨}-invariant quaternionic structure on 𝔪 (i.e. ad : 𝔨 → 𝔤𝔩(𝔪) normalizes Q_{eK}). Then there exists a unique G-invariant almost quaternionic structure Q on M (see Def. 24 and Def. 31) such that Q(eK) = Q_{eK}.

Finally, we need to discuss G-invariant connections on M = G/K.

Definition 35. A connection ∇ on a homogeneous supermanifold M = G/K is called G-invariant if

L_{X}(∇_{Y} S) = ∇_{[X,Y]}S + (−1)^{X̃Ỹ} ∇_{Y} (L_{X}S)

for all vector fields X and Y on M.

Let ∇ be a connection on a supermanifold M. For any vector field X on M one defines the A_{M}(M_{0})-linear operator

L_{X} := L_{X} − ∇_{X} .

We denote by L_{X}(p) ∈ Endℝ T_{p}M its value at p ∈ M_{0}; it is defined by L_{X}(p)Y (p) = (L_{X}Y)(p) for all vector fields Y on M.

For a G-invariant connection ∇ on a homogeneous supermanifold M = G/K as above we define the Nomizu map L = L(∇) : 𝔤 → End(T_{eK}M), x ↦ L_{x}, by the equation

L_{x} := L_{X}(eK) ,

where X is the fundamental vector field on M associated to x ∈ 𝔤. The operators L_{x} ∈ End(T_{eK}M) will be called Nomizu operators. They have the following properties:

L_{x} = dρ(x) for all x ∈ 𝔨 (24)

and

(25) L_{Ad_{k}x} = ρ(k)L_{x}ρ(k)^{−1} for all x ∈ 𝔤 , k ∈ K ,

where ρ : K → GL(T_{eK}M) is the isotropy representation (under the identification T_{eK}M ≅ 𝔪 the representation ρ is identified with adjoint representation of K on 𝔪).

Conversely, any even linear map L : 𝔤 → End(T_{[e]}M) satisfying (24) and (25) is the Nomizu map of a uniquely defined G-invariant connection ∇ = ∇(L) on M. Its torsion tensor T and curvature tensor R are expressed at eK by:

T (πx, πy) = −(L_{x}πy − (−1)^{x̃ỹ}L_{y}πx + π[x, y])

and

R(πx, πy) = [L_{x}, L_{y}] + L_{[x,y]} , x, y ∈ 𝔤

where π : 𝔤 → T_{eK}M is the canonical projection

x ↦ πx = X(eK) = \frac{d}{dt}|_{t=0} exp(tx)K .

Suppose now that we are given a G-invariant geometric structure S on M (e.g. a G-invariant almost quaternionic structure Q) defined by a corresponding K-invariant geometric structure S_{eK} on T_{eK}M. Then a G-invariant connection ∇ preserves S if and only if the corresponding Nomizu operators L_{x}, x ∈ 𝔤, preserve S_{eK}. So to construct a G-invariant connection preserving S it is sufficient to find a Nomizu map L : 𝔤 → End(T_{eK}M) such that L_{x} preserves S_{e} for all x ∈ 𝔤. We observe that, due to the K-invariance of S_{e}, the Nomizu operators L_{x} preserve S_{e} already for x ∈ 𝔨. The above considerations can be specialized as follows:

Proposition 17. Let Q be a G-invariant almost quaternionic structure on a homogeneous supermanifold M = G/K. There is a natural one-to-one correspondence between G-invariant almost quaternionic connections on (M, Q) and Nomizu maps L : 𝔤 → End(T_{eK}M), whose image normalizes Q(eK), i.e. whose Nomizu operators L_{x}, x ∈ 𝔤, belong to the normalizer n(Q) ≅ 𝔰𝔭(1)⊕𝔤𝔩(d, ℍ) (d = (m+n)/4) of the quaternionic structure Q(eK) in the super Lie algebra 𝔤𝔩(T_{eK}M).

Corollary 10. Let (M = G/K, Q) be a homogeneous almost quaternionic supermanifold and L : 𝔤 → End(T_{eK}M) a Nomizu map such that

(1) L_{x}πy − (−1)^{x̃ỹ}L_{y}πx = −π[x, y] for all x, y ∈ 𝔤 (i.e. T = 0) and

(2) L_{x} normalizes Q(eK) ⊂ End(T_{eK}M). Then ∇(L) is a G-invariant quaternionic connection on (M, Q) and hence Q is 1-integrable.

For use in 4, we give the formula for the Nomizu map L^{g} associated to the Levi-Civita connection ∇^{g} of a G-invariant pseudo-Riemannian metric g on a homogeneous supermanifold M = G/K. Let ⟨·, ·⟩ = g(eK) be the K-invariant nondegenerate supersymmetric bilinear form on T_{eK}M induced by g (the value of g at eK). Then L_{x}^{g} ∈ End(T_{eK}M), x ∈ 𝔤, is given by the following Koszul type formula:

−2⟨L_{x}^{g}πy, πz⟩ = ⟨π[x, y], πz⟩ − ⟨πx, π[y, z]⟩ − (−1)^{x̃ỹ}⟨πy, π[x, z]⟩ , x, y, z ∈ 𝔤 . (26)

Corollary 11. Let (M = G/K, Q, g) be a homogeneous almost quaternionic (pseudo-) Hermitian supermanifold and assume that L_{x}^{g} normalizes Q(eK) for all x ∈ 𝔤. Then the Levi-Civita connection ∇^{g} = ∇(L^{g}) is a G invariant quaternionic connection on (M, Q, g) and hence (M, Q, g) is a quaternionic (pseudo-) Kähler supermanifold if dim M_{0} > 4.

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