Alekseevskian Spaces

Vicente Cortés^{1} Mathematisches Institut der Universität Bonn Beringstr. 1 53115 Bonn

Keywords: homogeneous quaternionic Kählermanifolds MSC: 53C25, 53C30, 53C35

1Fax:+49-228-737916; e-mail: V.Cortes@math.uni-bonn.de; supported by the SFB 256 “nichtlineare partielle Differentialgleichungen” and the Graduiertenkolleg Mathematik at Bonn University

Abstract

D.V. Alekseevskiĭ’s classification of quaternionic Kählerian solvmanifolds (1975) is completed, confirming results which were obtained recently by the theoretical physicists B. de Wit and A. Van Proeyen in the context of supergravity. It is further shown that an Alekseevskiĭ space is symmetric, if and only if its sectional curvature is non-positive. This is an analogue of a well known theorem due to J.E. D’Atri and I. Dotti Miatello, which characterizes bounded symmetric domains by non-positive curvature. (MSC: 53C25, 53C30, 53C35)

Introduction

In this paper we study (non-flat) quaternionic Kähler manifolds which admit a simply transitive and real solvable group of isometries. We call them Alekseevskiĭ spaces after D.V. Alekseevskiĭ, who classified them and hereby discovered the first non-symmetric examples of quaternionic Kählerian manifolds (s. [Al 5]). He conjectured that every homogeneous quaternionic Kähler manifold of negative scalar curvature should be Alekseevskian.

In the first chapter we give an introduction to the theory of Alekseevskian spaces established in [Al 5].

As was kindly pointed out to me by A. Swann, the theoretical physicists B. de Wit and A. Van Proeyen have recently discovered, in the context of supergravity, that Alekseevskiĭ’s classification is incomplete.

In the second chapter we complete the classification (s. Thm. II.28) in line with Alekseevskiĭ’s original programme, confirming their discovery, without recourse to supergravity. Furthermore, some basic facts concerning the structure of Alekseevskian Lie algebras are presented.

In the third chapter we investigate the curvature of the Alekseevskiĭ spaces. The main discovery is the fact that an Alekseevskiĭ space is symmetric, if and only if its sectional curvature is non-positive (s. Thm. III.10). This is an analogue of a well known theorem due to J.E. D’Atri and I. Dotti Miatello, which characterizes bounded symmetric domains by non-positive curvature.

Contents

Introduction i

I Preliminaries 1 I.1 Alekseevskian Lie Algebras . . . . . . . . . . . . . . . . . . . . 1 I.2 Structure of Alekseevskiĭ’s Classification . . . . . . . . . . . . 4 I.3 W-Spaces, V-Spaces and Isometric Mappings . . . . . . . . . . 10

II Classification and Basic Facts 15 II.1 Rank 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 II.2 Rank 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 II.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III Characterization of Symmetric Spaces 52

Chapter I

Preliminaries

In the first chapter we shall provide a frame of reference for the reader to fall back on in the course of this paper. We review [Al 5], where one finds a wealth of technical details, which we omit here. After presenting the fundamental objects, namely Alekseevskian Lie algebras, in sec. I.1, we elucidate the conceptional framework of their classification in sec. I.2. In the process, we go into the notion of Q-representation in detail. Finally, in the third section, we define Alekseevskiĭ’s W- and V-spaces and explain the connection with isometric mappings.

I.1 Alekseevskian Lie Algebras

Definition I.1 A quaternionic Kähler manifold of non-trivial scalar curvature is said to be Alekseevskian or an Alekseevskiĭ space, if it admits a simply transitive, real^{1} solvable group of isometries.

In [Al 5], the classification of Alekseevskian spaces is interpreted as classification problem for metric Lie algebras and, apart from the minor refinements contributed in Ch. II, it is carried out.

1A solvable Lie algebra s is said to be real (or completely [Al 5] §1) solvable, if the endomorphisms ad_{X} (X ∈ s) have only real eigenvalues. A Lie group is said to be real solvable, if its Lie algebra is real solvable.

Definition I.2 A metric Lie algebra is a pair (s, ⟨, ⟩) consisting of a Lie algebra s and a scalar product ⟨, ⟩ on s. An isomorphism of metric Lie algebras is a Lie algebra isomorphism which is, at the same time, an isometry of Euclidean vector spaces.

For every metric Lie algebra (s, ⟨, ⟩) there is a unique (connected) simply connected Lie group S with Lie algebra s and a unique left-invariant Riemannian metric which coincides with ⟨, ⟩ on T_{e}S ≅ s. Thus, to every metric Lie algebra we have assigned a simply connected Riemannian manifold. Conversely, a Riemannian manifold M which admits a simply transitive group S of isometries can be regarded as a Lie group with left-invariant metric and we can consider the corresponding metric Lie algebra (s, ⟨, ⟩). We express this fact in the following by saying that M admits the metric Lie algebra (s, ⟨, ⟩). According to Alekseevskiĭ, a Riemannian manifold M admits at most one real solvable metric Lie algebra^{2} ([Al 4] Thm. 1, cf. [G-W] 4.3 Thm., 5.2 Thm. and 5.3 Cor.).

In the following, s denotes a metric Lie algebra and M the corresponding simply connected Riemannian manifold. We can transfer the notion of Levi-Civita connection and, for that reason, all derived notions from M to s or, equivalently, we can define it by the following Koszul formula (X, Y, Z ∈ s):

2⟨∇_{X}Y, Z⟩ = ⟨[X, Y], Z⟩ − ⟨X, [Y, Z]⟩ − ⟨Y, [X, Z]⟩ .

Canonical definitions (cf. [Al 5] Def. 1.5-8.), which will be frequently used in the course of this paper, now follow.

Definition I.3 Let s be a metric Lie algebra. The Kostant algebra kos is the Lie subalgebra of so(s) generated by the endomorphisms ∇_{X} (X ∈ s).

The holonomy algebra hol is the Lie subalgebra of kos generated by the expressions of the type

[∇_{X_{1}}, … [∇_{X_{k}}, R (X_{k+1}, X_{k+2})] …] , X_{1}, X_{2}, … X_{k+2} ∈ s , k = 0, 1, 2, …

2In general, there may be more than one simply transitive group of motions.

Accordingly, hol is the Lie algebra generated by the covariant derivatives of arbitrary order of the curvature endomorphisms, i.e. the Lie algebra of the holonomy group of M (s. [Ko-No 1] Ch. III Thm. 9.2 and Ch. II Thm. 10.8).

Definition I.4 Let V be a Euclidean vector space. A complex structure on V is a skew endomorphism J satisfying J^{2} = −Id.

A quaternionic structure on V is a Lie subalgebra q of so(V) generated by two anticommuting complex structures J_{1} and J_{2}.

We denote the centralizers of J and q = span{J_{1}, J_{2}, J_{3}} (J_{3} := J_{1}J_{2}) in so(V) by zent(J) and zent(q) respectively.

Definition I.5 A metric Lie algebra s with complex structure J is said to be Kählerian, if

hol ⊂ zent(J) .

A metric Lie algebra s of dimension > 4 (resp. = 4) with quaternionic structure q is said to be quaternionic Kählerian, if

hol ⊂ q ⊕ zent(q)

(resp. if q annihilates the curvature tensor). Non-Abelian, real solvable, quaternionic Kählerian Lie algebras are said to be Alekseevskian.

In other words, s is Kählerian resp. quaternionic Kählerian resp. Alekseevskian^{3}, if M is.

Alekseevskian spaces are Einsteinian with ric < 0 (s. [Al 5] p. 305, cf. [C1] p. 11) and, according to [Al 5] Thm. 1.3, simply connected. In particular, the following theorem holds, which is the starting point for Alekseevskiĭ’s classification, that will be discussed in the next section.

Theorem I.1 (D.V. Alekseevskiĭ [Al 5] §1) There is a one-to-one correspondence between Alekseevskian spaces and Alekseevskian Lie algebras.

3The condition “non-Abelian” in the definition of Alekseevskian Lie algebras guarantees that s has non-vanishing scalar curvature (s. [Al 5] Cor. 1.5. and Cor. 1.2., cf. [C1] II.1.2 Satz 5).

I.2 Structure of Alekseevskiĭ’s Classification

Let now (s, ⟨, ⟩, q) be an Alekseevskian Lie algebra and S the corresponding Alekseevskiĭ space. Being a non-Ricci-flat quaternionic Kähler manifold, S is also locally de-Rham-irreducible (s. [Be] 14.45 Thm. b). Homogeneous^{4} locally de-Rham-irreducible spaces of non-vanishing Ricci curvature satisfy kos = hol, according to an old result of Kostant/Lichnerowicz [Kos]. This justifies the following decomposition (X ∈ s):

∇_{X} = ∑_{α=1}^{3} ω_{α}(X)J_{α} + ¯∇_{X} ,  (I.1)

where ω_{α} are linear forms on s and ¯∇_{X} ∈ zent(q). On the other hand, owing to the general structure theorem^{5} for algebraic curvature tensors of type sp(1) ⊕ sp(m) (m = dim_{H} s = (dim_{R} s)/4) we have:

R = sk_{r}R_{0} + R̄ ,  (I.2)

where R_{0} is the curvature tensor of the quaternionic projective space P_{H}^{m} and R̄ is of type sp(m). sk_{r} denotes the reduced scalar curvature of R, which coincides with the scalar curvature sk, up to a dimension factor:

sk = 4m(m + 2)sk_{r} .  (I.3)

Combining the equations (I.1) and (I.2) yields the fundamental structure and integrability identities [Al 5] (∗) and (∗∗). These make possible a first important step of the classification: the study of (quaternionic) one-dimensional quaternionic subalgebras.

Definition I.6 ([Al 5] Def. 1.9. and 2.1.)Let s be a metric Lie algebra. A linear subspace s′ of s is said to be totally geodesic, if ∇_{s′}s′ ⊂ s′.

Suppose, in addition, that s is provided with a quaternionic structure q. A subalgebra s′ of s is said to be a quaternionic subalgebra, if qs′ ⊂ s′.

4The notion of Kostant algebra can be easily defined for general homogeneous spaces, e.g. by an appropriate Koszul formula.

5s. [Al 1] Table 1 or in more detail: [Sal 2] 9.3 Prop. and [Sal 1] Thm. 3.1.

Theorem I.2 (D.V. Alekseevskiĭ [Al 5] Lemma 2.1., Remark 2.1. , Prop. 3.1.-3.2. and Def. 3.2.) Let s be an Alekseevskian Lie algebra.

(i) Every quaternionic subalgebra of s is totally geodesic and has the same reduced scalar curvature as s.

(ii) s contains a quaternionic one-dimensional quaternionic subalgebra (this is canonically associated to the choice of a real one-dimensional ideal).

(iii) There are (up to scaling) exactly two quaternionic one-dimensional Alekseevskian Lie algebras; in Alekseevskiĭ’s notation: A_{1}^{1} and C_{1}^{1}. The corresponding Alekseevskiĭ spaces are the complex hyperbolic plane of reduced scalar curvature sk_{r} = −1/2 and the quaternionic hyperbolic line (sk_{r} = −1).

(iv) There is (up to scaling) a unique Alekseevskian Lie algebra of quaternionic dimension m with one-dimensional quaternionic subalgebra C_{1}^{1}. The corresponding Alekseevskiĭ space is the quaternionic hyperbolic space H_{H}^{m}.

To describe Alekseevskian Lie algebras with one-dimensional quaternionic subalgebra A_{1}^{1} we need further definitions. In the following we denote, in contrast to [Al 5], metric Lie algebras and their linear subspaces generally by small letters; vectors by capital letters. Unless otherwise stated, we use the notation of [Al 5]. The symbol “+” between Euclidean vector spaces always stands for the orthogonal sum.

Definition I.7 ([Al 5] Def. 6.1-5.) A metric Lie algebra f = span{G, H} with complex structure J is said to be a key algebra with root μ, if G = JH and [H, G] = μG with μ > 0.

A metric Lie algebra f + x with complex structure J is said to be an elementary Kählerian Lie algebra, if f = span{G, H} is a key subalgebra with root μ, ad_{H}|x = \frac{μ}{2}Id, ad_{G}|x = 0 and [X, Y] = μ⟨JX, Y ⟩G for X, Y ∈ x ([Al 5] Def. 6.2. contains a misprint).

A representation U ↦ T_{U} of a Lie algebra u with complex structure J on a Euclidean vector space (x, ⟨, ⟩) with complex structure J_{1} is said to be symplectic, if

1. T (continued as derivation of the tensor algebra of x) annihilates the Kähler form ρ_{1} = ⟨J_{1}·, ·⟩ of J_{1}: T_{U}ρ_{1} = 0 for all U ∈ u;

2. T_{JU}^{sym} = J_{1}T_{U}^{sym} for all U ∈ u, where T_{U}^{sym} denotes the symmetric part of T_{U}.

T is non-degenerate, if T_{u}x = x. The representation space x of a nondegenerate symplectic representation T of a key algebra f = span{G, H} with root μ admits a decomposition x = x_{+} +x_{−} with the following properties ([Al 5] Prop. 6.1. 3)):

x_{−} = J_{1}x_{+} , T_{H}|x_{±} = ±\frac{μ}{2}Id , T_{G}|x_{+} = 0 and T_{G}|x_{−} = −μJ_{1} .

This is called the weight decomposition of x with respect to the nondegenerate symplectic representation T.

In [Al 5] §5 it is shown that an Alekseevskian Lie algebra s with onedimensional quaternionic subalgebra A_{1}^{1} is necessarily of the form s = u+J_{2}u (orthogonal sum), where (u, J_{1}|u) is a totally geodesic Kählerian subalgebra: the so-called principal Kählerian subalgebra of s (s. [Al 5] Def. 5.1.). u contains a key subalgebra f_{0} = span{G_{0}, H_{0}} with root 1 such that f_{0} + J_{2}f_{0} is the (canonical) one-dimensional quaternionic subalgebra of s and

u = f_{0} + {S ∈ f_{0}^{⊥}| ad_{H_{0}}S = 0} ,
u^{⊥} = \{S ∈ s |ad_{H_{0}}S = \frac{1}{2}S \} .

The adjoint representation of s induces a representation of u on u^{⊥} = J_{2}u (addition of roots), which represents the model for the next definition.

Definition I.8 ([Al 5] Def. 5.2-3.) A Kählerian Lie algebra (u, J) is admissible, if u = f_{0} + u_{0} is a direct orthogonal sum of a key algebra f_{0} = span{G_{0}, H_{0}} with root 1 and a real solvable Kählerian Lie algebra u_{0}.

A representation U ↦ T_{U} of such a Lie algebra u on a Euclidean vector space ũ together with a vector space isometry ϕ : u → ũ is said to be a Q-re-presentation, if there is a linear form ω on u_{0} such that, with the notation Ũ := ϕ(U) (U ∈ u), J_{1} = −ϕJϕ^{−1}, Ĵ|ũ_{0} = J_{1} and Ĵ|f̃_{0} = −J_{1}, the following is true:

1. T_{H_{0}} = \frac{1}{2}Id and T_{G_{0}} = 0 (Q1).

2. The endomorphisms T_{U} (U ∈ u) have only real eigenvalues (Q8).

3. The endomorphisms T_{U} (U ∈ u_{0}) annihilate the Kähler form ρ̂ = ⟨Ĵ·, ·⟩ of the complex structure Ĵ (Q7) and satisfy the following conditions Q2-6, where Q3 and Q6 are only imposed on U_{0} ∈ u_{0}. T_{U}^{sk} and T_{U}^{sym} denote the skew and symmetric parts of T_{U} respectively; ∇ the covariant derivative of u:

(Q2) T_{U} H̃_{0} = \frac{1}{2} Ũ − 2ω(U)G̃_{0},

T_{U} G̃_{0} = \frac{1}{2} J̃U + 2ω(U)H̃_{0};

(Q3) T_{U} Ũ_{0} ≡ \frac{1}{2}⟨U, U_{0}⟩H̃_{0} + \frac{1}{2}⟨JU, U_{0}⟩G̃_{0} (mod ũ_{0}); (Q4) T_{U}^{sk} = ϕ∇_{U}ϕ^{−1} + 2ω(U)J_{1}; (Q5) T_{JU}^{sym}|ũ_{0} = J_{1}T_{U}^{sym}|ũ_{0}; (Q6) T_{U}^{sym}Ũ_{0} − T_{U_{0}}^{sym}Ũ = ⟨JU, U_{0}⟩G̃_{0}.

The numbering Q1-8 refers to [Al 5] Lemma 5.5.

We indicate how to construct an Alekseevskian Lie algebra out of a Qrepresentation.

Theorem I.3 (D.V. Alekseevskiĭ [Al 5] Lemma 5.5. and Prop. 5.1.) Q-re-presentations of an admissible Kählerian Lie algebra (u, J) and Alekseevskian Lie algebras with principal Kählerian subalgebra (u, J) are related by the following natural correspondence.

If s is an Alekseevskian Lie algebra with principal Kählerian subalgebra u, then the representation of u on J_{2}u induced by the adjoint representation of s is a Q-representation with ϕ = J_{2}|u : u →^{∼} u^{⊥} (ω = ω_{1}|u_{0}, s. eq. I.1).

Conversely let (T, ϕ) be a Q-representation of an admissible Kählerian Lie algebra (u, J) on the Euclidean vector space ũ = ϕ(u). A quaternionic structure q = span{J_{α}|α = 1, 2, 3} on the Euclidean vector space s := u + ũ (orthogonal sum) is defined as follows:

J_{1}|u := J , J_{1}|ũ := −ϕJϕ^{−1} , J_{2}|u := ϕ , J_{2}|ũ := −ϕ^{−1} , J_{3} := J_{1}J_{2} .

The following conditions define the structure of a Lie algebra on s:

1. u is a subalgebra of s;

2. ad_{U}|ũ := T_{U} for all U ∈ u;

3. [Ũ, Ṽ] := ρ̂(Ũ, Ṽ)G_{0} for all U, V ∈ u.

(Sketch of) proof: We only comment on the second part of the construction. That the above arrangements turn s = u + ũ into a metric Lie algebra (Jacobi identity) with quaternionic structure q, follows from the fact that T is a representation and from the conditions Q1 and Q7 alone. s is real solvable, thanks to u_{0} being real solvable and to Q8. The remaining conditions (Q2-6) ensure that (s, q) is quaternionic Kählerian. □

Theorem I.3 makes it possible to apply the structure theory of real solvable Kählerian Lie algebras and their symplectic representations, developed by S.G. Gindikin, I.I. Pjateckiĭ-S̆apiro and E.B. Vinberg, to the classification problem in question. Admissible Kählerian Lie algebras u = f_{0} + u_{0} which admit a Q-representation are non-degenerate^{6} and, hence^{7}, decompose as semi-direct sum of elementary Kählerian Lie algebras u_{0} = ∑_{i≥1}(f_{i} + x_{i}), i.e. [f_{i} + x_{i}, f_{j} + x_{j}] ⊂ f_{j} + x_{j} for i ≥ j, with symplectic representations ad_{f_{i}}|x_{j} for i > j and commuting key algebras: [f_{i}, f_{j}] = 0 (i ≠ j).

Definition I.9 ([Al 6] Def. 3.1.) The rank of a solvable Lie algebra s is the dimension of a Cartan subalgebra^{8} of s. The rank of a Alekseevskiĭ space S is the rank of its Alekseevskian Lie algebra s.

6s. [Al 5] Def. 1.8. d), Def. 5.5. and Cor. 5.1.

7s. [Al 5] Prop. 6.2., which contains two misprints in 4).

8A Cartan subalgebra is a nilpotent subalgebra which coincides with its normalizer.

According to [Var] Lemma 4.1.1, Cartan subalgebras are necessarily maximally nilpotent.

Remark: The rank of u = f_{0} + ∑_{i≥1}(f_{i} + x_{i}) coincides with the number of key algebras of u, because a := span{H_{i}|i = 0, 1, …} is obviously an (Abelian) Cartan subalgebra of u. Due to T_{H_{0}} = \frac{1}{2}Id (Q1), it is even true that a = norm_{s}(a) such that a is also a Cartan subalgebra of s, whence u and s are of the same rank.

Alekseevskiĭ shows that the existence of a Q-representation of u imposes severe restrictions on the rank of u and on the allowed roots of the key algebras occurring in the decomposition of u_{0}. For instance, the roots μ_{i} can only be 1, 1/\sqrt{2} or 1/\sqrt{3}. Correspondingly Alekseevskiĭ distinguishes between three types of admissible Kählerian Lie algebras u: u is of type α (α = 1, 2, 3), if 1/\sqrt{α} is the smallest root.

Type 3: There is a unique Lie algebra of type 3 which admits a Q-re-presentation: u = f_{0} + f_{1}. The Q-representation is unique and yields the symmetric Alekseevskiĭ space G_{2}^{(2)}/SO(4) (s. [Al 5] Prop. 8.1., in which the the formulas for T_{H} are incorrectly printed).

Type 2: Lie algebras of type 2 which admit Q-representations must have rank 3. We shall see in sec. II.1 that there is a countably infinite family u(p) (p = 0, 1, …) of such Lie algebras and that any of them admits a unique Q-representation. The corresponding Alekseevskiĭ spaces will be denoted by T (p) and will be described in sec. II.1. In the original classification ([Al 5] §8 Prop. on p. 329) only T (0) ≅ SO_{0}(3, 4)/(SO(3) × SO(4)) occurs, due to an error in the calculations.

Type 1: Apart from the hermitian symmetric Alekseevskiĭ spaces ([Al 5] Prop. 9.1.), this class contains the Alekeevskian spaces of rank 4: the Wand V-spaces which will be reviewed in the next section.

The rank is well defined, as, according to [Bl] Cor. 2.3., the Cartan subalgebras of a solvable Lie algebra are conjugate.

I.3 W-Spaces, V-Spaces and isometric map-

pings

The following concept (going back to Pjateckiĭ-S̆apiro) from the theory of real solvable Kählerian Lie algebras is all-important for the classification of Kählerian Lie algebras of type 1 and Rank > 2 which admit Q-representa-tions.

Definition I.10 ([Al 5] Def. 6.7. and 9.1.) Let x, y and z be Euclidean vector spaces. A bilinear mapping ψ : x × z → y is said to be isometric, if

⟨ψ(X, Z), ψ(X, Z)⟩ = ⟨X, X⟩⟨Z, Z⟩

for all X ∈ x and Z ∈ z. Isometric mappings ψ : x × z → y and ψ′ : x′ × z′ → y′ are said to be equivalent, if there are isomorphisms σ : x → x′, τ : z → z′ and υ : y → y′ of Euclidean vector spaces such that the following diagram commutes:

x × z ⟶^{ψ} y
↓ σ×τ  ↓ υ
x′ × z′ ⟶^{ψ′} y′

An isometric mapping ψ : x × z → y with k = dim x ≠ 0 is said to be special, if dim y = dim z ≠ 0. k is called the order of the special isometric mapping.

As Alekseevskiĭ notices, there is an intimate relation between special isometric mappings and Z_{2}-graded Clifford modules. This will be discussed in great detail in sec. II.2.

Let x_{−}, z_{−} and y_{−} be Euclidean vector spaces. Every isometric mapping ψ : x_{−} × z_{−} → y_{−} defines a Kählerian Lie algebra u(ψ) = (f_{0} + u_{0}, J) of type 1 and rank 4 by means of the following recipe ([Al 5] Prop. 9.3.).

1. u_{0} is a semi-direct sum u_{0} = (f_{1} + x_{1}) + (f_{2} + x_{2}) + f_{3} of elementary Kählerian Lie algebras with commuting key algebras with root 1.

2. x_{1} admits a (J-invariant) decomposition x_{1} = y + z such that the following is true for x := x_{2}, y and z: ad_{f_{3}}|y, ad_{f_{2}}|z and ad_{f_{3}}|x are non-degenerate symplectic representations with weight decompositions y = y_{+} + y_{−}, z = z_{+} + z_{−} and x = x_{+} + x_{−}, where y_{+} = Jy_{−}, z_{+} = Jz_{−} and x_{+} = Jx_{−}. Furthermore:

[f_{1}, x] = [f_{2}, y] = [f_{3}, z] = [y, z] = [x, z_{+}] = [x_{+}, y_{+}] = [x_{−}, y_{−}] = 0 .

3. The remaining Lie brackets are computed according to the rules^{9} (X ∈ x, X_{±} ∈ x_{±}, Y_{±} ∈ y_{±} and Z_{±} ∈ z_{±}):

[X_{−}, Z_{−}] = \frac{1}{\sqrt{2}}ψ(X_{−}, Z_{−}) , [JX, Z_{−}] = J[X, Z_{−}] ,

[x_{−}, y_{+}] ⊂ z_{+} , [X_{+}, Y_{−}] = [JX_{+}, JY_{−}] and

⟨[X_{−}, Y_{+}], Z_{+}⟩ = −\frac{1}{\sqrt{2}}⟨JY_{+}, ψ(X_{−}, JZ_{+})⟩ .

Theorem I.4 (D.V. Alekseevskiĭ [Al 5] Prop. 9.2-4.) The Kählerian Lie algebras of type 1 and rank > 2 which admit a Q-representation are given by the Lie algebras u = u(ψ) for which either

(i) x_{−} = 0 (hence ψ = 0) and u = u(p, q) ≅ u(q, p) is completely determined by the parameters p = dim y_{−} and q = dim z_{−} or

(ii) ψ is a special isometric mapping.

Every such Lie algebra u has a unique Q-representation T. The corresponding Alekseevskiĭ spaces are denoted by W(p, q) and V(ψ) in the cases (i) and (ii) respectively.

The definition of the Q-representation T of u(ψ) now follows. Set f := ∑_{i=0}^{3} f_{i}. The operators T_{H_{α}}|f̃ and T_{G_{α}}|f̃ are given, with respect to the orthonormal basis

P̃_{0} := \frac{1}{2} (H̃_{0} + H̃_{1} + H̃_{2} + H̃_{3}) ,
P̃_{α} := \frac{1}{2} (−H̃_{0} − H̃_{α} + H̃_{β} + H̃_{γ}) ,

9The factor 1/\sqrt{2} is apparently missing in [Al 5] Prop. 9.3.

Q̃_{i} := ĴP̃_{i} , ({α, β, γ} = {1, 2, 3} , i ∈ {0, 1, 2, 3})

of f̃, by the following formulas:

T_{H_{α}}|span \{P̃_{0}, P̃_{α}, Q̃_{β}, Q̃_{γ}\} = \frac{1}{2}Id ,
T_{H_{α}}|span \{Q̃_{0}, Q̃_{α}, P̃_{β}, P̃_{γ}\} = −\frac{1}{2}Id ,

T_{G_{α}} : \{ \begin{matrix}Q̃_{0} ↦ & P̃_{α} ↦ & 0 , \\ Q̃_{α} ↦ & P̃_{0} ↦ & 0 , \\ P̃_{β} ↦ & Q̃_{γ} ↦ & 0 . \end{matrix}

T_{f_{1}}|x̃, T_{f_{2}}|ỹ and T_{f_{3}}|z̃ are non-degenerate symplectic representations with weight decompositions x̃ = x̃_{+} + x̃_{−}, ỹ = ỹ_{+} + ỹ_{−} and z̃ = z̃_{+} + z̃_{−} and

T_{f_{1}} (ỹ + z̃) = T_{f_{2}} (x̃ + z̃) = T_{f_{3}} (x̃ + ỹ) = 0 .

In the following, let X_{±}, Y_{±} and Z_{±} denote arbitrary elements and (X_{±}^{i})_{i}, (Y_{±}^{i})_{i} and (Z_{±}^{i})_{i} arbitrary orthonormal bases of x_{±}, y_{±} and z_{±} respectively. U ∘ V := 2∇_{U}V defines a bilinear mapping from the product of two of the spaces x, y and z to the third. With these conventions the expressions^{10} for the remaining operators T_{U} (U ∈ y + z + x) read

T_{X_{+}} = P̃_{0} ⊗ X̃_{+} − X̃_{+} ⊗ P̃_{1} + J̃X_{+} ⊗ Q̃_{0} − Q̃_{1} ⊗ J̃X_{+}

+ ∑_{i} X̃_{+} ∘ Y_{−}^{i} ⊗ Ỹ_{−}^{i} + ∑_{i} X̃_{+} ∘ Z_{−}^{i} ⊗ Z̃_{−}^{i} ,

T_{Y_{+}} = P̃_{0} ⊗ Ỹ_{+} − Ỹ_{+} ⊗ P̃_{2} + J̃Y_{+} ⊗ Q̃_{0} − Q̃_{2} ⊗ J̃Y_{+}

+ ∑_{i} Ỹ_{+} ∘ X_{−}^{i} ⊗ X̃_{−}^{i} + ∑_{i} Ỹ_{+} ∘ Z_{−}^{i} ⊗ Z̃_{−}^{i} ,

T_{Z_{+}} = P̃_{0} ⊗ Z̃_{+} − Z̃_{+} ⊗ P̃_{3} + J̃Z_{+} ⊗ Q̃_{0} − Q̃_{3} ⊗ J̃Z_{+}

+ ∑_{i} Z̃_{+} ∘ X_{−}^{i} ⊗ X̃_{−}^{i} + ∑_{i} Z̃_{+} ∘ Y_{−}^{i} ⊗ Ỹ_{−}^{i} ,

T_{X_{−}} = −P̃_{2} ⊗ X̃_{−} − X̃_{−} ⊗ P̃_{3} − J̃X_{−} ⊗ Q̃_{2} − Q̃_{3} ⊗ J̃X_{−}

+ ∑_{i} X̃_{−} ∘ Y_{−}^{i} ⊗ Ỹ_{−}^{i} + ∑_{i} X̃_{−} ∘ Z_{+}^{i} ⊗ Z̃_{+}^{i} ,

10Let V and W be vectors of a Euclidean vector space. V ⊗W denotes, for the remainder of this paper, the linear mapping S ↦ V ⟨W, S⟩. Its domain should be clear from the corresponding context. In the above formulas, e.g., it is ũ.

T_{Y_{−}} = −P̃_{1} ⊗ Ỹ_{−} − Ỹ_{−} ⊗ P̃_{3} − J̃Y_{−} ⊗ Q̃_{1} − Q̃_{3} ⊗ J̃Y_{−}

+ ∑_{i} Ỹ_{−} ∘ X_{−}^{i} ⊗ X̃_{−}^{i} + ∑_{i} Ỹ_{−} ∘ Z_{+}^{i} ⊗ Z̃_{+}^{i} ,

T_{Z_{−}} = −P̃_{1} ⊗ Z̃_{−} − Z̃_{−} ⊗ P̃_{2} − J̃Z_{−} ⊗ Q̃_{1} − Q̃_{2} ⊗ J̃Z_{−}

+ ∑_{i} Z̃_{−} ∘ X_{−}^{i} ⊗ X̃_{−}^{i} + ∑_{i} Z̃_{−} ∘ Y_{+}^{i} ⊗ Ỹ_{+}^{i} .

Remark: The bilinear mapping “∘” introduced above is completely determined by the following relations^{11}, which are obtained from the Koszul formula. The first three formulas are, as usual, to be read selecting either all the upper or all the lower signs and ϵ ∈ {−, +}.

X_{ϵ} ∘ Y_{±ϵ} = ±Y_{±ϵ} ∘ X_{ϵ} ∈ z_{∓} ,
X_{ϵ} ∘ Z_{±ϵ} = ±ϵZ_{±ϵ} ∘ X_{ϵ} ∈ y_{∓} ,
Y_{ϵ} ∘ Z_{±ϵ} = Z_{±ϵ} ∘ Y_{ϵ} ∈ x_{∓} ,
⟨X_{+} ∘ Y_{+}, Z_{−}⟩ = −\frac{1}{\sqrt{2}}⟨JY_{+}, ψ(JX_{+}, Z_{−})⟩ ,
⟨X_{−} ∘ Y_{−}, Z_{−}⟩ = −\frac{1}{\sqrt{2}}⟨Y_{−}, ψ(X_{−}, Z_{−})⟩ ,
⟨X_{+} ∘ Y_{−}, Z_{+}⟩ = \frac{1}{\sqrt{2}}⟨Y_{−}, ψ(JX_{+}, JZ_{+})⟩ ,
⟨X_{−} ∘ Y_{+}, Z_{+}⟩ = −\frac{1}{\sqrt{2}}⟨JY_{+}, ψ(X_{−}, JZ_{+})⟩ ,
⟨X_{+} ∘ Z_{+}, Y_{−}⟩ = −\frac{1}{\sqrt{2}}⟨Y_{−}, ψ(JX_{+}, JZ_{+})⟩ ,
⟨X_{−} ∘ Z_{−}, Y_{−}⟩ = \frac{1}{\sqrt{2}}⟨Y_{−}, ψ(X_{−}, Z_{−})⟩ ,
⟨X_{+} ∘ Z_{−}, Y_{+}⟩ = \frac{1}{\sqrt{2}}⟨JY_{+}, ψ(JX_{+}, Z_{−})⟩ ,
⟨X_{−} ∘ Z_{+}, Y_{+}⟩ = \frac{1}{\sqrt{2}}⟨JY_{+}, ψ(X_{−}, JZ_{+})⟩ ,
⟨Y_{+} ∘ Z_{+}, X_{−}⟩ = −\frac{1}{\sqrt{2}}⟨JY_{+}, ψ(X_{−}, JZ_{+})⟩ ,
⟨Y_{−} ∘ Z_{−}, X_{−}⟩ = \frac{1}{\sqrt{2}}⟨Y_{−}, ψ(X_{−}, Z_{−})⟩ ,

11The last formula in 1) [Al 5] p. 334 contains a misprint and the factor 1/\sqrt{2} is missing at the end of the computation in 3) [Al 5] p. 334.

⟨Y_{+} ∘ Z_{−}, X_{+}⟩ = \frac{1}{\sqrt{2}}⟨JY_{+}, ψ(JX_{+}, Z_{−})⟩ ,
⟨Y_{−} ∘ Z_{+}, X_{+}⟩ = \frac{1}{\sqrt{2}}⟨Y_{−}, ψ(JX_{+}, JZ_{+})⟩ .

The classification of isometry classes of Alekseevskian V-spaces can now be reduced to the classification of equivalence classes of special isometric mappings (s. Prop. II.23). The latter and its consequences will be presented in sec. II.2. This will correct/refine [Al 5] Thm. 10.1.

Chapter II

Classification and Basic Facts

The theoretical physicists de Wit and Van Proeyen have discovered that Alekseevskiĭ’s classification [Al 5] is not complete^{1}. In [dW-VP] they classify certain homogeneous N = 2 supergravity models in d = 5 spacetime dimensions. Owing to a connection between N = 2 supergravity and Alekseevskiĭ spaces, which was used earlier by Cecotti [Ce], they conclude from the result of their classification that there should exist Alekseevskian spaces missing in [Al 5]. Furthermore they conjecture that the absence of these spaces is merely due to an error in the calculations of [Al 5].

Motivated by [dW-VP] we have examined [Al 5] with the intention of smoothing out possible errors and thus completing the original classification. Our results confirm de Wit’s and Van Proeyen’s discovery.

We found that the classification of Alekseevskian spaces of rank 3 is indeed flawed by a miscalculation. Correcting it and completing the classification guided by Alekseevskiĭ’s example one arrives at the result that the original classification includes only the first member of a family T (p), p = 0, 1, 2 …, of rank 3 spaces (s. sec. II.1).

The classification of rank 4 spaces is slightly obscured by a subtle conceptual error which leads to the identification of non-isomorphic spaces (s. sec. II.2).

We give the new examples explicitly as Alekseevskian Lie algebras, in contrast to [dW-VP], where the corresponding N = 2 supergravity theories in d = 5 are discussed.

1A very rough sketch of their argument is given in the appendix of [C2].

II.1 Rank 3

Alekseevskiĭ’s classification of the Kählerian Lie algebras of type 2 that admit a Q-representation (s. [Al 5] §8 Prop. on p. 329) is not complete. Before completing it by Prop. II.1, we remind the reader of the pertinent (correct) results obtained in [Al 5] §8 and point out the error in computation which cut short Alekseevskiĭ’s arguments.

From [Al 5] §8 we know that every Kählerian Lie algebra (u, J) of type 2 which posses a Q-representation is of the form u = f_{0} + u_{0}, where u_{0} is a semi-direct sum of two elementary Kählerian Lie algebras f_{1} + x_{1} and f_{2}, [f_{1}, f_{2}] = 0 and ad_{f_{2}}|x_{1} is a non-degenerate symplectic representation. One of the two key algebras f_{1}, f_{2} has root 1 and will be denoted by f′, in accordance with [Al 5] §8, the other has root 1/\sqrt{2} and will be denoted by f. Under the assumption that u has a Q-representation T, it is shown in [Al 5] §8 that the T_{U}|x̃_{1}^{⊥} for U ∈ f_{0} + f_{1} + f_{2} are uniquely determined by the properties of a Q-representation. With respect to the orthonormal basis

P̃_{+} := \frac{1}{2} (H̃_{0} + H̃′ + \sqrt{2}H̃) , Q̃_{+} := ĴP̃_{−} = \frac{1}{2} (G̃_{0} − G̃′ + \sqrt{2}G̃) ,

P̃_{0} := \frac{1}{\sqrt{2}} (G̃_{0} + G̃′) , Q̃_{0} := ĴP̃_{0} = \frac{1}{\sqrt{2}} (−H̃_{0} + H̃′) ,

P̃_{−} := \frac{1}{2} (H̃_{0} + H̃′ − \sqrt{2}H̃) , Q̃_{−} := ĴP̃_{+} = \frac{1}{2} (G̃_{0} − G̃′ − \sqrt{2}G̃)

of f̃_{0} + f̃_{1} + f̃_{2} they are given by the following formulas, valid on x̃_{1}^{⊥}.

T_{H′} = \frac{1}{2} (P̃_{+} ⊗ P̃_{+} + P̃_{0} ⊗ P̃_{0} + P̃_{−} ⊗ P̃_{−})

− \frac{1}{2} (Q̃_{+} ⊗ Q̃_{+} + Q̃_{0} ⊗ Q̃_{0} + Q̃_{−} ⊗ Q̃_{−}) ,

T_{G′} = −P̃_{0} ⊗ Q̃_{0} − P̃_{+} ⊗ Q̃_{+} − P̃_{−} ⊗ Q̃_{−} ,
T_{H} = \frac{1}{\sqrt{2}} (P̃_{+} ⊗ P̃_{+} − P̃_{−} ⊗ P̃_{−} + Q̃_{+} ⊗ Q̃_{+} − Q̃_{−} ⊗ Q̃_{−}) ,  (II.1)

T_{G} = Q̃_{0} ⊗ Q̃_{−} − Q̃_{+} ⊗ Q̃_{0} + P̃_{0} ⊗ P̃_{−} − P̃_{+} ⊗ P̃_{0} .

In addition, it is shown that T_{f′} x̃_{1} = 0. Finally, T_{H_{0}} = \frac{1}{2}Id and T_{G_{0}} = 0 holds for every Q-representation (Q1).

Under point 8. on p. 328 [Al 5] it is (erroneously) claimed that the above formulas cannot be extended to define a Q-representation unless x := x_{1} = 0. Accordingly Alekseevskiĭ finds only one Kählerian Lie algebra of type 2 which has a Q-representation (defined by the above formulas). The proof of this claim contains a miscalculation. The first line of the computation^{2} should read:

0 = (T_{G_{i}}T_{X_{+}} − T_{X_{+}}T_{G_{i}}) G̃_{0} = \frac{1}{2} (T_{G_{i}} J̃_{1}X_{+} + T_{X_{+}} H̃_{i}) − ω (G_{i}) X̃_{+} .

Therefore, the last line should read:

T_{G_{i}} J̃_{1}X_{+} + T_{H_{i}} X̃_{+} = \frac{μ_{i}}{2} X̃_{+} + 2ω (G_{i}) X̃_{+} = \frac{1}{2} (μ_{i} − \frac{1}{μ_{i}} ) X̃_{+} .

For f_{i} = f′, i. e. μ_{i} = 1, this does not imply anything new, since we know already that T_{f′} x̃ = 0. In particular, we cannot conclude that x = 0. The proposition on p. 329 in [Al 5] §8 should be replaced with the following.

Proposition II.1 Every Kählerian Lie algebra (u, J) of type 2 which admits a Q-representation is (up to scaling) of the form u = f_{0} + u_{0}, where

u_{0} = f_{1} + x_{1} + f_{2}

is a semi-direct orthogonal sum of the elementary Kählerian Lie algebras f_{1} + x_{1} and f_{2} with root 1 and 1/\sqrt{2} respectively. The key algebras f_{i} = span {G_{i}, H_{i}} commute. ad_{f_{2}}|x_{1} is a non-degenerate symplectic representation with weight decomposition x_{1} = x_{+} + x_{−}, x_{−} = Jx_{+}.

u has a unique Q-representation T : u → End(ũ). Let ∼: u → ũ denote, as usual, the corresponding isometry of Euclidean vector spaces. Then, with respect to the orthonormal basis

P̃_{+} := \frac{1}{2} (H̃_{0} + H̃_{1} + \sqrt{2}H̃_{2}) , Q̃_{+} := ĴP̃_{−} = \frac{1}{2} (G̃_{0} − G̃_{1} + \sqrt{2}G̃_{2}) ,

2Apart from condition Q2 of the definition of Q-representation (Def. I.8) we only use that ω(X_{+}) = 0 and ω(G_{i}) = −1/(4μ_{i}), which follows from the structure equation 4dω = ρ = ⟨J·, ·⟩ (s. [Al 5] Lemma 5.3.).

P̃_{0} := \frac{1}{\sqrt{2}} (G̃_{0} + G̃_{1}) , Q̃_{0} := ĴP̃_{0} = \frac{1}{\sqrt{2}} (−H̃_{0} + H̃_{1}) ,

P̃_{−} := \frac{1}{2} (H̃_{0} + H̃_{1} − \sqrt{2}H̃_{2}) , Q̃_{−} := ĴP̃_{+} = \frac{1}{2} (G̃_{0} − G̃_{1} − \sqrt{2}G̃_{2})

of f̃_{0} + f̃_{1} + f̃_{2}, T is given by

T_{H_{0}} = \frac{1}{2}Id , T_{G_{0}} = 0 ,

T_{H_{1}} = \frac{1}{2} (P̃_{+} ⊗ P̃_{+} + P̃_{0} ⊗ P̃_{0} + P̃_{−} ⊗ P̃_{−})

− \frac{1}{2} (Q̃_{+} ⊗ Q̃_{+} + Q̃_{0} ⊗ Q̃_{0} + Q̃_{−} ⊗ Q̃_{−}) ,

T_{G_{1}} = −P̃_{0} ⊗ Q̃_{0} − P̃_{+} ⊗ Q̃_{+} − P̃_{−} ⊗ Q̃_{−} ,
T_{H_{2}}|x̃_{1}^{⊥} = \frac{1}{\sqrt{2}} (P̃_{+} ⊗ P̃_{+} − P̃_{−} ⊗ P̃_{−} + Q̃_{+} ⊗ Q̃_{+} − Q̃_{−} ⊗ Q̃_{−}) ,
T_{G_{2}}|x̃_{1}^{⊥} = Q̃_{0} ⊗ Q̃_{−} − Q̃_{+} ⊗ Q̃_{0} + P̃_{0} ⊗ P̃_{−} − P̃_{+} ⊗ P̃_{0} ,

T_{H_{2}}|x̃_{+} = \frac{\sqrt{2}}{4}Id , T_{H_{2}}|x̃_{−} = −\frac{\sqrt{2}}{4}Id ,

T_{G_{2}}|x̃_{+} = 0 , T_{G_{2}}|x̃_{−} = −\frac{\sqrt{2}}{2}J_{1} ,

T_{X_{+}} = P̃_{+} ⊗ X̃_{+} + \frac{1}{\sqrt{2}}P̃_{0} ⊗ J̃X_{+} + J̃X_{+} ⊗ Q̃_{−} − \frac{1}{\sqrt{2}}X̃_{+} ⊗ Q̃_{0} ,
T_{X_{−}} = P̃_{−} ⊗ X̃_{−} + \frac{1}{\sqrt{2}}P̃_{0} ⊗ J̃X_{−} + J̃X_{−} ⊗ Q̃_{+} − \frac{1}{\sqrt{2}}X̃_{−} ⊗ Q̃_{0} .

In particular, T_{f_{2}}|x̃_{1} is a non-degenerate symplectic representation with weight decomposition x̃_{1} = x̃_{+} + x̃_{−}.

Proof: We take up our previous exposition. Next, we show that the conditions Q1-8 for Q-representations uniquely determine T_{X} and T_{F} X̃ for X ∈ x_{1} and F ∈ f. First, we compute the covariant derivative ∇_{X} of u for X ∈ x_{1} = x_{+} + x_{−}:

∇_{X_{±}} = \frac{1}{2} (μ_{1}H_{1} ± μ_{2}H_{2}) ∧ X_{±} + \frac{1}{2} (μ_{1}G_{1} ± μ_{2}G_{2}) ∧ JX_{±} .

Where μ_{1}, μ_{2} ∈ \{1, 1/\sqrt{2}\} are the corresponding roots of the key algebras f_{1}, f_{2}. V ∧ W (V, W ∈ u) denotes the skew endomorphism V ⊗ W − W ⊗ V of u (s. Ch. I footnote 10). From this we obtain immediately the skew part of the operators T_{X} by Q4 observing that ω(X) = 0 (s. structure equation in footnote 2).

T_{X_{±}}^{sk} = \frac{1}{2} (μ_{1}H̃_{1} ± μ_{2}H̃_{2}) ∧ X̃_{±} + \frac{1}{2} (μ_{1}G̃_{1} ± μ_{2}G̃_{2}) ∧ J̃X_{±}  (II.2)

Now we can already compute T_{F} for F ∈ f:

T_{H} X̃_{±} =^{(a)} T_{H}^{sym}X̃_{±} =^{Q6} T_{X_{±}}^{sym}H̃ =^{(a)} c_{±} [T_{H}, T_{X_{±}}^{sk}] H̃
= c_{±} \{T_{H} (−\frac{1}{c_{±}} X̃_{±}) − T_{X_{±}}^{sk} \frac{1}{2} (H̃_{0} + H̃′)\}
= −T_{H} X̃_{±} − \frac{c_{±}}{2} T_{X_{±}}^{sk}H̃′ ,

where

c_{±} = \{ \begin{matrix}\frac{4}{\sqrt{2}} , if & f = f_{1} \\ ±\frac{4}{\sqrt{2}} , if & f = f_{2} . \end{matrix}

(a) holds, because due to [Al 5] §7 A1 and Prop. 7.1. d) we have T_{H_{1}} = T_{H_{1}}^{sym}. We conclude

8T_{H} X̃_{±} = −2c_{±}T_{X_{±}}^{sk}H̃′ = \{ \begin{matrix}±c_{±}X̃_{±} , if & f = f_{1} \\ c_{±}X̃_{±} , if & f = f_{2} \end{matrix}
= ±2\sqrt{2}X̃_{±} .  (II.3)

According to [Al 5] Table A Cor. 7.2,

T_{G}^{sk} = −\frac{\sqrt{2}}{4}J_{1} on x̃_{1} .

Q5 and equation (II.3) give

T_{G}^{sym}X̃_{±} = J_{1}T_{H}^{sym}X̃_{±} =^{(a)} ±\frac{\sqrt{2}}{4}J_{1}X̃_{±} .

Adding these two expressions yields

T_{G}|x̃_{+} = 0 , T_{G}|x̃_{−} = −\frac{\sqrt{2}}{2}J_{1} .

The operators T_{X}^{sym} can now be computed using the following equation

T_{X}^{sym} = \frac{2}{μ_{1}} T_{[H_{1},X]}^{sym} =^{(a)} \frac{2}{μ_{1}} [T_{H_{1}}, T_{X}^{sk}] .  (II.4)

We have to discuss two cases.

1) First, we consider the case when f = f_{1} and f′ = f_{2}. In this case we will see that u has no Q-representation, if x_{1} ≠ 0.

With the help of the equations II.2, II.4 and the knowledge of T_{H_{1}} = T_{H} (equations II.1 and II.3) we compute:

T_{X_{+}} = −\frac{\sqrt{2}}{2}X̃_{+} ⊗ Q̃_{0} + J̃X_{+} ⊗ Q̃_{−} + P̃_{+} ⊗ X̃_{+} + \frac{\sqrt{2}}{2}P̃_{0} ⊗ J̃X_{+}
T_{JX_{+}} = J̃X_{+} ⊗ P̃_{−} − \frac{\sqrt{2}}{2}X̃_{+} ⊗ P̃_{0} − \frac{\sqrt{2}}{2}Q̃_{0} ⊗ J̃X_{+} − Q̃_{+} ⊗ X̃_{+} .

From this it follows that

[T_{X_{+}}, T_{JX_{+}}] J̃X_{+} = X̃_{+} .

But T being a representation implies

[T_{X_{+}}, T_{JX_{+}}] J̃X_{+} = T_{[X_{+},JX_{+}]}J̃X_{+} = \frac{\sqrt{2}}{2}T_{G_{1}} J̃X_{+} = −\frac{1}{2}X̃_{+} .

This is only possible, if x_{1} = 0.

2) Suppose now that f = f_{2} and f′ = f_{1}. From the equations II.2, II.4 and the knowledge of T_{H_{1}} = T_{H′} we obtain this time

T_{X_{±}} = P̃_{±} ⊗ X̃_{±} + \frac{\sqrt{2}}{2}P̃_{0} ⊗ J̃X_{±} + J̃X_{±} ⊗ Q̃_{∓} − \frac{\sqrt{2}}{2}X̃_{±} ⊗ Q̃_{0} .

Choose a Euclidean vector space ũ isometric to u and an isometry ∼: u → ũ. Then the formulas just derived for T define a linear mapping T : u → End(ũ). It is straightforward to check that T is a representation and that it fulfills the conditions Q1 and Q7. Therefore, we can make a metric Lie algebra out of t := u + ũ using the procedure explained in Thm. I.3. According to Thm. I.3, T is a Q-representation, if and only if (t, q) is an Alekseevskian Lie algebra. The latter statement is contained in the more precise information which we will derive in the following (s. Cor. II.4 and II.10 (ii)). □

Definition II.1 Let u = f_{0} + f_{1} + x_{1} + f_{2} be an admissible Kählerian Lie algebra as in Prop. II.1, p = dim_{C} x_{1} = dim_{R} x_{+} ∈ {0, 1, 2, …} and T : u → End(ũ) the representation given there. We will denote by (t(p), ⟨, ⟩, q) or short by (t, q) the metric Lie algebra with quaternionic structure defined at the end of the proof of Prop. II.1 on u + ũ with the help of T. The corresponding family of simply connected Lie groups with left-invariant metric will be denoted by T (p) (p = 0, 1, 2, …), and we will speak of T -spaces.

Proposition II.2 t = t(p) is 4-step solvable. The derived Lie algebra n = [t, t] is 7-step nilpotent, if p ≠ 0. For t(0) it is 5-step nilpotent.

Proof: Derived series D^{j+1}t = [D^{j}t, D^{j}t] (D^{0}t = t, j = 0, 1, 2, …) of t:

D^{1}t = n = a^{⊥} , where a = span{H_{i}|i = 0, 1, 2} .

For p ≠ 0 we get

D^{2}t = [n, n] = span {G_{0}, G_{1}} + x_{+} + span {P̃_{0}, P̃_{+}, P̃_{−}, Q̃_{0}, Q̃_{+}} + x̃_{1} ,

D^{3}t = span {G_{0}} + span {P̃_{0}, P̃_{+}, P̃_{−}} + x̃_{+} , D^{4}t = 0 .

For p = 0 we obtain

D^{2}t = span {G_{0}} + span {P̃_{0}, P̃_{+}, P̃_{−}, Q̃_{0}, Q̃_{+}} ,
D^{3}t = span {G_{0}} , D^{4}t = 0 .

Central series Z^{j+1}n = [n, Z^{j}n] (Z^{0}n = n, j = 0, 1, 2, …) of n:

Z^{1}n = [n, n] = D^{2}t .

For p ≠ 0:

Z^{2}n = [n, Z^{1}n] = span {G_{0}, G_{1}} + span {P̃_{0}, P̃_{+}, P̃_{−}, Q̃_{+}} + x̃_{1} ,
Z^{3}n = span {G_{0}} + span {P̃_{0}, P̃_{+}, P̃_{−}} + x̃_{+} ,

Z^{4}n = span {G_{0}} + span {P̃_{0}, P̃_{+}} ,

Z^{5}n = span \{G_{0}\} + span \left\{{ \atop P̃_{+}}\right\} ,

Z^{6}n = span {G_{0}} ,
Z^{7}n = 0 .

For p = 0:

Z^{2}n = span {G_{0}} + span {P̃_{0}, P̃_{+}, Q̃_{+}} ,

Z^{3}n = span \{G_{0}\} + span \left\{{ \atop P̃_{+}}\right\} ,

Z^{4}n = span {G_{0}} ,
Z^{5}n = 0 . □

Definition II.2 (s. [Wolt 2] Introduction) A solvable Lie algebra s is of Iwasawa type, if:

1. s = a + n, where a is an Abelian subalgebra and n = [s, s].

2. All ad_{A}, A ∈ a are semisimple.

3. There is an A^{0} ∈ a such that ad_{A^{0}}|n has only eigenvalues with positive real part.

Remark: The classic example of a Lie algebra of Iwasawa type is the solvable part of the Iwasawa decomposition of the isometry algebra of a symmetric space of non-compact type (cf. [He] Ch. VI §3, Ch. VII Lemma 2.20 and Ch. IX §1).

Proposition II.3 t(p) is of Iwasawa type, and all operators ad_{A}, A ∈ a, are diagonalizable over R.

Proof: We set a = span {H_{i}|i = 0, 1, 2}. The proof of Prop. II.2 shows that t is solvable and that n = a^{⊥}. Let A ∈ a, i.e. A = ∑_{i=0}^{2} a_{i}H_{i}, a_{i} ∈ R, and let μ_{i} ∈ \{1, 1/\sqrt{2}\} be the root of the key algebra f_{i}. ad_{A} has the following eigenspace decomposition:

ad_{A}|a = 0 ,
ad_{A}G_{i} = a_{i}μ_{i}G_{i} ,

ad_{A}|x_{±} = \frac{1}{2} (a_{1} ± \frac{\sqrt{2}}{2}a_{2}) Id ,
ad_{A}P̃_{0} = \frac{1}{2} (a_{0} + a_{1}) P̃_{0} ,
ad_{A}P̃_{±} = \frac{1}{2} (a_{0} + a_{1} ± \sqrt{2}a_{2}) P̃_{±} ,
ad_{A}Q̃_{0} = \frac{1}{2} (a_{0} − a_{1}) Q̃_{0} ,
ad_{A}Q̃_{±} = \frac{1}{2} (a_{0} − a_{1} ± \sqrt{2}a_{2}) Q̃_{±} ,
ad_{A}|x̃_{±} = \frac{1}{2} (a_{0} ± \frac{\sqrt{2}}{2}a_{2}) Id .

Finally, A^{0} := 3H_{0} + H_{2} + H_{1} satisfies the third condition of Def. II.2. □

Corollary II.4 t(p) is real solvable.

Proof: We give a basis B with respect to which the ad_{A}, A ∈ a, are real diagonal matrices and the ad_{N}, N ∈ n, are lower triangular matrices with zeroes on the diagonal. For a subspace y ⊂ t let B_{y} denote any basis of y.

B := (B_{a}, B_{x_{−}}, G_{2}, B_{x_{+}}, G_{1}, Q̃_{−}, P̃_{−}, Q̃_{0}, Q̃_{+}, Bx̃_{−}, Bx̃_{+}, P̃_{0}, P̃_{+}, G_{0}) □

We describe the decomposition t = a + n more accurately. For b ⊂ t let zent(b) (resp. norm(b) ) be the centralizer (resp. normalizer) of b in t.

Proposition II.5 The following holds for t:

(i) a is an (Abelian) Cartan subalgebra.

(ii) n is the nilradical.

(iii) zent(t) = 0.

In particular, t(p) has rank 3.

Proof: (i) follows e.g. from Def. II.2.

(ii): None of the endomorphisms ad_{A}|n, A ∈ a − {0}, is nilpotent, therefore n is the maximal nilpotent ideal.

(iii): On the one hand zent(t) ⊂ zent(a) = a by (i), on the other hand zent(t) ⊂ n by (ii), so zent(t) = 0. □

The roots of the Lie algebra t(p) with respect to the Cartan subalgebra a (s. Prop. II.5 (i)) can be read from the eigenspace decomposition given in the proof of Prop. II.3.

Definition II.3 Let s be a Lie algebra of Iwasawa type and Ω_{+} its set of roots with respect to the Cartan subalgebra a. We will say that the roots of s form a root system, if Ω := Ω_{+} ∪ −Ω_{+} is a root system.

Proposition II.6 The following holds in the sense of Def. II.3:

(i) The roots of t(0) form a root system isomorphic^{3} to the root system b_{3}.

(ii) For p ≠ 0 the roots of t(p) do not form a root system.

Proof: If we identify a and a^{∗} by means of the metric Lie algebra’s inner product, the roots Ω_{+}(p) of t(p) will be

Ω_{+}(0) = \{μ_{i}H_{i} (i = 0, 1, 2), \frac{1}{2} (H_{0} ± H_{1}) , P_{±}, −JQ_{±}\} and
Ω_{+}(p) = \{μ_{i}H_{i}, \frac{1}{2} (μ_{i}H_{i} ± μ_{j}H_{j}) (0 ≤ i < j ≤ 2), P_{±}, −JQ_{±}\} ,

if p ≠ 0. We recall:

P_{±} = \frac{1}{2} (H_{0} + H_{1} ± \sqrt{2}H_{2}) , −JQ_{±} = \frac{1}{2} (H_{0} − H_{1} ± \sqrt{2}H_{2}) .

If Ω = Ω_{+} ⋃ −Ω_{+} is a root system, then for every root λ ∈ Ω there is a reflection along λ that leaves Ω invariant (s. [He] Ch. X §3 1. Def. on p. 456). We suppose that Ω(p), p ≠ 0, is a root system and will derive a contradiction. Let σ be the reflection along λ = (H_{0} + 1/\sqrt{2} H_{2}) /2 which leaves Ω(p) invariant. Since neither λ + H_{1} nor λ − H_{1} lies in Ω, σH_{1} = H_{1} must hold (s. [He] Ch. X Lemma 3.4.). As well, λ±P_{−} ∉ Ω and λ±JQ_{−} ∉ Ω imply σP_{−} = P_{−} and σJQ_{−} = JQ_{−}. Therefore, σ is the Euclidean reflection with respect to λ^{⊥} and one obtains e.g. σH_{0} = −(H_{0} + 2\sqrt{2}H_{2})/3 ∉ Ω. This shows (ii).

3The isomorphism type of the root system of t(0) is indicated in [Al 5] Table 1.

(i): It is easy to check that the Euclidean reflection σ_{λ} with respect to λ^{⊥} leaves the set Ω(0) invariant for all λ ∈ Ω(0). In addition, the numbers c(μ, λ) defined by

σ_{λ}μ = μ − c(μ, λ)λ , λ, μ ∈ Ω(0)  (II.5)

are integers, which can be checked easily using the formula

c(μ, λ) = 2\frac{⟨μ, λ⟩}{|λ|^{2}}

(s. [He] Ch. X §3 1. Lemma 3.3.). Thus, we see that Ω(0) is a (reduced) root system. In order to determine its isomorphism type we give a basis B of the root system and compute the corresponding Cartan matrix (c(λ, μ))_{λ,μ∈B}. As basis we choose the simple roots of Ω_{+}:

B = (H_{1}, −JQ_{−}, \frac{\sqrt{2}}{2}H_{2}) .

The Cartan matrix of Ω(0) with respect to B reads

( \begin{matrix}2 & −1 & 0 \\ −1 & 2 & −2 \\ 0 & −1 & 2 \end{matrix} )

Therefore, Ω(0) ≅ b_{3} (s. [He] Ch. X §3 3. p. 463). □

Next, we study the Levi-Civita connection of t. From the formulas for the covariant derivative one sees, on the one hand, that (t, q) is quaternionic Kählerian, on the other hand, they serve as a basis for curvature computations in Ch. III.

Proposition II.7 The following subalgebras of t are totally geodesic:

a , ∑_{i∈I} f_{i} , u_{0} , u , I ⊂ {0, 1, 2} .

Proof: Trivial check using the Koszul formula. □

Proposition II.8 t(p′) is a quaternionic subalgebra of t(p), if p′ ≤ p. In particular, the inclusion t(p′) ⊂ t(p) is totally geodesic.

Proof: Choose x′_{−} ⊂ x_{−} with dim x′_{−} = p′. Set x′ := x′_{−} + Jx′_{−} and u′ := ∑_{i} f_{i} + x′. Then t(p′) ≅ u′ + ũ′ is a quaternionic subalgebra. Quaternionic subalgebras are totally geodesic, according to Thm. I.2 (i). □

Proposition II.9 The covariant derivative of t is given by the following formulas (A ∈ a, X_{±} ∈ x_{±}).

∇_{A} = 0 , ∇_{G_{0}} = −G_{0} ∧ H_{0} − \frac{1}{2}Ĵ ∘ Prũ ,

∇_{G_{1}} = −G_{1} ∧ H_{1} − \frac{1}{2}J ∘ Pr_{x} + \frac{1}{2} (G̃_{0} ∧ H̃_{0} − G̃_{1} ∧ H̃_{1} + G̃_{2} ∧ H̃_{2}) ,

∇_{G_{2}} = −\frac{\sqrt{2}}{2} (G_{2} ∧ H_{2} + \frac{1}{2}J ∘ Pr_{x} − G̃_{0} ∧ H̃_{0} − G̃_{1} ∧ H̃_{1} + \frac{1}{2}J_{1} ∘ Prx̃) ,

∇_{X_{±}} = \frac{1}{2} ((H_{1} ± \frac{\sqrt{2}}{2}H_{2}) ∧ X_{±} + (G_{1} ± \frac{\sqrt{2}}{2}G_{2}) ∧ JX_{±}) +

\frac{1}{2} ((G̃_{1} ± \frac{\sqrt{2}}{2}G̃_{2}) ∧ J̃X_{±} + (H̃_{1} ± \frac{\sqrt{2}}{2}H̃_{2}) ∧ X̃_{±}) ,

∇H̃_{0} = −\frac{1}{2}J_{2} , ∇G̃_{0} = \frac{1}{2}J_{3} ,

∇H̃_{1} = −\frac{1}{2} (H̃_{1} ∧ H_{0} + H̃_{0} ∧ H_{1} + H̃_{2} ∧ H_{2} − G̃_{1} ∧ G_{0} − G̃_{0} ∧ G_{1} − G̃_{2} ∧ G_{2}) ,

∇H̃_{2} = −\frac{1}{2} (H̃_{2} ∧ (H_{0} + H_{1}) + (H̃_{0} + H̃_{1}) ∧ H_{2} − G̃_{2} ∧ (G_{0} + G_{1}))

+ \frac{1}{2} ((G̃_{0} + G̃_{1}) ∧ G_{2} − \frac{\sqrt{2}}{2}J_{2} ∘ (Pr_{x_{+}+x̃_{+}} − Pr_{x_{−}+x̃_{−}})) ,

∇G̃_{1} = −\frac{1}{2} (G̃_{1} ∧ H_{0} + G̃_{0} ∧ H_{1} − G̃_{2} ∧ H_{2})

− \frac{1}{2} (H̃_{1} ∧ G_{0} + H̃_{0} ∧ G_{1} − H̃_{2} ∧ G_{2}) ,

∇G̃_{2} = −\frac{1}{2} (G̃_{2} ∧ (H_{0} − H_{1}) + (G̃_{0} − G̃_{1}) ∧ H_{2} + H̃_{2} ∧ (G_{0} − G_{1}))

− \frac{1}{2} ((H̃_{0} − H̃_{1}) ∧ G_{2} + \frac{\sqrt{2}}{2}J_{3} ∘ (Pr_{x_{+}+x̃_{−}} − Pr_{x_{−}+x̃_{+}})) ,

∇X̃_{±} = −\frac{1}{2} (X̃_{±} ∧ (H_{0} ± \frac{\sqrt{2}}{2}H_{2}) − J̃X_{±} ∧ (G_{0} ± \frac{\sqrt{2}}{2}G_{2})

+ (H̃_{0} ± \frac{\sqrt{2}}{2}H̃_{2}) ∧ X_{±} − (G̃_{0} ± \frac{\sqrt{2}}{2}G̃_{2}) ∧ JX_{±}) .

Where Pr_{v} denotes the orthogonal projection on the subspace v ⊂ t, and V ∧ W = V ⊗ W − W ⊗ V denotes the skew endomorphism S ↦ V ⟨W, S⟩ − W ⟨V, S⟩ of t.

Proof: We have computed the covariant derivatives ∇_{U}, U ∈ u, by means of the Koszul formula, observing Prop. II.7 and II.8. The operators ∇Ũ|u result from the vanishing of the torsion (U_{1}, U_{2} ∈ u):

∇Ũ_{1}U_{2} = −T_{U_{2}} Ũ_{1} + ∇_{U_{2}} Ũ_{1} .

[ũ, ũ] ⊂ u yields ∇ũũ ⊂ u, by the Koszul formula. Therefore, we can compute the remaining operators ∇Ũ|ũ by using the compatibility of the covariant derivative and the metric (U, U_{1}, U_{2} ∈ u):

⟨∇Ũ_{1} Ũ_{2}, U⟩ = − ⟨Ũ_{2}, ∇_{Ũ1}U⟩ . □

Corollary II.10 The following holds in the sense of Def. I.5:

(i) (u, J) is Kählerian.

(ii) (t, q) is quaternionic Kählerian.

Proof: (i): From the formulas for the covariant derivative of the totally geodesic subalgebra u we see that kos(u) ⊂ zent(J).

(ii): The formulas in Prop. II.9 allow us to infer immediately that

1. ∇_{A} = 0 for A ∈ a, 2. ∇H̃_{0}, ∇G̃_{0} ∈ q,

3. ∇_{W} ∈ zent(q) for W ∈ x_{1} + ũ_{0},

4. ∇_{G_{i}} ∈ zent(J_{1}), i = 0, 1, 2.

In particular, (1-3.), we have ∇_{W} ∈ q + zent(q) for W ∈ (Ja)^{⊥}.

Claim:

∇_{G_{i}} ∈ span {J_{1}} + zent(q) ⊂ q + zent(q) .

We define ¯∇_{G_{i}} by:

∇_{G_{i}} = −\frac{1}{4}J_{1} + ¯∇_{G_{i}} , i = 0, 1 ,
∇_{G_{2}} = −\frac{\sqrt{2}}{4}J_{1} + ¯∇_{G_{2}} .

We have to show that ¯∇_{G_{i}} ∈ zent(q). Because of 4. we only need to show ¯∇_{G_{i}} ∈ zent(J_{2}). Obviously it is sufficient to check [ ¯∇_{G_{i}}, J_{2}]|u = 0.

[∇_{G_{0}}, J_{2}] |u = −\frac{1}{2}ĴJ_{2} − (−G̃_{0} ∧ H_{0})
= −\frac{1}{2}J_{3} = [−\frac{1}{4}J_{1}, J_{2}] ,

[∇_{G_{1}}, J_{2}] |u = \frac{1}{2} (G̃_{0} ∧ H_{0} − G̃_{1} ∧ H_{1} + G̃_{2} ∧ H_{2})

− (−G̃_{1} ∧ H_{1} + \frac{1}{2}J_{3} ∘ Pr_{x}) = −\frac{1}{2}J_{3} ,

[∇_{G_{2}}, J_{2}] |u = \frac{\sqrt{2}}{2} (G̃_{0} ∧ H_{0} + G̃_{1} ∧ H_{1} − \frac{1}{2}J_{3} ∘ Pr_{x})

+ \frac{\sqrt{2}}{2} (G̃_{2} ∧ H_{2} − \frac{1}{2}J_{3} ∘ Pr_{x})

= −\frac{\sqrt{2}}{2}J_{3} = [−\frac{\sqrt{2}}{4}J_{1}, J_{2}] . □

Cor. II.4, Prop. II.5 and Cor. II.10 imply that (t, q) is an Alekseevskian Lie algebra of rank 3. That finishes the proof of Prop. II.1, and the classification of Alekseevskiĭ spaces of rank 3 goes as follows.

Theorem II.11 There exists a family T (p), p = 0, 1, 2, …, of Alekseevskian spaces of rank 3 and of quaternionic dimension p+3. Any Alekseevskiĭ space of rank 3 is (up to scaling) isometric to one of these spaces.

T (0) is isometric to the symmetric space SO_{0}(3, 4)/ (SO(3) × SO(4)) of scalar curvature −30. All the other T (p) are not symmetric.

Proof: The only question that is left to be answered is which of the spaces T (p) are symmetric. The 12-dimensional non-compact Grassmann manifold SO_{0}(3, 4)/ (SO(3) × SO(4)) is the only symmetric space of noncompact type which is quaternionic Kählerian and has rank 3 (s. [Be] 14.52 Table). According to Prop. II.1, T (0) is the only Alekseevskiĭ space of rank 3 and of real dimension 3 · 4 = 12. Thus, up to scaling,

T (0) ≅ \frac{SO_{0}(3, 4)}{SO(3) × SO(4)}

and all the other T (p) are not symmetric.

The reduced scalar curvature sk_{r} of T (0) coincides, due to Thm. I.2 (i) and (iii), with the reduced scalar curvature sk_{r} = −1/2 of the canonical onedimensional quaternionic subalgebra f_{0} + f̃_{0}. So, in accordance with equation I.3, the scalar curvature of T (0) is

sk = 4 · 3(3 + 2) · (−\frac{1}{2}) = −30 . □

II.2 Rank 4

The Alekseevskian spaces of rank 4 are the W- and V-spaces reviewed in the sections I.2 and I.3. The main aim of this section is to clarify the classification of V-spaces. Nevertheless, with the purpose of rounding off our exposition, we begin with a discussion of W-spaces similar to that of T -spaces in the previous section.

The fact that the metric Lie algebras w(p, q) and v(ψ) are Alekseevskian is due to the circumstance that they are constructed from Q-representations (s. Thm. I.3 and Thm. I.4). Alternatively one could check directly that they are real solvable and quaternionic Kählerian. We have pursued this approach for the T -spaces in sec. II.1 (cf. Cor. II.4, Prop. II.9 and Prop. II.10) and for the W-spaces in [C1] II.3-4. We shall not insist on this stand-point in order to avoid too much redundancy.

Proposition II.12 w = w(p, q) is 4-step solvable. The derived Lie algebra n = [w, w] is 7-step nilpotent, if (p, q) ≠ (0, 0). For w(0, 0) it is 5-step nilpotent.

Proof: Derived series:

Dw = [w, w] = span{G_{i}|i = 0, 1, 2, 3} + x_{1} + ũ .

D^{2}w = span{G_{0}, G_{1}} + y_{+} + z_{+} + span{P̃_{i}, Q̃_{α}|i = 0, 1, 2, 3; α = 1, 2, 3} + x̃_{1} ,

if (p, q) ≠ (0, 0).

In the case p = q = 0 one obtains

D^{2}w = span{G_{0}} + span{P̃_{i}, Q̃_{α}|i = 0, 1, 2, 3; α = 1, 2, 3}

and thus

D^{3}w = span{G_{0}} and D^{4}w = 0 .

For (p, q) ≠ (0, 0) one has

D^{3}w = span{G_{0}} + span{P̃_{0}, Q̃_{2}, Q̃_{3}} + ỹ_{+} + z̃_{+} ,
D^{4}w = 0 .

Central series of n:

Zn = [n, n] = D^{2}w , Z^{2}n = [n, Zn] .

For p = q = 0:

Z^{2}n = span{G_{0}} + span{P̃_{0}, Q̃_{α}|α = 1, 2, 3} ,
Z^{3}n = span{G_{0}} + span{P̃_{0}} ,
Z^{4}n = span{G_{0}} ,

Z^{5}n = 0 .

For (p, q) ≠ (0, 0):

Z^{2}n = span{G_{0}, G_{1}} + span{P̃_{0}, P̃_{1}, Q̃_{α}|α = 1, 2, 3} + x̃_{1} ,
Z^{3}n = span{G_{0}} + span{P̃_{0}, P̃_{1}, Q̃_{2}, Q̃_{3}} + ỹ_{+} + z̃_{+} ,
Z^{4}n = span{G_{0}} + span{P̃_{0}, Q̃_{2}, Q̃_{3}} ,
Z^{5}n = span{G_{0}} + span{P̃_{0}} ,
Z^{6}n = span{G_{0}} ,
Z^{7}n = 0 . □

Proposition II.13 w(p, q) is of Iwasawa type and the ad_{A}, A ∈ a, have the following eigenspace decomposition

ad_{A}a = 0 ,
ad_{A}G_{i} = a_{i}G_{i} ,
ad_{A}|y_{±} = \frac{1}{2} (a_{1} ± a_{3}) Id ,
ad_{A}|z_{±} = \frac{1}{2} (a_{1} ± a_{2}) Id ,
ad_{A}P̃_{0} = \frac{1}{2} (a_{0} + a_{1} + a_{2} + a_{3}) P̃_{0} ,
ad_{A}P̃_{α} = \frac{1}{2} (a_{0} + a_{α} − a_{β} − a_{γ}) P̃_{α} ,
ad_{A}Q̃_{0} = \frac{1}{2} (a_{0} − a_{1} − a_{2} − a_{3}) Q̃_{0} ,
ad_{A}Q̃_{α} = \frac{1}{2} (a_{0} − a_{α} + a_{β} + a_{γ}) Q̃_{α} ,
ad_{A}|ỹ_{±} = \frac{1}{2} (a_{0} ± a_{2}) Id ,
ad_{A}|z̃_{±} = \frac{1}{2} (a_{0} ± a_{3}) Id ,

Where A = ∑_{i=0}^{3} a_{i}H_{i}, a_{i} ∈ R, and {α, β, γ} = {1, 2, 3}.

Proof: A^{0} := 7H_{0} + 3H_{1} + 2H_{2} + H_{3} satisfies the third condition of Definition II.2. □

Proposition II.14 The following holds for w:

1. a is an (Abelian) Cartan subalgebra.

2. n is the nilradical.

3. zent(w) = 0.

In particular, w has rank 4.

Proof: Same proof as for Prop. II.5 □.

The following information together with Prop. II.22 serves to recognize the symmetric spaces in the list of Alekseevskian spaces of rank 4. The corresponding discussion is postponed until Thm. II.28.

Proposition II.15 (s. [Al 5] Table 1) The roots of w(p, q) with respect to the Cartan subalgebra a (cf. Def. II.3):

(i) form a root system isomorphic to d_{4}, if p = q = 0;

(ii) form a root system isomorphic to b_{4}, if either p = 0 or q = 0.

(iii) do not form a root system, if p ≠ 0 and q ≠ 0.

Proof: The roots Ω_{+}(p, q) can be read from the eigenspace decomposition of Prop. II.13. If p ≠ 0 and q ≠ 0 they are (i = 0, 1, 2, 3; α = 1, 2, 3):

\{H_{i}, \frac{1}{2}(H_{1}±H_{3}), \frac{1}{2}(H_{1}±H_{2}), P_{0}, −P_{α}, −JQ_{0}, JQ_{α}, \frac{1}{2}(H_{0}±H_{2}), \frac{1}{2}(H_{0}±H_{3})\} .

For q = 0 (H_{1} ± H_{2})/2 and (H_{0} ± H_{3})/2 have to be dropped. For p = 0 (H_{1} ± H_{3})/2 and (H_{0} ± H_{2})/2 have to be dropped.

First, we assume that Ω(p, q) (p ≠ 0 and q ≠ 0) forms a root system and deduce a contradiction. Let σ be the reflection along λ := (H_{1} +H_{3})/2 which leaves Ω(p, q) invariant. Since neither λ + (H_{1} + H_{2})/2 nor λ − (H_{1} + H_{2})/2 lies in Ω(p, q), we must have σ(H_{1} + H_{2}) = H_{1} + H_{2}. By the same token σ(H_{1} − H_{2}) = H_{1} − H_{2} and σ(H_{0} ± H_{3}) = H_{0} ± H_{3}. Thus, σ = Id, which is not a reflection. This shows (iii).

For λ ∈ a − {0} let σ_{λ} denote the Euclidean reflection with respect to λ^{⊥}. We observe (i = 0, 1, 2, 3):

1. σ_{H_{i}}, σ_{P_{i}} and σ_{JQ_{i}} leave Ω(p, q) invariant.

2. σ_{H_{1}±H_{3}} and σ_{H_{0}±H_{2}} leave Ω(p, 0), p ≠ 0, invariant.

3. σ_{H_{1}±H_{2}} and σ_{H_{0}±H_{3}} leave Ω(0, q), q ≠ 0, invariant.

The condition c(μ, λ) ∈ Z may be checked for λ, μ ∈ Ω(p, 0) as indicated in the proof of Prop. II.6. This proves that Ω(p, q) is a root system, if p = 0 or q = 0. Next, we compute the isomorphism type of the root system. A basis of Ω(p, 0) (p ≠ 0) is

B_{1,0} := (H_{1}, −JQ_{3}, H_{0}, −\frac{1}{2}(H_{0} − H_{2})) .

A basis of Ω(0, 0) is

B_{0,0} := (−JQ_{0}, P_{0}, −JQ_{1}, −JQ_{2}) .

The Cartan matrix of B_{1,0} reads

( \begin{matrix}2 & −1 & 0 & 0 \\ −1 & 2 & −1 & 0 \\ 0 & −1 & 2 & −2 \\ 0 & 0 & −1 & 2 \end{matrix} ) ,

and that of B_{0,0} is

( \begin{matrix}2 & −1 & 0 & 0 \\ −1 & 2 & −1 & −1 \\ 0 & −1 & 2 & 0 \\ 0 & −1 & 0 & 2 \end{matrix} ) .

This determines the isomorphism type to be b_{4} and d_{4} respectively (s. [He] Ch. X §3 3. pp. 462-464). □

Next we compute the Levi-Civita connection of w part of which we will use for curvature computations in Ch. III.

Proposition II.16 The following subalgebras of w are totally geodesic:

a , ∑_{i∈I} f_{i} , f_{1} + f_{3} + y , f_{1} + f_{2} + z , u_{0} u , I ⊂ {0, 1, 2, 3}.

Proof: Easy computation using the Koszul formula. □ Proposition II.17 w(p′, q′) is a quaternionic subalgebra of w(p, q), if p′ ≤ p and q′ ≤ q. In particular, the corresponding inclusion is totally geodesic.

Proof: Cf. proof of Prop. II.8 □ Proposition II.18 The covariant derivative of w(p, q) is given by the following formulas (A ∈ a, X_{±} ∈ x_{±}, Y_{±} ∈ y_{±} and Z_{±} ∈ z_{±}).

∇_{A} = 0 , ∇_{G_{i}}|f_{i} = −J , ∇_{G_{0}}|u_{0} = 0 ,

∇_{G_{1}}|x_{1} = −\frac{1}{2}J , ∇_{G_{1}}|f_{0} + f_{2} + f_{3} = 0 ,

∇_{G_{2}}|z = −\frac{1}{2}J , ∇_{G_{2}}|f_{0} + f_{1} + f_{3} + y = 0 ,

∇_{G_{3}}|y = −\frac{1}{2}J , ∇_{G_{3}}|f_{0} + f_{1} + f_{2} + z = 0 ,

∇_{G_{i}}|f̃_{i} = \frac{1}{2}J_{1} , ∇_{G_{0}}|ũ_{0} = −\frac{1}{2}J_{1} ,

∇_{G_{1}}|x̃_{1} = 0 , ∇_{G_{1}}|f̃_{0} + f̃_{2} + f̃_{3} = −\frac{1}{2}J_{1} ,

∇_{G_{2}}|z̃ = 0 , ∇_{G_{2}}|f̃_{0} + f̃_{1} + f̃_{3} + ỹ = −\frac{1}{2}J_{1} ,

∇_{G_{3}}|ỹ = 0 , ∇_{G_{3}}|f̃_{0} + f̃_{1} + f̃_{2} + z̃ = −\frac{1}{2}J_{1} .

∇_{X_{1}} ∈ zent(q), if X_{1} ∈ x_{1} = y + z. The operators ∇_{X_{1}}, X_{1} ∈ x_{1}, are thus already determined by the following restrictions:

∇_{Y_{−}}|a + y_{−} + z_{−} = \frac{1}{2}(H_{1} − H_{3}) ∧ Y_{−} ,

∇_{Y_{+}}|a + y_{+} + z_{+} = \frac{1}{2}(H_{1} + H_{3}) ∧ Y_{+} ,

∇_{Z_{−}}|a + y_{−} + z_{−} = \frac{1}{2}(H_{1} − H_{2}) ∧ Z_{−} ,

∇_{Z_{+}}|a + y_{+} + z_{+} = \frac{1}{2}(H_{1} + H_{2}) ∧ Z_{+} .

∇H̃_{0} = −\frac{1}{2}J_{2} , ∇G̃_{0} = \frac{1}{2}J_{3} .

∇Ṽ ∈ zent(q), if V ∈ u_{0}. Therefore, it suffices to give the following restrictions ({α, β, γ} = {1, 2, 3}):

∇H̃_{α}|a = −\frac{1}{2}(H̃_{α} ⊗ H_{0} + H̃_{0} ⊗ H_{α} + H̃_{γ} ⊗ H_{β} + H̃_{β} ⊗ H_{γ}) ,

∇H̃_{1}|y_{−} + z_{−} = 0 ,

∇H̃_{2}|y_{−} = \frac{1}{2}J_{2} , ∇H̃_{2}|z_{−} = 0 ,

∇H̃_{3}|z_{−} = \frac{1}{2}J_{2} , ∇H̃_{3}|y_{−} = 0 ,

∇G̃_{α}|a = −\frac{1}{2}(G̃_{α} ⊗ H_{0} + G̃_{0} ⊗ H_{α} − G̃_{γ} ⊗ H_{β} − G̃_{β} ⊗ H_{γ}) ,

∇G̃_{1}|y_{−} + z_{−} = 0 ,

∇G̃_{2}|y_{−} = \frac{1}{2}J_{3} , ∇G̃_{2}|z_{−} = 0 ,

∇G̃_{3}|z_{−} = \frac{1}{2}J_{3} , ∇G̃_{3}|y_{−} = 0 ,

∇Ỹ_{−}|a + ỹ_{−} + z̃_{−} = \frac{1}{2}(H_{0} − H_{2}) ∧ Ỹ_{−} ,

∇Ỹ_{+}|a + ỹ_{+} + z̃_{+} = \frac{1}{2}(H_{0} + H_{2}) ∧ Ỹ_{+} ,

∇Z̃_{−}|a + ỹ_{−} + z̃_{−} = \frac{1}{2}(H_{0} − H_{3}) ∧ Z̃_{−} ,

∇Z̃_{+}|a + ỹ_{+} + z̃_{+} = \frac{1}{2}(H_{0} + H_{3}) ∧ Z̃_{+} .

Proof: Similar to that of Prop. II.9 □

For more details concerning W-spaces the reader is referred to [C1], where e.g. the automorphism group of the metric Lie algebras w(p, q) is determined.

We have explained in sec. I.3 how the Alekseevskian V-spaces V(ψ) are constructed out of special isometric mappings ψ. It is clear that equivalent (s. Def. I.10) special isometric mappings yield isometric V-spaces by this construction.

The classification of V-spaces is based on that of special isometric mappings and this is in turn based on the classification of Z_{2}-graded Clifford modules. The correction which we shall note concerns [Al 5] Thm. 10.1. (not to be mixed up with Prop. 10.1.), i.e. the classification of special isometric mappings up to equivalence which we carry out in Prop. II.26 (cf. Prop. II.25). Before going into that, we will mention some basic facts which are independent of the choice of special isometric mapping ψ.

Proposition II.19 The Lie algebra v = v(ψ) is 5-step solvable. Its derived Lie algebra n = [v, v] is 11-step nilpotent.

Proof: Derived series:

D^{1}v = [v, v] = a^{⊥} , where a = span {H_{i}|i = 0, 1, 2, 3} .

In the following we use that ψ(X_{−}, ·) : z_{−} → y_{−} (X_{−} ∈ x_{−} − {0}) is an isomorphism and that the bilinear mappings (V, W) ↦ [V, W] and (V, W) ↦ V ∘ W = 2∇_{V} W, from the product of two of the spaces x, y, z into the third, are completely determined by ψ, according to the remark on p. 13. We use e.g. [x_{−}, z_{−}] = x_{−} ∘ z_{−} = y_{−}.

D^{2}v = [n, n] = span {G_{i}|i = 0, 1, 2} + y + z_{+} + x_{+} +
span {P̃_{i}, Q̃_{α}|i = 0, 1, 2, 3; α = 1, 2, 3} + ỹ + z̃ + x̃ ,

D^{3}v = span {G_{0}, G_{1}} + z_{+} + span {P̃_{α}, Q̃_{β}|α = 0, 1; β = 1, 2, 3} + ỹ+ z̃+ x̃_{+} ,

D^{4}v = span {G_{0}} + span {P̃_{0}, Q̃_{3}} + x̃_{+} , D^{5}v = 0 .

Central series of n:

Z^{1}n = [n, n] = D^{2}v ,
Z^{2}n = [n, D^{2}v] = span {G_{i}|i = 0, 1, 2} + y_{+} + z_{+} +
span {P̃_{i}, Q̃_{α}|i = 0, 1, 2; α = 1, 2, 3} + ỹ + z̃ + x̃ ,

Z^{3}n = span {G_{0}, G_{1}}+z_{+} +span {P̃_{i}, Q̃_{α}|i = 0, 1, 2; α = 1, 2, 3}+ỹ+z̃+x̃_{+} ,

Z^{4}n = span {G_{0}, G_{1}} + span {P̃_{i}, Q̃_{α}|i = 0, 1; α = 1, 2, 3} + ỹ_{+} + z̃ + x̃_{+} ,

Z^{5}n = span {G_{0}} + span {P̃_{i}, Q̃_{α}|i = 0, 1; α = 2, 3} + ỹ_{+} + z̃_{+} + x̃_{+} ,

Z^{6}n = span {G_{0}} + span {P̃_{0}, Q̃_{α}|α = 2, 3} + ỹ_{+} + x̃_{+} ,
Z^{7}n = span {G_{0}} + span {P̃_{0}, Q̃_{3}} + x̃_{+} ,

Z^{8}n = span {G_{0}} + span {P̃_{0}, Q̃_{3}} ,

Z^{9}n = span \{G_{0}\} + span \left\{{ \atop P̃_{0}}\right\} ,

Z^{10}n = span {G_{0}} ,
Z^{11}n = 0 . □

Proposition II.20 v = a + [v, v] is of Iwasawa type and the ad_{A}, A ∈ a, have the following eigenspace decomposition

ad_{A}a = 0 ,
ad_{A}G_{i} = a_{i}G_{i} ,
ad_{A}|y_{±} = \frac{1}{2} (a_{1} ± a_{3}) Id ,
ad_{A}|z_{±} = \frac{1}{2} (a_{1} ± a_{2}) Id ,
ad_{A}|x_{±} = \frac{1}{2} (a_{2} ± a_{3}) Id ,
ad_{A}P̃_{0} = \frac{1}{2} (a_{0} + a_{1} + a_{2} + a_{3}) P̃_{0} ,
ad_{A}P̃_{α} = \frac{1}{2} (a_{0} + a_{α} − a_{β} − a_{γ}) P̃_{α} ,
ad_{A}Q̃_{0} = \frac{1}{2} (a_{0} − a_{1} − a_{2} − a_{3}) Q̃_{0} ,
ad_{A}Q̃_{α} = \frac{1}{2} (a_{0} − a_{α} + a_{β} + a_{γ}) Q̃_{α} ,
ad_{A}|ỹ_{±} = \frac{1}{2} (a_{0} ± a_{2}) Id ,
ad_{A}|z̃_{±} = \frac{1}{2} (a_{0} ± a_{3}) Id ,
ad_{A}|x̃_{±} = \frac{1}{2} (a_{0} ± a_{1}) Id .

Where A = ∑_{i=0}^{3} a_{i}H_{i}, a_{i} ∈ R, and {α, β, γ} = {1, 2, 3}.

Proof: A^{0} := 7H_{0} + 3H_{1} + 2H_{2} + H_{3} satisfies the third condition of Definition II.2. □

Proposition II.21 The following holds for v:

1. a is an (Abelian) Cartan subalgebra.

2. n is the nilradical.

3. zent(v) = 0.

In particular, v has rank 4.

Proof: Same proof as for Prop. II.5 □.

The remark preceding Prop. II.15 refers to the following proposition as well.

Proposition II.22 (s. [Al 5] Table 1) The roots of v(ψ) with respect to the Cartan subalgebra a form a root system isomorphic to f_{4}.

Proof: The roots Ω_{+} of v(ψ) can be read from the eigenspace decomposition of Prop. II.20:

Ω_{+} = \{H_{i}, \frac{1}{2} (H_{i} ± H_{j}) (0 ≤ i < j ≤ 3), P_{0}, −JQ_{0}, −P_{α}, JQ_{α} (α = 1, 2, 3)\}

As in the proof of Prop. II.6, one can check that for every λ ∈ Ω_{+} the Euclidean reflection with respect to λ^{⊥} leaves Ω = Ω_{+} ⋃ −Ω_{+} invariant and that c(μ, λ) ∈ Z for all μ, λ ∈ Ω. Thus, Ω is a root system.

If we choose the simple roots

B = (−JQ_{0}, H_{3}, \frac{1}{2} (H_{2} − H_{3}) , \frac{1}{2} (H_{1} − H_{2}))

of Ω_{+} as basis of the root system, then the Cartan matrix reads

( \begin{matrix}2 & −1 & 0 & 0 \\ −1 & 2 & −2 & 0 \\ 0 & −1 & 2 & −1 \\ 0 & 0 & −1 & 2 \end{matrix} ) .

Therefore, Ω ≅ f_{4} (s. [He] Ch. X §3 3. p. 474). □

Next, we settle the question in which cases two special isometric mappings define isometric V-spaces. The adequate concept is that of equivalence (s. Def. I.10).

Proposition II.23 Two V-spaces V(ψ) and V(ψ′) are isometric, if and only if the special isometric mappings ψ and ψ′ are equivalent.

Proof: The notation for vectors H_{i}, G_{i}, … and subspaces f_{i}, u, … of v(ψ) will be as up to now. Vectors and subspaces of v(ψ′) will be marked out by a prime: H′_{i}, G′_{i}, f′_{i}, u′, … As already mentioned (s. sec. I.1 p. 2) V(ψ) and V(ψ′) are isometric, if and only if the corresponding metric Lie algebras v = v(ψ) and v′ = v(ψ′) are isomorphic. It is clear that equivalent special isometric mappings yield isomorphic metric Lie algebras.

Conversely, let ϕ : v → v′ be an isomorphism of metric Lie algebras. We show that ψ and ψ′ are equivalent via ϕ. a = [v, v]^{⊥} (together with a′ = [v′, v′]^{⊥}) implies ϕ(a) = a′, and ϕ maps the root spaces of v with respect to a onto the root spaces of v′.

1. Claim: ϕ (G_{0}) ∈ {−G′_{0}, G′_{0}}, ϕ (H_{0}) = H′_{0} and ϕ (u_{0}) = u′_{0}.

Z^{10}n = span {G_{0}} (s. proof of Prop. II.19) implies ϕ (G_{0}) = ±G′_{0} and, as well, ϕ (H_{0}) = H′_{0}, due to span {H_{0}} = (ker ad_{G_{0}})^{⊥} and [H_{0}, G_{0}] = G_{0}. ϕ (u_{0}) = u′_{0} follows from u_{0} = (ker ad_{H_{0}}) ∩ H_{0}^{⊥} (we recall: u_{0} = f_{0}^{⊥} ∩ u).

2. Claim: ϕ (G_{1}) ∈ {−G′_{1}, G′_{1}} and ϕ (H_{1}) = H′_{1}.

From span {G_{1}} = (Z^{4}n)∩u_{0} we obtain ϕ (G_{1}) = ±G′_{1} and by virtue of span {H_{1}} = (ker ad_{G_{1}})^{⊥} ∩ u ( and [H_{1}, G_{1}] = G_{1}) we get ϕ (H_{1}) = H′_{1} as well.

3. Claim: ϕ (G_{2}) ∈ {−G′_{2}, G′_{2}}, ϕ (H_{2}) = H′_{2}, ϕ(y) = y′, ϕ (z_{±}) = z′_{±} and ϕ(x) = x′. y + z is the eigenspace of ad_{H_{1}}|u corresponding to the eigenvalue 1/2, thus ϕ(y+z) = y′+z′. (Z^{2}n)∩u_{0}∩(G_{1})^{⊥}∩(y+z)^{⊥} = span {G_{2}} implies ϕ (G_{2}) = ±G′_{2}, and (ker ad_{G_{2}})^{⊥} ∩ u ∩ (y + z)^{⊥} = span {H_{2}} implies ϕ (H_{2}) = H′_{2}. Further, z_{±} is the eigenspace of ad_{H_{2}}|y + z corresponding to the eigenvalue ±1/2 and hence ϕ (z_{±}) = z′_{±} and ϕ(y) = y′. Finally, x is the eigenspace of ad_{H_{2}}|z^{⊥} ∩ u corresponding to the eigenvalue 1/2.

4. Claim: ϕ (G_{3}) ∈ {−G′_{3}, G′_{3}}, ϕ (H_{3}) = H′_{3}, ϕ (y_{±}) = y′_{±}, ϕ (z_{±}) = z′_{±} and ϕ (x_{±}) = x′_{±}. The two first relations result from span {G_{3}} = u_{0} ∩ (a + x + y + z)^{⊥} ∩ G_{1}^{⊥} ∩ G_{2}^{⊥} and span {H_{3}} = span {H_{i}|i = 0, 1, 2}^{⊥} ∩ a. The rest follows from the three last equations of 3. and from the fact that y = y_{+} + y_{−} and x_{+} + x_{−} are weight decompositions for the representations ad_{H_{3}}|y and ad_{H_{3}}|x.

The special isometric mapping ψ : x_{−} × z_{−} → y_{−} (analogously ψ′) is proportional to the restriction of the Lie product:

ψ (X_{−}, Z_{−}) = \sqrt{2} [X_{−}, Z_{−}] , X_{−} ∈ x_{−}, Z_{−} ∈ z_{−} .

From the fact that ϕ is a Lie algebra isomorphism which transfers y_{−}, z_{−} and x_{−} to y′_{−}, z′_{−} and x′_{−} respectively we can now conclude

ϕ^{−1} (ψ′ (ϕ (X_{−}) , ϕ (Z_{−}))) = ψ (X_{−}, Z_{−}) . □

Prop. II.23 should be ascribed to D. V. Alekseevskiĭ, although it is not proved in [Al 5], since the one-to-one correspondence between isometry classes of Alekseevskian V-spaces on the one hand and isomorphism classes of special isometric mappings on the other hand is needed to conclude the classification of V spaces up to isometry. One may prove similarly that two W-spaces W(p, q) and W(p′, q′) are isometric, if and only if {p, q} = {p′, q′}.

We shall now derive the classification of special isometric mappings up to equivalence from the well known classification of Z_{2}-graded Clifford modules. For that we will state the connection between special isometric mappings and Z_{2}-graded Clifford modules discovered by Alekseevskiĭ.

Besides the notion of equivalence we introduce the finer notion of isomorphism for didactic reasons.

Definition II.4 Let ϕ : x → x′ be an isomorphism of Euclidean vector spaces. Two isometric mappings ψ : x × z → y and ψ′ : x′ × z′ → y′ are said to be ϕ-isomorphic, if there are isomorphisms τ : z → z′ and υ : y → y′ of Euclidean vector spaces such that the following diagram commutes:

x × z ⟶^{ψ} y
↓ ϕ×τ  ↓ υ
x′ × z′ ⟶^{ψ′} y′

If x = x′ and ϕ = Id, we simply speak of isomorphic isometric mappings ψ : x × z → y and ψ′ : x × z′ → y′.

Consequently two isometric mappings ψ : x × z → y and ψ′ : x′ × z′ → y′ are equivalent, if and only if there is an isomorphism ϕ : x → x′ such that ψ and ψ′ are ϕ-isomorphic. If we consider a fixed isomorphism ϕ, then the relation of ϕ-isomorphism implies that of equivalence, but, in general, we cannot expect equivalent isometric mappings to be ϕ-isomorphic for the given choice of ϕ.

As Alekseevskiĭ has observed, a special isometric mapping ψ : x × z → y defines in a canonical way a Z_{2}-graded module M_{ψ} = M_{0} ⊕ M_{1} over the Clifford algebra Cℓ(x). In fact, set M_{0} := z, M_{1} := y and define Ψ : x → End(M_{ψ}) by

Ψ(X)Z := ψ(X, Z) , Ψ(X)Y := −ψ^{t}(X, Y) (X ∈ x, Y ∈ y, Z ∈ z)

(s. Def. II.6), then Ψ satisfies the relation Ψ(X)^{2} = −⟨X, X⟩Id.

Definition II.5 (cf. [Al 5] §10) Let x and x′ be Euclidean vector spaces of dimension k ≥ 1. Two Z_{2}-graded modules M = M_{0} ⊕M_{1} and M′ = M′_{0} ⊕M′_{1} over Cℓ(x) and Cℓ(x′) respectively are equivalent, if there is an isomorphism σ : Cℓ(x) → Cℓ(x′) of Clifford algebras and an isomorphism τ : M → M′ of Z_{2}-graded vector spaces such that the following diagram commutes:

Cℓ(x) × M ⟶ M
↓ σ×τ  ↓ τ
Cℓ(x′) × M′ ⟶ M′

k is the order of the Clifford module M. Let ϕ : x → x′ be an isomorphism of Euclidean vector spaces. M and M′ are ϕ-isomorphic, if there is an isomorphism τ : M → M′ of Z_{2}-graded vector spaces such that the following diagram commutes:

x × M_{0} ⟶ M_{1}
↓ ϕ×τ|M_{0}  ↓ τ|M_{1}
x′ × M′_{0} ⟶ M′_{1}

The horizontal arrows denote the corresponding restriction of Clifford multiplication. If x = x′ and ϕ = Id, then the Z_{2}-graded Cℓ(x)-modules M and M′ are said to be isomorphic.

We recall the classification (up to isomorphism) of Z_{2}-graded Clifford modules. Let k ∈ {1, 2, …}. We set Cℓ_{k} := Cℓ(R^{k}), where R^{k} is equipped with the standard scalar product. It is well known (s. e.g. [L-M] Ch. I Thm. 5.8., Prop. 5.20. and Prop. 5.4.) that for every k ≢ 0 (mod 4) there is (up to isomorphism) a unique irreducible Z_{2}-graded module M_{k} over Cℓ_{k} and that for every k ≡ 0 (mod 4) there are exactly two irreducible Z_{2}-graded modules M_{k}^{+}, M_{k}^{−} over Cℓ_{k}. Every Z_{2}-graded Clifford module over Cℓ_{k} is a sum of irreducible Z_{2}-graded Clifford modules. As a result, an arbitrary (nontrivial) Z_{2}-graded module over Cℓ_{k} is of the form (p, q ∈ {0, 1, …}; l, p + q ∈ {1, 2, …}):

1. M_{l,k} := l · M_{k} = M_{k} ⊕ ⋯ ⊕ M_{k}, if k ≢ 0 (mod 4);

2. M_{p,q;k} := p · M_{k}^{+} ⊕ q · M_{k}^{−}, if k ≡ 0 (mod 4).

The relation between the notions of equivalence and ϕ-isomorphism is the same as for special isometric mappings.

Proposition II.24 Let x and x′ be Euclidean vector spaces of dimension ≥ 1. Two Z_{2}-graded modules M and M′ over Cℓ(x) and Cℓ(x′) respectively are equivalent, if and only if there is an isomorphism ϕ : x → x′ such that M and M′ are ϕ-isomorphic.

Proof: It is clear that ϕ-isomorphic modules are equivalent. Conversely, suppose M and M′ are equivalent and σ : Cℓ(x) → Cℓ(x′) is an algebra isomorphism establishing the equivalence (s. Def. II.5).

If k = dim x ≢ 0 (mod 4), then, according to our previous remarks, the isomorphism type of the module M is already determined by its dimension. Consequently, M and M′ are ϕ-isomorphic for every isomorphism ϕ : x → x′, simply for dimensional reasons.

If k ≡ 0 (mod 4), the automorphism −Id : x → x induces an automorphism α of the Clifford algebra Cℓ(x) which interchanges the two isomorphism classes of irreducible Z_{2}-graded Cℓ(x)-modules (cf. [L-M] Prop. 5.9.). We decompose M′ into irreducible summands M′^{j}. Let σ^{∗}M′^{j} denote the (irreducible) Z_{2}-graded Cℓ(x)-module induced by σ. Let ϕ : x → x′ be an isomorphism of Euclidean vector spaces and ϕ_{#} : Cℓ(x) → Cℓ(x′) the induced algebra isomorphism. According to what we have just said, it is evident that either the Z_{2}-graded Cℓ(x)-module ϕ_{#}^{∗}M′^{j} or (−ϕ)_{#}^{∗}M′^{j} is isomorphic to σ^{∗}M′^{j}. Without restriction of generality, we assume that ϕ_{#}^{∗}M′^{j} is isomorphic to σ^{∗}M′^{j} for the summand M′^{j} in question. Let M′^{i} be an other summand and θ : Cℓ(x) → Cℓ(x′) an algebra isomorphism. If M′^{i} and M′^{j} are isomorphic, then so are θ^{∗}M′^{i} and θ^{∗}M′^{j}. In particular, ϕ_{#}^{∗}M′^{i} is isomorphic to ϕ_{#}^{∗}M′^{j} and σ^{∗}M′^{i} is isomorphic to σ^{∗}M′^{j}, so ϕ_{#}^{∗}M′^{i} and σ^{∗}M′^{i} are isomorphic. If M′^{i} and M′^{j} are not isomorphic, then neither are θ^{∗}M′^{i} and θ^{∗}M′^{j}. In particular, ϕ_{#}^{∗}M′^{i} is not isomorphic to ϕ_{#}^{∗}M′^{j} and σ^{∗}M′^{i} is not isomorphic to σ^{∗}M′^{j}. Since there are only two isomorphism types of Z_{2}-graded Cℓ(x)- modules, ϕ_{#}^{∗}M′^{i} and σ^{∗}M′^{i} have to be isomorphic in this case as well. In summary, ϕ_{#}^{∗}M′, σ^{∗}M′ and M are isomorphic, hence M′ and M are ϕisomorphic. □

Now we give a slightly refined version of [Al 5] Prop. 10.1. which reduces the classification of special isometric mappings up to equivalence to the classification of Z_{2}-graded Clifford modules up to equivalence. Clifford modules will be tacitly assumed to be non-trivial and of order k > 0.

Proposition II.25 (cf. [Al 5] Prop. 10.1.)

(i) Fix a Euclidean vector space x. There is a one-to-one correspondence between special isometric mappings ψ : x × z → y considered up to isomorphism and Z_{2}-graded Cℓ(x)-modules up to isomorphism.

(ii) There is a one-to-one correspondence between special isometric mappings of order k considered up to equivalence and Z_{2}-graded Clifford modules of order k up to equivalence.

In both cases the correspondence is induced by the map ψ ↦ M_{ψ}.

Proof: Let Ψ : Cℓ(x) → End(M), M = M_{0} ⊕ M_{1}, be a Z_{2}-graded Clifford module. We choose a (positive definite) scalar product ⟨, ⟩ on M invariant under the compact group^{4} Ψ(P in(x)). Set z := (M_{0}, ⟨, ⟩|M_{0} × M_{0}) and y := (M_{1}, ⟨, ⟩|M_{1} × M_{1}), then

ψ_{M}(X, Z) := Ψ(X)Z (X ∈ x, Z ∈ z)

defines a special isometric mapping ψ_{M} : x × z → y (cf. pf. of [Al 5] Prop. 10.1.). The isomorphism type of ψ_{M} does not depend on the choice of invariant scalar product which can be seen by decomposing Ψ|P in(x) into irreducible summands. What we have just described is a map which assigns an isomorphism class (x is fixed) [ψ_{M}] of special isometric mappings to every Z_{2}-graded Cℓ(x)-module M. ψ ↦ M_{ψ} and M ↦ [ψ_{M}] factorize modulo isomorphism relation to mutual inverses, from which (i) follows.

(ii) results from the following remark. Let ϕ : x → x′ be an isomorphism of Euclidean vector spaces. Two special isometric mappings ψ : x × z → y and ψ′ : x′ × z′ → y′ are ϕ-isomorphic, if and only if M_{ψ} and M_{ψ′} are. □

Corresponding to the operations defined for Z_{2}-graded Clifford modules, sums and Z_{2}-graded tensor products of special isometric mappings can be defined.

Definition II.6 (cf. [Al 5] §10) Let ψ : x × z → y be a special isometric mapping. The transpose of ψ is the special isometric mapping ψ^{t} : x×y → z defined by

⟨ψ^{t}(X, Y), Z⟩ = ⟨Y, ψ(X, Z)⟩ .

4We recall: P in(x) is the subgroup of the multiplicative group of invertible elements of Cℓ(x) generated by the elements X ∈ x with |X| = 1.

Let ψ̄ : x̄ × z̄ → ȳ be a second special isometric mapping and ¯ x̄ := x + x̄, ¯z̄:= z ⊗ z̄ + y ⊗ ȳ and ¯ ȳ:= z ⊗ ȳ + y ⊗ z̄. The Z_{2}-graded tensor product ψ ˆ⊗ψ̄ is the special isometric mapping ψ ˆ⊗ψ̄ : ¯ x̄ × ¯ z̄ → ¯ ȳ defined by

ψ ˆ⊗ψ̄ (X + X̄, Z ⊗ Z̄ + Y ⊗ Ȳ ) :=

ψ(X, Z) ⊗ Z̄ − ψ^{t}(X, Y) ⊗ Ȳ + Z ⊗ ψ̄ (X̄, Z̄) + Y ⊗ ψ̄^{t} (X̄, Ȳ ) .

Suppose now that dim x = dim x̄ and fix an isomorphism ϕ : x →^{∼} x̄. The sum ψ ⊕_{ϕ} ψ̄ is the special isometric mapping ψ ⊕_{ϕ} ψ̄ : x × (z + z̄) → y + ȳ defined by

ψ ⊕_{ϕ} ψ̄ (X, Z + Z̄) := ψ(X, Z) + ψ̄ (ϕ(X), Z̄) .

The equivalence class of ψ ⊕_{ϕ} ψ̄ depends, in general, on the isomorphism ϕ. In particular, the sum is not a well defined operation between equivalence classes of special isometric mappings of a fixed order. This detail has been overlooked in [Al 5] §10, for which reason the statement in Thm. 10.1. b) is not well defined and misleading. Herewith, we have essentially anticipated the refinement of [Al 5] Thm. 10.1. which will be presented subsequently.

Now we give the correct classification of Z_{2}-graded Clifford modules up to equivalence. Due to Prop. II.25, this canonically provides the classification of special isometric mappings up to equivalence, which in turn implies the classification of V-spaces up to isometry, thanks to Prop. II.23. We replace [Al 5] Thm. 10.1. by the following proposition.

Proposition II.26 Let k ∈ {1, 2, …}.

(i) Two Z_{2}-graded modules over Cℓ_{k}, k ≢ 0 (mod 4), are equivalent, if and only if they are isomorphic.

(ii) Two Z_{2}-graded modules M_{p,q;k} and M_{p′,q′;k} over Cℓ_{k}, k ≡ 0 (mod 4), are equivalent, if and only if {p, q} = {p′, q′}.

In particular, for every k ∈ {1, 2, …} there is exactly one equivalence class of irreducible Z_{2}-graded modules over Cℓ_{k}.

Proof: (i) is true, since for k ≢ 0 (mod 4) non-isomorphic Z_{2}-graded modules over Cℓ_{k} differ in dimension.

(ii): As the categories of Z_{2}-graded modules over Cℓ_{k} and of ungraded modules over Cℓ_{k−1} are equivalent (s. [L-M] Ch. I Prop. 5.20.), we can equally well discuss the induced notion of equivalence for ungraded Clifford modules. Assume k ≡ 0 (mod 4), so according to [L-M] Ch. I Thm. 5.8.

Cℓ_{k−1} ≅ K(n) ⊕ K(n) (as algebras) ,

where K(n) denotes the algebra of n × n-matrices over the field K, and both n and K ∈ {R, H} depend on k as indicated there. We denote by M_{k−1}^{±} and M_{p,q;k−1} the ungraded Clifford modules corresponding to M_{k}^{±} and M_{p,q;k} respectively. M_{k−1}^{+} resp. M_{k−1}^{−} is given by the standard representation of the first resp. the second K(n)-summand and the trivial representation of the second resp. the first K(n)-summand on K^{n}. The automorphism A⊕B ↦ B⊕A of Cℓ_{k−1} interchanges the K(n)-summands, consequently M_{p,q;k−1} and M_{q,p;k−1} are equivalent. On the other hand, every automorphism of Cℓ_{k−1} permutes the summands K(n), because K(n) is simple. Therefore, it is sufficient to discuss the effect of an automorphism ϕ : K(n) → K(n) on the standard representation ρ of K(n) on K^{n}. ρ ∘ ϕ is irreducible, hence isomorphic to ρ, since simple algebras have only one irreducible representation (cf. [La] Ch. XVII §4 Cor. 2.). So, for every automorphism α : Cℓ_{k−1} → Cℓ_{k−1} the Clifford module α^{∗}M_{p,q;k−1} induced by α is isomorphic to M_{p,q;k−1} or M_{q,p;k−1}, depending on whether α preserves or interchanges the K(n)-summands. □

It should be taken into account that, in line with Prop. II.26, sums of Z_{2}- graded Cℓ_{4m}-modules with equivalent summands may fail to be equivalent, e.g. it is true that

M_{4m}^{+} ⊕ M_{4m}^{+} ∼ M_{4m}^{−} ⊕ M_{4m}^{−} but ≁ M_{4m}^{+} ⊕ M_{4m}^{−} .

This has been apparently overlooked in [Al 5], which results in not distinguishing between modules M_{p,q;4m} and M_{p′,q′;4m} with p + q = p′ + q′ = l. Consequently, the corresponding Alekseevskiĭ spaces are denoted by V(l, 4m) in [Al 5] §10. This notation is ambiguous for l = p + q ≥ 2 and should be replaced by V(p, q; 4m). Therefore, in the following, we will denote by V(p, q; 4m) the V-space corresponding to M_{p,q;4m}. We set V(1, 4m) := V(0, 1; 4m) in harmony with [Al 5]. We conclude this section with a precise formulation of this observation.

Corollary II.27 For m, l ∈ {1, 2, …} let V_{l,4m} denote the family (of isometry classes) of Alekseevskian V-spaces corresponding to Z_{2}-graded Cℓ_{4m}- modules of (real) dimension 2lN(4m), where 2N(k) is the dimension of an irreducible Z_{2}-graded module over Cℓ_{k}. Then

V_{l,4m} = \{V(p, q; 4m)|0 ≤ p ≤ q, p + q = l\} , ♯V_{l,4m} = 1 + [\frac{l}{2}] .

In particular, ♯V_{l,4m} ≥ 2 for l ≥ 2.

Proof: According to Prop. II.26, the modules M_{p,q;4m} with p + q = l and 0 ≤ p ≤ q are pairwise inequivalent, and altogether they represent all the equivalence classes of Z_{2}-graded Cℓ_{4m}-modules of dimension 2lN(4m). □

II.3 Summary

We summarize the analysis undertaken in the sections II.1 and II.2 in the form of a corrected classification theorem.

Theorem II.28 (Classification of Alekseevskiĭ spaces)

(i) Alekseevskian spaces have rank ≤ 4. If their rank is ≤ 2, they are symmetric.

(ii) the spaces of rank 3 are (up to scaling) exactly the spaces

T (p) , p ≥ 0 .

(iii) The spaces of rank 4 are exactly the spaces

W(p, q) , 0 ≤ p ≤ q;

V(l, k) , l, k ≥ 1 , k ≢ 0 (mod 4) ;
V(p, q; 4m) , q, m ≥ 1 , 0 ≤ p ≤ q .

(iv) The symmetric Alekseevskiĭ spaces of rank ≥ 3 are T (0), W(0, q) (q ≥ 0), V(1, 1), V(1, 2), V(1, 4) = V(0, 1; 4) and V(1, 8) = V(0, 1; 8).

This and further information is compiled in the following table. The rank and the quaternionic dimension (dim_{R} /4) of the Alekseevskian spaces is listed in the third and in the fourth columns respectively. 2N(k) denotes the dimension of an irreducible Z_{2}-graded Cℓ_{k}-module and is computable from the recursion formula

N(8s + t) = (16)^{s}N(t) (1 ≤ t ≤ 8, s ≥ 0) , N(1) = 1 , N(2) = 2 ,

N(3) = N(4) = 4 , N(5) = N(6) = N(7) = N(8) = 8 .

From the last column one can decide whether the roots of the Alekseevskian Lie algebra in question form a root system in the sense of Def. II.3. If this is the case, the isomorphism type of the root system is indicated in the usual notation (s. e.g. [He] Ch. X §3 3.).

Alekseevskiĭ space symm. space ? rk.

dim_{H}

root system?

H_{H}^{m}  \frac{Sp(m, 1)}{Sp(m)Sp(1)}  1  m ≥ 1  a_{1}

hermitian symm.

\frac{SU(1, 2)}{U(2)} 1  1  a_{1}

Alekseevskiĭ spaces

\frac{SU(m, 2)}{S (U(m) × U(2))} 2  m ≥ 2  b_{2}

“type 3” (s. [Al 5] §8)  \frac{G_{2}^{(2)}}{SO(4)}  2  2  g_{2}
T (0)  \frac{SO_{0}(3, 4)}{SO(3) × SO(4)}  3  3  b_{3}
T (p), p ≥ 1  no  3  p + 3  no
W(0, 0)  \frac{SO_{0}(4, 4)}{SO(4) × SO(4)}  4  4  d_{4}
W(0, q), q ≥ 1  \frac{SO_{0}(q + 4, 4)}{SO(q + 4) × SO(4)} 4  q + 4  b_{4}
W(p, q), 1 ≤ p ≤ q  no  4  4 + p + q  no
V(1, 1)  \frac{F_{4}^{(4)}}{Sp(3)Sp(1)}  4  7  f_{4}
V(1, 2)  \frac{E_{6}^{(2)}}{SU(6)Sp(1)}  4  10  f_{4}
V(1, 4)  \frac{E_{7}^{(−5)}}{Spin(12)Sp(1)}  4  16  f_{4}
V(1, 8)  \frac{E_{8}^{(−24)}}{E_{7}Sp(1)}  4  28  f_{4}

\begin{matrix}V(l, k), k ≢ 0 (mod 4), \\ (l, k) ∉ \{(1, 1); (1, 2)\} \end{matrix}  no  4 4 + k + 2lN(k)  f_{4}

\begin{matrix}V(p, q; k), k ≡ 0 (mod 4), \\ (p + q, k) ∉ \{(1, 4); (1, 8)\} \end{matrix}  no  4 \begin{matrix}4 + k + 2lN(k) \\ (l = p + q) \end{matrix}  f_{4}

Proof: First of all Alekseevskiĭ’s classification [Al 5] (cf. Ch. I of the present exposition) together with the refinements noted in Thm. II.11 and Cor. II.27) provides the Alekseevskian spaces as list of Alekseevskian Lie algebras. Some of these metric Lie algebras are the solvable part of the Iwasawa decomposition of the isometry algebra of a symmetric space of noncompact type; they correspond to the symmetric Alekseevskiĭ spaces. For deciding which Alekseevskiĭ spaces are symmetric the following argument due to Alekseevskiĭ is enough (cf. [C1] II.3.4 for details). Quaternionic Kählerian symmetric spaces of non-compact type (s. [Be] 14.52 Table for a list) are Alekseevskian (owed to the Iwasawa decomposition), so for every such symmetric space there must be at least one Alekseevskian Lie algebra s of same dimension, same rank and same root system^{5}. From the list of Alekseevskian Lie algebras one gathers (having determined the root systems) that there is always exactly one^{6}. The remaining Alekseevskian Lie algebras necessarily correspond to the non-symmetric Alekseevskiĭ spaces (which also follows from our curvature computations in Ch. III). In [Al 5] Table 1, Alekseevskiĭ gives (without computation) the isomorphism type of the root systems for the Lie algebras he constructed, in case a root system is on hand. For reasons of completeness we discussed the roots for the T -spaces in Prop. II.6 and even for the W- and V-spaces in Prop. II.15 and II.22. □

Proposition II.29 Alekseevskian Lie algebras are of Iwasawa type. In particular, they admit — independently of the scalar product ⟨, ⟩ considered up to now — scalar products ⟨, ⟩_{np} of non-positive sectional curvature. The T - and W-spaces are 4-step, the V-spaces 5-step solvable Lie groups.

Proof: Since Lie algebras of Iwasawa type admit non-positively curved scalar products (s. [Wolt 1] pp. 17-20.), the first statements flow from Prop.

5It is meant that the roots of s have to form a root system (in the sense of Def. II.3)

which is isomorphic to the restricted root system of the symmetric space. In particular, an Alekseevskiĭ space whose roots do not form a root system cannot be symmetric. A list of the restricted root systems of the symmetric spaces of non-compact type is printed in [He] Ch. X Table VI on pp. 532-534.

6In some cases, an Alekseevskiĭ space is already uniquely determined by its rank and its dimension alone, e.g. in the case of the spaces of rank 3 (cf. Thm. II.11).

II.3, II.13 and II.20. The derived series are computed in Prop. II.2, II.12 and II.19. □

Nevertheless we will go on studying (t(p), ⟨, ⟩), because at present we are not interested in non-positively curved metrics but in quaternionic Kählerian ones. We will see in Ch. III that the non-symmetric Alekseevskiĭ spaces have both negatively curved and positively curved tangent planes.

Chapter III

Characterization of Symmetric Spaces

The main result of this chapter is that the symmetric Alekseevskiĭ spaces are completely characterized by the property of being non-positively curved (Theorem III.10).

First, we make use of the formulas provided for the covariant derivative of the T - and W-spaces to compute curvatures. Let s be a metric Lie algebra. We use the abbreviation R_{S} for the operator R_{S} : s → s defined as R_{S}T := R(T, S)S (S, T ∈ s) by means of the curvature tensor.

Proposition III.1 The curvature of T (p) satisfies:

R_{A} = −ad_{A}^{2} for A ∈ a .

In particular, the eigenspace decomposition of the operators R_{A} ensues from the root space decomposition in the proof of Prop. II.3.

Let p ≥ 1. The operators R_{X_{+}} (X_{+} ∈ x_{+}, |X_{+}| = 1) have the following eigenspace decomposition:

R_{X_{+}}| (H_{1} + \frac{\sqrt{2}}{2}H_{2})^{⊥}∩ a = 0 , R_{X_{+}} (H_{1} + \frac{\sqrt{2}}{2}H_{2}) = −\frac{3}{8} (H_{1} + \frac{\sqrt{2}}{2}H_{2}) ,

R_{X_{+}}| (G_{1} − \frac{\sqrt{2}}{2}G_{2})^{⊥}∩ Ja = 0 , R_{X_{+}} (G_{1} − \frac{\sqrt{2}}{2}G_{2}) = −\frac{3}{8} (G_{1} − \frac{\sqrt{2}}{2}G_{2}) ,

R_{X_{+}}| (H̃_{0} + \frac{\sqrt{2}}{2}H̃_{2})^{⊥} ∩ ã = 0, R_{X_{+}} (H̃_{0} + \frac{\sqrt{2}}{2}H̃_{2}) = −\frac{3}{8} (H̃_{0} + \frac{\sqrt{2}}{2}H̃_{2}) ,

R_{X_{+}}| (G̃_{0} − \frac{\sqrt{2}}{2}G̃_{2})^{⊥} ∩ J̃a = 0, R_{X_{+}} (G̃_{0} − \frac{\sqrt{2}}{2}G̃_{2}) = −\frac{3}{8} (G̃_{0} − \frac{\sqrt{2}}{2}G̃_{2}) ,

R_{X_{+}}X_{+} = R_{X_{+}} J̃X_{+} = 0 , R_{X_{+}}JX_{+} = −\frac{3}{4}JX_{+} , R_{X_{+}} X̃_{+} = −\frac{3}{4}X̃_{+} ,

R_{X_{+}}X′_{+} = −\frac{3}{8}X′_{+} , R_{X_{+}} J̃X′_{+} = \frac{1}{8}J̃X′_{+} ,

R_{X_{+}}JX′_{+} = −\frac{1}{8}JX′_{+} , R_{X_{+}} X̃′_{+} = −\frac{1}{8}X̃′_{+} ,

where X′_{+} ∈ x_{+} and X′_{+} ⊥ X_{+}. In particular, if X′_{+} ≠ 0 (which requires p ≥ 2),

K (X_{+} ∧ J̃X′_{+}) = +\frac{1}{8} .  (III.1)

Analogous formulas for the operators R_{J_{α}A} and R_{J_{α}X_{+}} (α = 1, 2, 3) arise from the Sp(1)-invariance of the curvature.

Proof: The first statement follows immediately from ∇_{A} = 0. The operators R_{X}|a (X ∈ x_{+} ∪ x_{−} ∪ x̃_{+} ∪ x̃_{−}) are easy to compute, due to ∇_{A} = 0 (A ∈ a). From this one obtains expressions for R_{X}|J_{α}a (α = 1, 2, 3) using the Sp(1)-invariance of the curvature (eq. I.2). The remaining formulas result from Prop. II.9 after a short computation. □

Corollary III.2 The sectional curvature of T (1) satisfies the sharp estimate

−1 ≤ K(S ∧ T) ≤ 0

for all planes with S ∈ t(1) and T ∈ t(0).

Proof: Since T (0) is totally geodesic, it is enough to show the estimate for S ∈ t(0) ∪ t(0)^{⊥}. The Sp(1)-invariance of the curvature allows us further to assume that S ∈ a ∪ x_{+}. Now the estimate follows from Prop. III.1. □

Corollary III.3 The sectional curvature of the symmetric space T (0) satisfies the sharp estimate

−1 ≤ K ≤ 0 .

Proof: This results e.g. from Cor. III.2, as T (0) is totally geodesic in T (1). □

Now we repeate the same kind of computation for the W-spaces (for more details concerning their curvature s. [C2] II.6).

Proposition III.4 The curvature of W(p, q) satisfies:

R_{A} = −ad_{A}^{2} for A ∈ a .

In particular, the eigenspace decomposition of the operators R_{A} ensues from the root space decomposition in Prop. II.13.

Let p ≥ 1. The operators R_{Y_{−}} (Y_{−} ∈ y_{−}, |Y_{−}| = 1) have the following eigenspace decomposition:

R_{Y_{−}}(H_{1} − H_{3}) = −\frac{1}{2}(H_{1} − H_{3}) , R_{Y_{−}}(H_{1} − H_{3})^{⊥} ∩ a = 0 ,

R_{Y_{−}}(G_{1} + G_{3}) = −\frac{1}{2}(G_{1} + G_{3}) , R_{Y_{−}}(G_{1} + G_{3})^{⊥} ∩ Ja = 0 ,

R_{Y_{−}}(H̃_{0} − H̃_{2}) = −\frac{1}{2}(H̃_{0} − H̃_{2}) , R_{Y_{−}}(H̃_{0} − H̃_{2})^{⊥} ∩ ã = 0 ,

R_{Y_{−}}(G̃_{0} + G̃_{2}) = −\frac{1}{2}(G̃_{0} + G̃_{2}) , R_{Y_{−}}(G̃_{0} + G̃_{2})^{⊥} ∩ J̃a = 0 ,

R_{Y_{−}}Y_{−} = 0 , R_{Y_{−}}JY_{−} = −\frac{1}{2}JY_{−} , R_{Y_{−}}|y_{−}∩Y_{−}^{⊥} = −\frac{1}{2}Id , R_{Y_{−}}|y_{+}∩JY_{−}^{⊥} = 0 ,

R_{Y_{−}}|z = −\frac{1}{4}Id ,

R_{Y_{−}} Ỹ_{−} = −\frac{1}{2}Ỹ_{−} , R_{Y_{−}} J̃Y_{−} = −\frac{1}{2}J̃Y_{−} , R_{Y_{−}}|Ỹ_{−}^{⊥} ∩ J̃Y_{−}^{⊥} ∩ ỹ = 0 ,

R_{Y_{−}}|z̃_{±} = ±\frac{1}{4}Id .

In particular,

K(Y_{−} ∧ Z̃_{+}) = +\frac{1}{4} ,  (III.2)

if Y_{−} ∈ y_{−} − {0} and Z_{−} ∈ z_{−} − {0}, which requires p ≥ 1 and q ≥ 1.

For q ≥ 1 the eigenspace decomposition of R_{Z_{−}} (Z_{−} ∈ z_{−}, |Z_{−}| = 1) can be obtained by simultaneously interchanging the roles of y and z and the indices 2 and 3 in the expressions for R_{Y_{−}}. Further eigenspace decompositions flow immediately from the Sp(1)-invariance of the curvature.

Proof: The proof is similar to that of Prop. III.1. The penultimate statement arises from the standard isomorphism w(p, q) →^{∼} w(q, p). □

Corollary III.5 The sectional curvature of w(p, q) satisfies the following sharp estimates for A ∈ a, Y_{−} ∈ y_{−} and W ∈ w arbitrary

−1 ≤ K(A ∧ W) ≤ 0 ,

−\frac{1}{2} ≤ K(Y_{−} ∧ W) ≤ \frac{1}{4} .

As before Y_{−} may be replaced by Z_{−} ∈ z_{−} and further estimates follow from Sp(1)-invariance.

Proof: This follows from the eigenspace decomposition of Prop. III.4 (cf.

[C1] p. 45 for trivial details) □

Corollary III.6 The sectional curvature of the symmetric spaces W(p, 0) ≅ W(0, p) satisfies the sharp estimate

−1 ≤ K ≤ 0 .

Proof: The linear isotropy representation of a symmetric space is transitive on the flat totally geodesic (infinitesimal) subspaces of maximal dimension (s. [He] Ch.V §6 Theorem 6.2). Therefore, it is sufficient to consider sectional curvatures K(A ∧ W) for A ∈ a. Now the claim follows from Cor. III.5. □

In dealing with the V-spaces we will make use of the following two lemmas.

Lemma III.7 With the notations introduced in sec. I.3, the V-spaces satisfy

K(Y_{−} ∧ Z̃_{+}) = \frac{1}{4} (1 − 3 |Y_{−} ∘ Z_{+}|^{2})

for Y_{−} ∈ y_{−} and Z_{+} ∈ z_{+} with |Y_{−}| = |Z_{+}| = 1. Furthermore,

|Y_{−} ∘ Z_{+}|^{2} = \frac{1}{2} ∑_{i}⟨Y_{−}, ψ (X_{−}^{i}, JZ_{+})⟩^{2}

for every orthonormal basis (X_{−}^{i})_{i=1,...,k} of x_{−}.

Proof: The following holds because the covariant derivative is compatible with the metric:

K(Y_{−} ∧ Z̃_{+}) = −⟨∇Z̃_{+} Z̃_{+}, ∇_{Y_{−}}Y_{−}⟩ + ⟨∇_{Y_{−}} Z̃_{+}, ∇Z̃_{+}Y_{−}⟩ − ⟨∇_{T_{Y_{−}} Z̃_{+}} Z̃_{+}, Y_{−}⟩ .

Using the Koszul formula we compute

∇_{Y_{−}}Y_{−} = \frac{1}{2} (H_{1} − H_{3}) and then − ⟨∇Z̃_{+} Z̃_{+}, ∇_{Y_{−}}Y_{−}⟩ = \frac{1}{4} .

In dealing with the second and the third term we use the vanishing of the covariant derivative’s torsion, the definition of the representation T (s. Thm. I.4) and, at the places labeled by (∗), the identity ∇_{Y_{−}} Z̃_{+} = ˜ ∇_{Y_{−}}Z_{+}, which results from the Koszul formula.

⟨∇_{Y_{−}} Z̃_{+}, ∇Z̃_{+}Y_{−}⟩ = |∇_{Y_{−}} Z̃_{+}|^{2} − ⟨∇_{Y_{−}} Z̃_{+}, T_{Y_{−}} Z̃_{+}⟩
=^{(∗)} |∇_{Y_{−}}Z_{+}|^{2} − ⟨∇_{Y_{−}} Z̃_{+}, Ỹ_{−} ∘ Z_{+}⟩
=^{(∗)} − |∇_{Y_{−}}Z_{+}|^{2} = −\frac{1}{4} |Y_{−} ∘ Z_{+}|^{2} .

In addition, for the last term we use again the compatibility of the covariant derivative with the metric and, at the place labeled by (a), the fact that Y_{−} ∘ Z_{+} ∈ x_{+} and T_{Y_{−}}x_{+} = 0.

−⟨∇_{T_{Y_{−}} Z̃_{+}} Z̃_{+}, Y_{−}⟩ = −⟨∇_{Y_{−}∘Z_{+}} Z̃_{+}, Y_{−}⟩ = ⟨Z̃_{+}, ∇_{Y_{−}∘Z_{+}}Y_{−}⟩
= −⟨Z̃_{+}, T_{Y_{−}} Ỹ_{−} ∘ Z_{+}⟩ + ⟨Z̃_{+}, ∇_{Y_{−}} Ỹ_{−} ∘ Z_{+}⟩
=^{(a)} −⟨∇_{Y_{−}} Z̃_{+}, Ỹ_{−} ∘ Z_{+}⟩ =^{(∗)} −\frac{1}{2} |Y_{−} ∘ Z_{+}|^{2} .

The second statement is a trivial consequence of the remark on p. 13:

|Y_{−} ∘ Z_{+}|^{2} = ∑_{i}⟨Y_{−} ∘ Z_{+}, JX_{−}^{i}⟩^{2} = \frac{1}{2} ∑_{i}⟨Y_{−}, ψ (X_{−}^{i}, JZ_{+})⟩^{2} . □

Lemma III.8 V-spaces which correspond to sums or Z_{2}-graded tensor products of (at least two non-trivial) Z_{2}-graded Clifford modules (of order k ≥ 1) show positively curved tangent planes.

Proof: Sums: In this case we have z_{−} = v^{0} + w^{0} and y_{−} = v^{1} + w^{1} (orthogonal sums), where v^{0} + v^{1} and w^{0} + w^{1} are Z_{2}-graded modules over Cℓ(x_{−}) ≅ Cℓ_{k}. ψ : x_{−} × z_{−} → y_{−} is the restriction of Clifford multiplication. Choosing JZ_{+} ∈ v^{0} − {0} and Y_{−} ∈ w^{1} − {0} entails ψ(X_{−}, JZ_{+}) ∈ v^{1} ⊥ w^{1} and hence K(Y_{−} ∧ Z̃_{+}) = 1/4 > 0, due to Lemma III.7.

Tensor products: In this case the special isometric mapping ψ defining the V-space is of the form

ψ = ψ̄ˆ⊗¯ψ̄ : x_{−} × z_{−} → y_{−} , where
ψ̄ : x̄ × z̄ → ȳ and ¯ ψ̄ : ¯ x̄ × ¯ z̄ → ¯ ȳ

are special isometric mappings and

x_{−} = x̄ + ¯ x̄ , z_{−} = z̄ ⊗ ¯ z̄ + ȳ ⊗ ¯ ȳ and y_{−} = z̄ ⊗ ¯ ȳ + ȳ ⊗ ¯ z̄.

If we choose

Y_{−} = Ȳ ⊗ ¯ Z̄ ∈ ȳ ⊗ ¯ z̄ ⊂ y_{−} and JZ_{+} = Z̄ ⊗ ¯ Z̄′ ∈ z̄ ⊗ ¯ z̄ ⊂ z_{−}

such that Y_{−} ≠ 0, Z_{+} ≠ 0 and ¯ Z̄ ⊥ ¯ Z̄′, then the following holds for every X_{−} = X̄ + ¯ X̄ ∈ x̄ + ¯ x̄ = x_{−}, according to the definition of ψ̄ˆ⊗¯ψ̄ (s. Def. II.6)

⟨Y_{−}, ψ (X_{−}, JZ_{+})⟩ = ⟨Ȳ ⊗ ¯ Z̄, ψ̄ (X̄, Z̄) ⊗ ¯ Z̄′⟩ = 0 .

Hence, K(Y_{−} ∧ Z̃_{+}) = 1/4 > 0, again by the previous lemma. □

Corollary III.9 The following spaces show positively curved tangent planes:

(i) V(l, k), if l > 1 or k > 8.

(ii) V(p, q; 4m), if p + q > 1 or 4m > 8.

Proof: The Z_{2}-graded Clifford module defining V(l, k) resp. V(p, q; 4m) is, by definition, a sum of l resp. p + q irreducible Z_{2}-graded Clifford modules. Therefore if l > 1 resp. p + q > 1, then the corresponding V-space shows positively curved tangent planes, owing to Lemma III.8.

By virtue of the periodicity modulo 8 of Z_{2}-graded Clifford modules (s. [A-B-S] Cor. 6.6), every irreducible Z_{2}-graded module over Cℓ_{k} for k > 8 is a non-trivial Z_{2}-graded tensor product of Z_{2}-graded Clifford modules. Therefore, again by Lemma III.8, V-spaces corresponding to Clifford modules of order > 8 show positively curved tangent planes. □

According to a theorem of J.E. D’Atri and I. Dotti Miatello [DA-DM], a bounded homogeneous domain is symmetric, if and only if its sectional curvature (with respect to the Bergmann metric) is non-positive. We proof an analogue of this statement for Alekseevskiĭ spaces.

Theorem III.10 Let S be an Alekseevskiĭ space. Then the following conditions are equivalent:

(i) S is symmetric.

(ii) The sectional curvature of S is non-positive.

Proof: For every non-symmetric Alekseevskiĭ space we will exhibit tangent planes of positive curvature. There are three families to treat; T -, Wand V-spaces.

T -spaces: Let T (p) be non-symmetric, i.e. p ≥ 1. We can choose X_{+} ∈ x_{+} with |X_{+}| = 1, since p = dim x_{+}. We claim that the tangent plane

τ = (H_{0} + X_{+}) ∧ (G̃_{0} − J̃X_{+})

is positively curved. With the help of the propositions III.1 and II.9 and considering the fact that t(0) is totally geodesic in t(1) we obtain

4K(τ) = |τ|^{2}K(τ) = K(H_{0} ∧ G̃_{0}) + K(H_{0} ∧ J̃X_{+}) + K(G̃_{0} ∧ X_{+}) + K(X_{+} ∧ J̃X_{+}) − 2 (⟨R(H_{0}, G̃_{0})J̃X_{+}, X_{+}⟩ + ⟨R(H_{0}, J̃X_{+})G̃_{0}, X_{+}⟩)

= −\frac{1}{4} − \frac{1}{4} − \frac{1}{4} + 0 − 2 (−\frac{1}{4} − \frac{1}{4}) = \frac{1}{4} > 0 .

W-spaces: Let W(p, q) be non-symmetric, i.e. p ≥ 1 and q ≥ 1. We can choose Y_{−} ∈ y_{−} − {0} and Z_{+} ∈ z_{+} − {0}, since p = dim y_{−} and q = dim z_{+}. According to eq. III.2 in Prop. III.4, the plane Y_{−} ∧ Z̃_{+} is positively curved.

V-spaces: Thanks to Cor. III.9, we only need to exhibit positively curved tangent planes in the case of v(1, k) for k = 3, 5, 6 and 7. Let ψ_{k} be the special isometric mapping determined up to equivalence by v(1, k), i.e. one which defines an irreducible Z_{2}-graded module over Cℓ_{k}. It is well known ([Al 5] §10) and obvious that ψ_{3} : x_{−} × z_{−} → y_{−} can be realized restricting quaternion multiplication:

z_{−} = y_{−} = H and x_{−} = span{1, i, j} ⊂ H .

Where H is supplied with the standard scalar product. Set Y_{−} = k, JZ_{+} = 1 and (X_{−}^{1}, X_{−}^{2}, X_{−}^{3}) = (1, i, j). Then Lemma III.7 gives

|Y_{−} ∘ Z_{+}|^{2} = \frac{1}{2} (⟨k, 1⟩ + ⟨k, i⟩ + ⟨k, j⟩) = 0 ,

hence K(Y_{−} ∧ Z̃_{+}) = 1/4 > 0. The cases k = 5, 6, 7 will be treated analogously. Let Ca denote the Cayley octaves provided with the standard scalar product,

z_{−} = y_{−} = Ca and x_{−} ⊂ Ca a k-dimensional subspace .

ψ_{k} : x_{−} × z_{−} → y_{−} is obtained by restricting the Cayley product. Choosing Y_{−} ∈ x_{−}^{⊥} − {0}, JZ_{+} = 1 and any orthonormal basis (X_{−}^{i})_{i=1,...,k} of x_{−} and applying Lemma III.7 gives

|Y_{−} ∘ Z_{+}|^{2} = \frac{1}{2} ∑_{i=1}^{k}⟨Y_{−}, X_{−}^{i}⟩ = 0 ,

hence again K(Y_{−} ∧ Z̃_{+}) = 1/4 > 0. □

Remark: In eq. (III.1) of Prop. III.1 we had already exhibited positively curved tangent planes of T (p) for p ≥ 2. For T (1) such planes, as given in the proof of Thm. III.10, are less easy to find.

Acknowledgements: This paper is based on my doctoral thesis written under the supervision of W. Ballmann and U. Hamenstädt, whom I would like to thank for continuous encouragement. I am also grateful to O. Baues, R. Briegel and T. Kakolewski, who read preliminary versions of the (German) manuscript. Finally, I would like to thank A. Swann for drawing my attention to the paper [dW-VP] of B. de Wit and A. Van Proeyen.

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