Isometry Groups of Homogeneous Quaternionic Kähler Manifolds

D.V. Alekseevsky^{∗} Max-Planck-Institut

für Mathematik Gottfried-Claren-Str. 26

D-53225 Bonn

V. Cortés^{†}

Mathematisches Institut der Universität Bonn Beringstr. 6 D-53115 Bonn

November 11, 1999

Abstract

A general method for calculation of the full isometry group of a Riemannian solvmanifold is presented.

Using it we determine the full isometry group of the non-symmetric quaternionic Kähler solvmanifolds M: T -, W- and V-spaces.

As an application we prove that the isometry group acts transitively on the twistor space and on the SO(3)-principal (“3-Sasakian”) bundle of M and that the manifold M does not admit quotients of finite volume.

As other application, we give a simple description of the quaternionic Kähler solvmanifolds in terms of a certain spinorial module S of the group Spin(3, 3 + k). The Lie bracket is defined by means of the unique embedding of the vector module V = ℝ^{3,3+k} into ∧^{2}S. We describe also the group of isometries which preserves the principal Kähler submanifold U ⊂ M.

Introduction

We recall that a quaternionic Kähler manifold is a Riemannian manifold M^{4n} with holonomy group Hol contained in Sp(1)Sp(n). These are Einstein manifolds if n > 1. The only known examples of complete quaternionic Kähler manifolds of positive scalar curvature are symmetric and in one-to-one correspondence with the simple compact Lie algebras (Wolf spaces). On the other hand, many examples of non-symmetric quaternionic Kähler manifolds of negative scalar curvature are known. The first such examples were constructed in [A 2], where the classification of quaternionic Kähler manifolds admitting a transitive splittable solvable group of isometries was given. One additional series of such manifolds, missing in the classification [A 2] was discovered by the physicists B. de Wit and A. Van Proeyen in the context of supergravity models and the original algebraic classification was completed in [C]. The final result is that there exist three such series of non-symmetric homogeneous quaternionic Kähler manifolds depending on integer parameters: T (p), W(p, q) and V(l, k) (k ≢ 0 (4)), V(p, q; k) (k ≡ 0 (4)). We will call them quaternionic Kähler solvmanifolds.

∗e-mail: daleksee@mpim-bonn.mpg.de; partially supported by Erwin Schrödinger Institut (Vienna) and Max-Planck-Institut für Mathematik (Bonn).

†Fax: +49-228-737916; e-mail: vicente@math.uni-bonn.de; partially supported by ESI (Vienna) and SFB 256 (Bonn University).

Quaternionic Kähler solvmanifolds and the associated “principal Kähler submanifolds” are studied by physicists (s. [Ce], [C-F-G], [dW-VP 0-3] and [dW-V-VP]) in the context of N = 2 supergravity. They describe the coupling of n vector multiplets to N = 2 supergravity in dimension d = 4 in terms of a meromorphic function F (X) on ℂ^{n+1} which is homogeneous of degree 2. They associate with F a Kähler metric g on (an open subset of) the projective space P ℂ^{n+1} with the Kähler potential

K = ln [ ∑_{i,j=0}^{n}(∂_{i}∂_{j}F + \overline{∂_{i}∂_{j}F})\frac{X^{i}}{X^{0}} \overline{(\frac{X^{j}}{X^{0}} )}] .

The corresponding metric is called special Kähler. Using dimensional reduction from d = 4 to d = 3, to every special Kähler metric g they associate a metric g̃, which appear to be quaternionic Kähler and is also called special. The metric g̃ is explicitely expressed in terms of F, it is homogeneous if g is homogeneous. If g̃ is the metric of a quaternionic Kähler solvmanifold, then g is the metric of the essential part U_{0} of the principal Kähler submanifold U = F_{0} × U_{0} (associated with the principal Kählerian subalgebra 𝔲 = 𝔣_{0}+𝔲_{0}). In [dW-VP 0-3] and [dW-V-VP] B. de Wit, F. Vanderseypen and A. Van Proeyen studied the isometries of special geometries and in particular described the full isometry algebra of quaternionic Kähler solvmanifolds in terms of symmetries of the Lagrangian associated to F.

In the present paper we determine the full isometry algebra and the full isometry group of the quaternionic Kähler solvmanifolds using the purely algebraic approach of Lie group theory. It seems that our results are consistent with that of B. de Wit, F. Vanderseypen and A. Van Proeyen.

Now we sketch the structure of the paper. In section 1 a general method for calculation of the full isometry Lie algebra and the full isometry Lie group of Riemannian solvmanifolds is presented. For simplicity we assume that the simply transitive solvable Lie group satisfies some additional conditions which are fulfilled for the quaternionic Kähler solvmanifolds (also for homogeneous bounded domains and for homogeneous spaces of non-positive curvature).

Let (M, g) be a Riemannian manifold with a simply transitive group of isometries L. We can equip the Lie algebra 𝔩 of L with the Euclidean scalar product ⟨, ⟩ induced by g. If M is simply connected it can be reconstructed from the metric Lie algebra (𝔩, ⟨, ⟩).

Under some conditions we prove that the description of the full isometry Lie algebra 𝔤(M) reduces to the description of so-called suitable SRdecompositions 𝔩 = 𝔩_{s} + 𝔩_{r} of (𝔩, ⟨, ⟩), s. Thm. 1.1. An SR-decomposition is a semi-direct orthogonal decomposition into an ideal (𝔩_{r}, ⟨, ⟩) and the Iwasawa algebra (𝔩_{s}, ⟨, ⟩) of some semi-simple Lie algebra 𝔰 with scalar product associated to a symmetric Riemannian metric on the corresponding manifold L_{s}. It is suitable if the adjoint representation ad : 𝔩_{s} → 𝔡𝔢𝔯(𝔩_{r}) is extended to a representation ρ : 𝔰 → 𝔡𝔢𝔯(𝔩_{r}) with some properties.

If 𝔩 = 𝔩_{s} + 𝔩_{r} is a suitable SR-decomposition with maximal dimension of 𝔩_{s}, then 𝔤(M) = (𝔰 + 𝔡_{0}(𝔩_{r}))+ ∋𝔩_{r}, where 𝔡_{0}(𝔩_{r}) denotes the Lie algebra of skew-symmetric derivations of 𝔩_{r} which commute with ρ(𝔰).

Starting from the description of the full isometry algebra, we can describe the full isometry group in terms of outer automorphisms of the semisimple Lie algebra 𝔰 (s. Thm. 1.5, Cor. 1.8 and Cor. 1.9).

In section 2 we apply this algorithm to the quaternionic Kähler solvmanifolds and determine the full isometry algebra of the T -, W- and V-spaces (s. Thm. 2.5, Thm. 2.11 and Thm 2.18). In all the cases the maximal semisimple Lie subalgebra of non-compact type 𝔰 ⊂ 𝔤(M) has the form 𝔰 = 𝔰𝔬(3, 3 + k).

In section 3 we determine the automorphism group of the Lie algebra 𝔰𝔬(3, 3 + k) (s. Prop. 3.1) and use this to determine the full (not necessarily connected) isometry group (s. Thm. 3.6, Thm. 3.7 and Thm. 3.9).

Our results have the following consequences:

1. The isometry group of the quaternionic Kählerian solvmanifold M acts transitively on the twistor space and on the SO(3)-principal bundle of M (s. Thm. 3.10).

2. A quaternionic Kähler solvmanifold M admits a (smooth) quotient M/Γ of finite volume if and only if M is symmetric (s. Cor. 1.10 and Cor. 1.11); here Γ is a discrete group of isometries.

Theorems 3.6, 3.7 and 3.9 give a new simple description of the quaternionic Kähler solvmanifolds in terms of certain spinorial representation of Spin(3, 3 + k).

For example, we can describe the quaternionic Kähler manifold T (p), p ≥ 1, as follows. Let (V = ℝ^{4}, ω_{0}) be Euclidean 4-space with standard symplectic form ω_{0}. Then ∧^{2}V as Sp(V)-module can be decomposed as

∧^{2}V = ∧_{0}^{2}V + ℝω^{0} ,

where ω^{0} = ω_{0}^{−1} ∈ ∧^{2}V. We define a Lie algebra structure on the vector space

𝔱_{r} = V ⊗ ℝ^{p} + ∧_{0}^{2}V + ℝω^{0}

by

ad_{ω^{0}}|V ⊗ ℝ^{p} = \frac{1}{2}Id, ad_{ω^{0}}| ∧_{0}^{2} V = Id , [∧_{0}^{2}V, V ⊗ ℝ^{p} + ∧_{0}^{2}V] = 0 ;

[v ⊗ x, w ⊗ y] = (v ∧ w)_{0}⟨x, y⟩ , v, w ∈ V, x, y ∈ ℝ^{p} ,

where the subindex 0 denotes the natural projection ∧^{2}V → ∧_{0}^{2}V and ⟨, ⟩ is the standard scalar product on ℝ^{p}. We denote by ⟨, ⟩ the scalar product on 𝔱_{r} induced by the given scalar products on V and ℝ^{p}. By definition the decomposition of 𝔱_{r} is orthogonal,

⟨v ∧ w, v ∧ w⟩ = ⟨v, v⟩⟨w, w⟩ − ⟨v, w⟩^{2} and ⟨ω^{0}, ω^{0}⟩ = 2 .

The symplectic group Sp(V) = Sp(4, ℝ) acts on the Lie algebra 𝔱_{r} as group of automorphisms. Its maximal compact subgroup U(2) preserves the scalar product on 𝔱_{r}. Consider the linear group Sp(V) · T_{r} ⊂ Aut(𝔱_{r}) with Lie algebra 𝔰𝔭(V)+ ∋𝔱_{r} ⊂ 𝔡𝔢𝔯(𝔱_{r}). Then

T (p) ≅ \frac{Sp(V) · T_{r}}{U(2)} .

Let 𝔰𝔭(V) = 𝔲(2) + 𝔪 be a Cartan decomposition and ⟨, ⟩_{𝔪} be a certain 𝔲(2)-invariant scalar product on 𝔪. Then the quaternionic Kähler metric on T (p) = (Sp(V) · T_{r})/U(2) is induced by the scalar product ⟨, ⟩_{𝔪} + ⟨, ⟩ on 𝔪 + 𝔱_{r} ≅ T_{U(2)}T (p).

Remark that in this case we may identify Sp(4, ℝ) = Spin_{0}(3, 2); then V is the semi-spinor module of Spin(3, 2) and ∧_{0}^{2}V = ℝ^{3,2} the vector representation of Spin(3, 2). A similar description is given for the other two series.

Acknowledgements This paper was started, when both authors participated in the Quaternionic Program at Erwin Schrödinger Institut in Vienna. It was finished during the stay of the first author at Max-Planck-Institut in Bonn. We are very grateful to ESI, MPI and SFB 256 for the stimulating atmosphere, hospitality and support.

We thank Bernard de Wit and Antoine Van Proeyen for useful discussions and introduction to supergravity. The second author also thanks Werner Ballmann and Ursula Hamenstädt for continuous encouragement and interest in this work.

Contents

Introduction i

1 Isometry Groups of Riemannian Solvmanifolds 1 1.1 Admissible Lie Algebras . . . . . . . . . . . . . . . . . . . . . 1 1.2 Holonomy Conditions . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Riemannian Homogeneous Spaces associated with an admissi-

ble Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Full Isometry Group and Algebra . . . . . . . . . . . . . . . . 5

2 Full Isometry Algebra of Quaternionic Kähler Solvmanifolds 12 2.1 T -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 W-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 V-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Full Isometry Group of Quaternionic Kähler Solvmanifolds 33 3.1 Automorphism Group of 𝔰 = 𝔰𝔬(3, 3 + k) . . . . . . . . . . . . 33 3.2 T -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 W-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 V-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References 38

1 Isometry Groups of Riemannian Solvmanifolds

1.1 Admissible Lie Algebras

Definition 1.1 A metric Lie algebra is a Lie algebra 𝔩 together with a Euclidean scalar product ⟨, ⟩. We say that a metric Lie algebra is irreducible if it cannot be decomposed into a direct orthogonal sum of two metric Lie algebras. A solvable metric Lie algebra (𝔩, ⟨, ⟩) is said to be admissible if

(i) it is an orthogonal sum of an Abelian Cartan subalgebra 𝔞 and of its derived Lie algebra 𝔫 = [𝔩, 𝔩] :

𝔩 = 𝔞 + 𝔫  (1)

(ii) the operators ad_{A}|𝔫, A ∈ 𝔞 − {0}, are non-zero and symmetric and (iii) there exists an element A^{0} ∈ 𝔞 such that ad_{A^{0}}|𝔫 > 0.

We denote by R ⊂ 𝔞^{∗} the set of roots of 𝔩 with respect to the Cartan subalgebra 𝔞. We have the following orthogonal root space decomposition

𝔩 = 𝔞 + ∑_{α∈R} 𝔫_{α}, 𝔫_{α} = {X ∈ 𝔫| ad_{A}X = α(A)X}.

Remark 1: 𝔩 has trivial center: 𝔷𝔢𝔫𝔱(𝔩) = 0. The nilradical of 𝔩 is 𝔫.

Example 1: Let 𝔰 be a semisimple Lie algebra of non-compact type and 𝔰 = 𝔨 + 𝔞 + 𝔫 its Iwasawa decomposition, where 𝔨 is a maximal compact subalgebra, that is the Lie algebra of a maximal compact subgroup of the adjoint group, 𝔫 is a nilpotent subalgebra and 𝔞 is a maximal ad-diagonalizable subalgebra normalizing 𝔫. Then 𝔦(𝔰) = 𝔞 + 𝔫 equipped with the scalar product ⟨, ⟩_{B} induced on 𝔦(𝔰) ≅ 𝔰/𝔨 by the Cartan-Killing-form B is admissible. The metric Lie algebra (𝔦(𝔰), ⟨, ⟩_{B}) is unique up to isomorphism (of metric Lie algebras). More generally we will consider Euclidean scalar products ⟨, ⟩ on 𝔦(𝔰) which are obtained from ⟨, ⟩_{B} via scaling along the irreducible summands of (𝔦(𝔰), ⟨, ⟩_{B}). We say that the admissible Lie algebra (𝔦(𝔰), ⟨, ⟩) is a symmetric Iwasawa algebra associated with 𝔰. We remark that the scalar products ⟨, ⟩ described above are precisely the Euclidean scalar products such that (𝔦(𝔰), ⟨, ⟩) is symmetric in the sense of Def. 1.2. In this case R ∪ −R is a root system, the system of restricted roots of the real Lie algebra 𝔰.

Example 2: Let 𝔩 = 𝔩_{1}∈ +𝔩_{2} be an orthogonal semidirect sum of admissible Lie algebras 𝔩_{i} = 𝔞_{i} + 𝔫_{i} (i = 1, 2), i.e. [𝔩_{1}, 𝔩_{2}] ⊂ 𝔩_{1}. A necessary and sufficient condition for 𝔩 to be admissible is that the operators ad_{A}, A ∈ 𝔞_{1} + 𝔞_{2}, be symmetric.

1.2 Holonomy Conditions

The notion of covariant derivative is defined for metric Lie algebras by means of the following Koszul formula (X, Y, Z ∈ 𝔩):

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

Definition 1.2 Let 𝔩 be a metric Lie algebra. The Kostant algebra 𝔨𝔬𝔰 is the Lie subalgebra of 𝔰𝔬(𝔩) generated by the endomorphisms ∇_{X} (X ∈ 𝔩).

The holonomy algebra 𝔥𝔬𝔩 is the ideal 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} ∈ 𝔩 , k = 0, 1, 2, ….

𝔩 is said to be symmetric if the curvature tensor is annihilated by 𝔥𝔬𝔩.

Definition 1.3 Let V be a Euclidean vector space. A complex structure on V is a skew-symmetric 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}.

Definition 1.4 A metric Lie algebra 𝔩 with complex structure J is said to be Kählerian, if

[𝔥𝔬𝔩, J] = 0 .

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

[𝔥𝔬𝔩, 𝔮] ⊂ 𝔮

(resp. if 𝔮 annihilates the curvature tensor).

Example 3: The basic examples of admissible Kählerian Lie algebras are the following.

A key algebra 𝔣 = span{G, H} with root μ > 0 is defined in terms of the orthonormal basis G = JH, H by the formula [H, G] = μG. Up to scaling 𝔣 = 𝔦(𝔰𝔲(1, 1)).

Given 𝔣 and a Euclidean vector space 𝔵 with complex structure, 𝔢 = 𝔣 + 𝔵 carries a canonical Euclidean scalar product ⟨, ⟩ and complex structure J. The structure of an elementary Kählerian Lie algebra with key subalgebra 𝔣 is defined on 𝔢 by the formulas

ad_{H}|𝔵 = \frac{μ}{2}Id, ad_{G}|𝔵 = 0 and [X, Y] = μ⟨JX, Y ⟩G for X, Y ∈ 𝔵 .

The metric Lie algebra 𝔢 is determined up to isomorphism by n = dimℂ 𝔵 and μ. If we wish to specify these parameters, we shall write 𝔢 = 𝔢(n + 1, μ). Up to scaling 𝔢(n + 1, μ) = 𝔦(𝔰𝔲(1, n + 1)).

Elementary Kählerian Lie algebras can be used as building blocks for the construction of more complicated admissible Kählerian Lie algebras via semidirect sums (v. Example 4).

Example 4: Let 𝔩 = 𝔩_{1}∈ +_{φ}𝔩_{2} be an admissible orthogonal semidirect sum of admissible Kählerian Lie algebras (𝔩_{i}, ⟨, ⟩_{i}, J_{i}), i = 1, 2 (v. Example 2). The admissible Lie algebra 𝔩 is Kählerian with complex structure J = J_{1} + J_{2} if and only if

1. φ : 𝔩_{2} → 𝔡𝔢𝔯(𝔩_{1}) preserves the Kähler form ρ_{1} = ⟨J_{1}·, ·⟩ of 𝔩_{1}, i.e. ρ_{1}(φ(X)Y, Z) + ρ_{1}(Y, φ(X)Z) = 0 for all X ∈ 𝔩_{2}, Y, Z ∈ 𝔩_{1} and 2. φ(J_{2}X)^{sym} = J_{1}φ(X)^{sym} for all X ∈ 𝔩_{2}, where φ(X)^{sym} denotes the

symmetric part of φ(X).

Such symplectic representations were studied for elementary Kählerian Lie algebras by S.G. Gindikin, I.I. Pyateckiĭ-Shapiro and E.B. Vinberg (v. e.g. [G-PS-V] and [PS]).

According to I.I. Pyatetskiĭ-Shapiro [PS] every homogeneous bounded domain in ℂ^{n} with Bergmann-metric admits a simply transitive group of isometries such that the corresponding metric Lie algebra 𝔩 can be decomposed as 𝔩 = 𝔢_{1} ∈ +^{⊥} 𝔢_{2} ∈ +^{⊥} ⋯ ∈ +^{⊥} 𝔢_{k}, where the 𝔢_{i} are elementary and the semidirect sum is of the type presented in the example.

Example 5: The classification of admissible quaternionic Kählerian Lie algebras follows from the classification of quaternionic Kählerian solvmanifolds which is due to the first author [A 2] and was completed by B. de Wit, A. Van Proeyen [dW-VP 1] and the second author [C]. Besides the (non-Abelian) symmetric examples, which are precisely 𝔦(𝔰𝔬(p+4, 4)) (p ≥ −1), 𝔦(𝔰𝔲(m, 2)), 𝔦(𝔰𝔭(m, 1)) (m ≥ 1), 𝔦(𝔤_{2}^{(2)}), 𝔦(𝔣_{4}^{(4)}), 𝔦(𝔢_{6}^{(2)}), 𝔦(𝔢_{7}^{(−5)}) and 𝔦(𝔢_{8}^{(−24)}), this classification yields also infinitely many non-symmetric admissible quaternionic Kählerian Lie algebras. The classification reduces essentially to the classification of admissible Kählerian Lie algebras (𝔲, ⟨, ⟩, J) of the type 𝔲 =

𝔣_{0} ⊕^{⊥} 𝔲_{0} (𝔣_{0} = span{G_{0}, H_{0}} = 𝔢(1, 1), 𝔲_{0} = 𝔢(n_{1} + 1, μ_{1}) ∈ +^{⊥} 𝔢(n_{2} + 1, μ_{2}) ∈ +^{⊥}

⋯ ∈ +^{⊥} 𝔢(n_{k} + 1, μ_{k})) which admit a Q-representation. We recall that a Qrepresentation is a representation T : 𝔲 → End(𝔲̃) which satisfies a list of technical conditions (v. [A 2] Def. 5.3). 𝔲̃ is isometric to 𝔲 via an isometry ˜ : 𝔲 → 𝔲̃ and carries a natural symplectic structure ρ̂ = ⟨Ĵ·, ·⟩, which is given by the complex structure Ĵ: ĴŨ_{0} = −J̃U_{0}, ĴF̃_{0} = J̃F_{0}, U_{0} ∈ 𝔲_{0}, F_{0} ∈ 𝔣_{0}. T |𝔲_{0} is symplectic with respect to ρ̂.

Given a Q-representation, 𝔩 = 𝔲 +^{⊥} 𝔲̃ carries a canonical structure of admissible quaternionic Kählerian Lie algebra such that

(i) 𝔲 is a subalgebra of 𝔩 (called principal Kählerian subalgebra);

(ii) ad_{U}|𝔲̃ = T_{U} for all U ∈ 𝔲 and

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

The quaternionic structure 𝔮 = span{J_{α}|α = 1, 2, 3} is defined as follows:

J_{1}U := JU , J_{1}Ũ := −J̃U , J_{2}U := Ũ, J_{2}Ũ := −U , J_{3} = J_{1}J_{2} ,

for U ∈ 𝔲. The main examples of Q-representations will be presented in section 2.

1.3 Riemannian Homogeneous Spaces associated with an admissible Lie Algebra

Let an admissible Lie algebra (𝔩, ⟨, ⟩) be given and let L be the corresponding simply connected Lie group. We denote by g the left-invariant Riemannian metric on L defined by ⟨, ⟩. We say that (L, g) is the Riemannian homogeneous space associated with (𝔩, ⟨, ⟩). Then the definitions of section 1.2 may be reformulated in terms of the homogeneous Riemannian space (L, g) as follows: The Koszul formula gives the covariant derivative of a left-invariant vector field Y^{∗} on L corresponding to Y ∈ 𝔩 in the point e ∈ L under the identification T_{e}L = 𝔩. The holonomy algebra 𝔥𝔬𝔩 is the Lie algebra of the holonomy group in the point e of the manifold (L, g). Symmetric (resp. Kähler, resp. quaternionic Kähler) metric Lie algebras correspond exactly to symmetric (resp. Kähler, resp. quaternionic Kähler) manifolds (L, g). Moreover the Riemannian manifold (L, g) is irreducible if and only if the admissible Lie algebra (𝔩, ⟨, ⟩) is irreducible (v. [A 1]). Remark that for arbitrary metric Lie algebras this is not true.

We will describe an algorithm to determine the full isometry group I(L, g) and the full isometry algebra Lie I(L, g) of the homogeneous Riemannian manifold (L, g).

1.4 Full Isometry Group and Algebra

Definition 1.5 An SR-decomposition of an admissible Lie algebra 𝔩 is an orthogonal semidirect sum 𝔩 = 𝔩_{s} + 𝔩_{r}, where

(i) 𝔩_{s} = 𝔦(𝔰) is a symmetric Iwasawa algebra of a semisimple Lie algebra 𝔰 of non-compact type (i.e. without compact ideals ≠ 0).

(ii) 𝔩_{r} is some admissible ideal or trivial.

It was proved by S.I. Araki [Ar] that the Lie algebra 𝔦(𝔰) uniquely determines 𝔰. Note that if 𝔩 = 𝔞 + 𝔫, 𝔩_{s} = 𝔞_{s} + 𝔫_{s} and 𝔩_{r} = 𝔞_{r} + 𝔫_{r} are the orthogonal decompositions (1) then 𝔞 = 𝔞_{s} + 𝔞_{r}, 𝔫 = 𝔫_{s} + 𝔫_{r}, [𝔞_{r}, 𝔩_{s}] = 0, 𝔫_{r} = ∑_{α∈R_{r}} 𝔫_{α} and 𝔫_{s} = ∑_{α∈R_{s}} 𝔫_{α}, where R_{r} = {α ∈ R| α|𝔞_{r} ≠ 0}, R_{s} = {α ∈ R| α|𝔞_{r} = 0} and the set of roots R has the decomposition R = R_{r} ∪ R_{s}.

We denote by 𝔡𝔢𝔯(𝔩) respectively 𝔡(𝔩) the Lie algebra of derivations respectively skew-symmetric derivations of 𝔩. Remark that the semidirect sum 𝔡(𝔩)+ ∋𝔩 is an isometry Lie algebra on (L, g), this means that it is the Lie algebra of some Lie group of isometries of the Riemannian manifold (L, g) (v. [K-N 1] Ch. VI Thm. 3.6. and Thm. 7.2.). But in general it is not the full isometry algebra, i.e. the Lie algebra of the full isometry group. For an SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r} we denote by 𝔡_{0}(𝔩_{r}) the Lie algebra of skew-symmetric derivations of 𝔩_{r} which can be extended to derivations of 𝔩 vanishing on 𝔩_{s}.

Definition 1.6 An SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r}, 𝔩_{s} = 𝔦(𝔰) of an irreducible admissible Lie algebra 𝔩 is called suitable, if the representation ad : 𝔩_{s} → 𝔡𝔢𝔯(𝔩_{r}) is extended to a representation ρ : 𝔰 → 𝔡𝔢𝔯(𝔩_{r}) such that 𝔨^{ρ} := ρ(𝔰) ∩ 𝔡(𝔩_{r}) is a maximal compact subalgebra in ρ(𝔰). A suitable SR-decomposition is called maximal if dim 𝔩_{s} is maximal.

Remark 2: From the irreducibility it follows that ρ is faithful.

The following theorem gives an algorithm which allows to compute the full isometry algebra of the homogeneous Riemannian manifold (L, g) associated with an irreducible admissible Lie algebra (𝔩, ⟨, ⟩).

Theorem 1.1 Let (𝔩, ⟨, ⟩) be an irreducible admissible Lie algebra. Any suitable SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r}, 𝔩_{s} = 𝔦(𝔰) (with representation ρ) defines an isometry algebra 𝔤 ⊃ 𝔩 of the corresponding homogeneous Riemannian manifold (L, g), which is described as follows:

𝔤 = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r} ,  (2)

𝔩_{r} is an ideal with the natural action of the Lie algebra 𝔡_{0}(𝔩_{r}) and the action of 𝔰 defined by the representation ρ : 𝔰 → 𝔡𝔢𝔯(𝔩_{r}) and [𝔰, 𝔡_{0}(𝔩_{r})] = 0. Moreover we have

𝔤 = 𝔨 + 𝔩 , 𝔰 = 𝔨_{s} + 𝔩_{s} and 𝔨 ∩ 𝔩 = 0 ,

where 𝔨 = 𝔨_{s} + 𝔡_{0}(𝔩_{r}) is a maximal compact subalgebra of 𝔤 and 𝔨_{s} = ρ^{−1}(𝔨^{ρ}) is a maximal compact subalgebra of 𝔰.

A maximal suitable SR-decomposition is unique and defines the full isometry algebra 𝔤.

Proof. Let 𝔩 = 𝔩_{s} + 𝔩_{r} be a suitable SR-decomposition and 𝔤 the associated Lie algebra described in the theorem. It is immediate that 𝔤 = 𝔨 + 𝔩 with stated properties.

To prove that 𝔤 is an isometry Lie algebra on (L, g) it is sufficient to construct a connected Lie group G with Lie G = 𝔤 such that the connected subgroup K with Lie K = 𝔨 is compact and hence closed. Then we may identify L ≅ G/K. For this we need Lemma 1.2.

Lemma 1.2 The adjoint representation ρ : 𝔤 → 𝔡𝔢𝔯(𝔩_{r}) is faithful.

Proof. Suppose that 𝔥 = ker ρ ≠ 0. By remarks 1 and 2 𝔥∩𝔩_{r} = 𝔷𝔢𝔫𝔱(𝔩_{r}) = 0 and 𝔥 ∩ (𝔰 + 𝔡_{0}(𝔩_{r})) = 0. Hence, the projection 𝔟 of 𝔥 onto 𝔰 + 𝔡_{0}(𝔩_{r}) is a non-trivial ideal of 𝔰 + 𝔡_{0}(𝔩_{r}). Since any semisimple Lie algebra has a nontrivial compact subalgebra and 𝔡_{0}(𝔩_{r}) is compact , there exists X ∈ 𝔟 such that ρ(X) ≠ 0 has purely imaginary eigenvalues. Choose Y ∈ 𝔩_{r} such that ρ(X + Y) = 0. Then ρ(Y) ≠ 0 has purely imaginary eigenvalues. This is impossible because 𝔩_{r} is admissible. □

We define G as the group of automorphisms of the Lie algebra 𝔩_{r} generated by the linear Lie algebra ρ(𝔤) ⊂ 𝔡𝔢𝔯(𝔩_{r}). By Lemma 1.2 the adjoint representation ρ : 𝔤 → 𝔡𝔢𝔯(𝔩_{r}) is faithful, hence Lie G = 𝔤.

Denote by S, A^{0}(𝔩_{r}) and L_{r} the connected linear Lie groups which correspond to the summands of 𝔤 = (𝔰 ⊕ 𝔡_{0}(𝔩_{r}))+ ∋𝔩_{r}. They are closed in GL(𝔩_{r}) because 𝔰 is semisimple and hence algebraic, 𝔡_{0}(𝔩_{r}) is algebraic by definition and L_{r} is splittable solvable with trivial center. This implies that G = SA^{0}(𝔩_{r})L_{r} is a closed linear group. Denote by K the subgroup of G with Lie K = ρ(𝔨) ≅ 𝔨_{s} +𝔡_{0}(𝔩_{r}). Since ρ(𝔨) consists of skew-symmetric operators the closure \overline{K} of K in GL(𝔩_{r}) (and hence in G) is compact. On the other hand K = K_{s}A^{0}(𝔩_{r}) is closed in G since the second factor is compact and K_{s} (the Lie group generated by 𝔨_{s}) is a closed subgroup of S and hence of G. This shows that K is a maximal compact subgroup of G. This proves that every suitable SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r} gives rise to an isometry algebra 𝔤 ⊃ 𝔩 of (L, g).

To prove the last statement let (L, g) be the Riemannian homogeneous space associated with the irreducible admissible Lie algebra (𝔩, ⟨, ⟩). Denote by 𝔤 the full isometry algebra of (L, g) and by 𝔨 the isotropy subalgebra of the point e ∈ L in G; it is a maximal compact subalgebra of 𝔤. Then 𝔤 = 𝔨+𝔩 and 𝔨 ∩ 𝔩 =0, because L acts simply transitively. Now we make use of two lemmas.

Lemma 1.3 There exists a decomposition 𝔤 = 𝔨 + 𝔩 = 𝔰 + 𝔰_{c} + 𝔯, where 𝔯 is the radical and 𝔰 + 𝔰_{c} is the splitting of a maximal semisimple subalgebra into a direct sum of a non-compact and a maximal compact ideal, such that

a) 𝔩 = 𝔩_{s} + 𝔩_{r}, 𝔩_{s} = 𝔦(𝔰) = 𝔩 ∩ 𝔰, 𝔩_{r} = 𝔩 ∩ 𝔯, is a suitable SR-decomposition. As following Def. 1.5 the corresponding decompositions (1) are denoted by 𝔩 = 𝔞 + 𝔫, 𝔩_{s} = 𝔞_{s} + 𝔫_{s} and 𝔩_{r} = 𝔞_{r} + 𝔫_{r}.

b) 𝔨 = 𝔨_{s} + 𝔰_{c} + 𝔨_{r}, where 𝔨_{s} = 𝔨 ∩ 𝔰 and 𝔨_{r} = 𝔨 ∩ 𝔯 are maximal compact subalgebras of 𝔰 and 𝔯 respectively.

c) 𝔰 = 𝔨_{s} + 𝔩_{s}, 𝔯 = 𝔨_{r} + 𝔩_{r}, 𝔤 = 𝔨 + 𝔩 = (𝔨_{s} + 𝔰_{c} + 𝔨_{r}) + 𝔩_{s} + 𝔩_{r}; d) [𝔨_{r}, 𝔰 + 𝔰_{c} + 𝔨_{r}] = [𝔰, 𝔰_{c}] = 0;

e) [𝔨 + 𝔰, 𝔞_{r}] = 0 and 𝔫_{r} is the nilradical of 𝔤.

Proof. The proof can be found in the literature, s. [Wolt] 2.1 Satz, cf. [A 1] and [A-W 2]. In [Wolt] non-positive curvature is assumed. This ensures the existence of a simply transitive solvable group of isometries which gives rise to a Lie algebra of “non-positive curvature type”. In our paper the existence of such a group is assumed from the very beginning, since admissible Lie algebras are of non-positive curvature type. □

Lemma 1.4 Under the notation of the preceding lemma:

1) The adjoint representation ρ : 𝔤 → 𝔡𝔢𝔯(𝔩_{r}) is faithful.

2) ρ(𝔰_{c} + 𝔨_{r}) = 𝔡_{0}(𝔩_{r}), where 𝔡_{0}(𝔩_{r}) is defined after Def. 1.5.

Proof. The proof of 1) is as the proof of Lemma 1.2. To prove 2) we remark that

ρ(𝔰_{c} + 𝔨_{r}) ⊂ 𝔡_{0}(𝔩_{r}) .

Indeed, ad(𝔰_{c} + 𝔨_{r}) is a Lie algebra of derivations of 𝔤 which acts trivially on 𝔰 by Lemma1.3 d), in particular it acts trivially on 𝔩_{s} and hence it acts on 𝔩 and ad(𝔰_{c} + 𝔨_{r})|𝔩_{r} = ρ(𝔰_{c} + 𝔨_{r}) ⊂ 𝔡_{0}(𝔩_{r}).

Now it is sufficient to check that

ρ(𝔨_{s}) ∩ 𝔡_{0}(𝔩_{r}) = 0 .

Let X ∈ 𝔨_{s} be such that ρ(X) ∈ 𝔡_{0}(𝔩_{r}), i.e. ρ(X) can be extended to D ∈ 𝔡(𝔩) with D|𝔩_{s} = 0. Since 𝔤 is the full isometry algebra, such a derivation defines an element Y ∈ 𝔨 such that ad_{Y} |𝔩 = D. The difference X − Y is in ker ρ = 0 (v. 1)). Hence ad_{X}𝔩 ⊂ 𝔩 and ad_{X}|𝔩_{s} = 0. This implies that X = 0, because the isotropy representation of the infinitesimal symmetric space 𝔰/𝔨_{s} is faithful. □

From Lemma 1.3 and 1.4 it follows that the full isometry algebra of (L, g) admits a decomposition (2), where 𝔰 is the non-compact part of some Levi subalgebra 𝔰 + 𝔰_{c} and 𝔡_{0}(𝔩_{r}) = 𝔰_{c} + 𝔨_{r}, where 𝔨_{r} is some maximal compact subalgebra of the radical 𝔯 = 𝔨_{r} + 𝔩_{r}. Moreover, all the properties stated in the theorem are fulfilled. Hence we proved that the full isometry algebra is associated with some suitable SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r}. Let 𝔩 = 𝔩′_{s} + 𝔩′_{r} be an other suitable SR-decomposition and 𝔤′ = 𝔰′ + 𝔡_{0}(𝔩′_{r}) + 𝔩′_{r} the associated isometry algebra. We may embed 𝔤′ ⊂ 𝔤 such that 𝔰′ ⊂ 𝔰 due to the conjugacy of maximal semisimple subalgebras. Then 𝔩′_{s} = 𝔰′ ∩ 𝔩 ⊂ 𝔩_{s} = 𝔰 ∩ 𝔩 and dim 𝔩′_{s} ≤ dim 𝔩_{s}. If dim 𝔩′_{s} = dim 𝔩_{s} then 𝔩′_{s} = 𝔩_{s} and 𝔩′_{r} = 𝔩_{r} as orthogonal complement of 𝔩_{s} in 𝔩. This finishes the proof of Theorem 1.1. □

Remark 3: Note that a subalgebra of a semisimple Lie algebra 𝔰 which generates a compact subgroup of Ad S may generate a non-compact subgroup of a Lie group S with Lie S = 𝔰.

Let 𝔩 = 𝔩_{s} + 𝔩_{r} be the maximal suitable SR-decomposition of an irreducible admissible Lie algebra (𝔩, ⟨, ⟩). We identify the full isometry algebra 𝔤(𝔩) = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r}, provided by Theorem 1.1, with the linear algebra ρ(𝔤(𝔩)) ⊂ 𝔡𝔢𝔯(𝔩_{r}), using the faithful representation ρ and denote by G(𝔩) ⊂ Aut(𝔩_{r}) the corresponding linear group. It is the connected component of the unity of the full isometry group I(L, g)

Now we describe I(L, g). Denote by K(𝔩) the group of all orthogonal automorphisms of the metric Lie algebra 𝔩_{r} which normalize the subalgebra 𝔰 ⊂ 𝔡𝔢𝔯(𝔩_{r}). Note that K(𝔩) normalizes 𝔤(𝔩).

Theorem 1.5 Let (𝔩, ⟨, ⟩) be an irreducible admissible Lie algebra and 𝔩 = 𝔩_{s} + 𝔩_{r} its maximal suitable SR-decomposition. Assume that the maximal semisimple subalgebra 𝔰 of non-compact type of the full isometry algebra 𝔤(𝔩) = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r} is simple. Then the (full) isometry group is

I(L, g) = K(𝔩) · G(𝔩) ⊂ Aut(𝔩_{r}) .

Moreover K(𝔩) is identified with the stabilizer K of the point e ∈ L and K(𝔩) ∩ G(𝔩) is the connected component of unity in K(𝔩).

Remark 4: The theorem remains true also when 𝔰 = 𝔰_{1} ⊕ ⋯ ⊕ 𝔰_{k} is not simple under the condition that for any two isomorphic ideals 𝔰_{i}, 𝔰_{j} the corresponding Iwasawa ideals 𝔦(𝔰_{i}), 𝔦(𝔰_{j}) of 𝔩_{s} are isomorphic as metric Lie algebras.

Proof. Denote by K the stabilizer of the point e ∈ L in I(L, g).

Lemma 1.6 1) The adjoint representation of K on 𝔤(𝔩) preserves the decomposition 𝔤(𝔩) = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r}.

2) The adjoint representation of K on 𝔩_{r} is faithful.

Proof. 1) The stabilizer K_{0} of the point e in G(𝔩) is a (connected) maximal compact subgroup of G(𝔩). It may be written as

K_{0} = K_{s} · A^{0}(𝔩_{r}) ,

where K_{s} and A^{0}(𝔩_{r}) are the subgroups of G(𝔩) corresponding to 𝔨_{s} (maximal compact subalgebra of 𝔰) and 𝔡_{0}(𝔩_{r}). It is clear that K_{0} preserves the decomposition 𝔤(𝔩) = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r}. Any k ∈ K preserves 𝔨 = Lie K and transforms the polar decomposition 𝔤(𝔩) = 𝔨 + 𝔩 into a polar decomposition 𝔤(𝔩) = 𝔨+Ad_{k}𝔩. Since any two polar decompositions are conjugated (s. [A 1]), there exists k_{0} ∈ K_{0} such that Ad_{kk_{0}} preserves 𝔨 and 𝔩. Then it preserves also the decompositions

𝔩 = 𝔩_{s} + 𝔩_{r}, 𝔤(𝔩) = (𝔨_{s} + 𝔩_{s}) + 𝔡_{0}(𝔩_{r}) + 𝔩_{r} .

Hence, Ad_{k} preserves the decomposition 𝔤(𝔩) = 𝔰 + 𝔡_{0}(𝔩_{r}) + 𝔩_{r}. This proves 1).

2) Let Ad_{k}|𝔩_{r} = Id for some k ∈ K. Then for any A ∈ 𝔰 + 𝔡_{0}(𝔩_{r}), X ∈ 𝔩_{r} we have

Ad_{k}[A, X] = [Ad_{k}A, X] = [A, X]

therefore [A − Ad_{k}A, X] = 0. This implies Ad_{k}A = A and Ad_{k}|𝔤(𝔩) = Id, since the representation of 𝔰 + 𝔡_{0}(𝔩_{r}) on 𝔩_{r} is faithful. From the faithfulness of the isotropy representation it follows now that k = id. □

The lemma shows that K ⊂ K(𝔩).

Lemma 1.7 K_{0} = K(𝔩)_{0}

Proof. The group Ad K(𝔩)_{0}|𝔰 is a compact connected group of automorphisms of a semisimple Lie algebra, hence it consists of inner automorphisms. On the other hand it contains Ad K_{s}, where K_{s} is maximal compact in S. This implies Ad K(𝔩)_{0}|𝔰 = Ad K_{s}. Consider a one-parameter subgroup φ(t) = exp tX in K(𝔩)_{0}. Modulo K_{s} ⊂ K_{0} = K_{s} · A^{0}(𝔩_{r}) we may assume that φ(t) commutes with 𝔰. This implies that X ∈ 𝔡_{0}(𝔩_{r}) (s. Def. 1.5) and hence φ(t) ∈ A^{0}(𝔩_{r}) ⊂ K_{0}. □

As before, due to the conjugacy of polar decompositions, any φ ∈ K(𝔩) after multiplication by some element from K_{0} = K(𝔩)_{0} preserves 𝔩. Then it preserves 𝔩_{r} = 𝔩∩𝔯, where 𝔯 is the radical of 𝔤(𝔩) (s. Lemma 1.3), and 𝔩_{s} = 𝔩∩𝔰.

Now we prove that φ preserves the scalar product on 𝔩. By definition φ is orthogonal on 𝔩_{r}. Identify 𝔩_{s} ≅ 𝔰/𝔨_{s} and note that ⟨, ⟩ on 𝔩_{s} is the unique (up to scaling) 𝔨_{s}-invariant scalar product on 𝔩_{s}, since 𝔰 is simple. This scalar product is invariant under φ, because φ together with K_{0} generates a compact Lie group with Lie algebra 𝔨 = 𝔨_{s} + 𝔡_{0}(𝔩_{r}) which preserves 𝔨_{s} = 𝔨 ∩ 𝔰. So φ is an orthogonal automorphism of (𝔩, ⟨, ⟩). By [K-N 1] Ch. 6, Th. 3.6 and Thm. 7.2, it is identified with some element in the stabilizer K. Hence we proved that K(𝔩) = K.□

Corollary 1.8 If all automorphisms of 𝔰 are inner, i.e. Aut(𝔰) = Int(𝔰), then

I(L, g) = G^{s} = S · A^{s}(𝔩_{r}) · L_{r} ,

where A^{s}(𝔩_{r}) is the group of orthogonal automorphisms of 𝔩_{r} which commute with 𝔰.

Corollary 1.9 Assume that Aut(𝔰) = Int(𝔰) ∪ φInt(𝔰) and that φ preserves the maximal compact subalgebra 𝔨_{s} ⊂ 𝔰. If φ can be extended to an automorphism φ̃ ∈ Aut(𝔰+ ∋𝔩_{r}) which preserves the decomposition 𝔰 + 𝔩_{r} and is orthogonal on 𝔩_{r} then

I(L, g) = G^{s} ∪ φ̃G^{s} .

In the opposite case I(L, g) = G^{s}.

Remark 5: It is no restriction of generality to assume that φ𝔨_{s} = 𝔨_{s}, since any two maximal compact subalgebras of 𝔰 are conjugated by an inner automorphism.

Remark 6: A similar statement holds if Aut(𝔰) has more than two connected components.

Proof. It is sufficient to prove that if φ cannot be extended, then K = K(𝔩) = K_{s} ·A^{s}(𝔩_{r}). Let k ∈ K(𝔩). Then k preserves 𝔰 , 𝔨 = Lie K and 𝔨_{s} = 𝔰∩𝔨. From the asumption we know that Ad_{k}|𝔰 = φAd_{a} or Ad_{a}, where a is an element of the normalizer N_{S}(𝔨_{s}) = N_{S}(K_{s}). Now we prove that N_{S}(K_{s}) = K_{s}. Without restriction we may assume that the symmetric space S/K_{s} is irreducible. Suppose that a ∈ N_{S}(K_{s}) − K_{s}. This implies that aK_{s} ∈ S/K_{s} is a fixpoint of K_{s} and hence K_{s} fixes the unique geodesic through the points K_{s} and aK_{s}, which contradicts the irreducibility of S/K_{s}. Hence, we proved that a ∈ K_{s}. In the first case we can write φ = Ad_{ka^{−1}}|𝔰. This shows that φ can be extended to an automorphism Ad_{ka^{−1}} with the stated properties, which contradicts the assumption. Hence we have Ad_{k}|𝔰 = Ad_{a}|𝔰 and therefore b = ka^{−1} ∈ A^{s}(𝔩_{r}). This shows k = ba ∈ A^{s}(𝔩_{r}) · K_{s} and K = A^{s}(𝔩_{r}) · K_{s}. □

Corollary 1.10 Let (L, g) be the Riemannian homogeneous space associated to an admissible Lie algebra (𝔩, ⟨, ⟩). Then the following conditions are equivalent.

1) The full isometry group I(L, g) is unimodular.

2) (L, g) is symmetric.

3) (L, g) has a compact quotient (as Riemannian manifold).

4) (L, g) has a quotient of finite volume.

Proof. It is well known that 2) ⇒ 3) ⇒ 4) ⇒ 1). We have to show 1) ⇒ 2). We may assume that 𝔩 is irreducible. Let 𝔩 = 𝔩_{s} + 𝔩_{r} be a maximal suitable SR-decomposition, in particular 𝔩_{r} is admissible. According to Theorem 1.1 𝔩_{r} is an ideal in the full isometry algebra 𝔤 = (𝔰 ⊕ 𝔡_{0}(𝔩_{r}))+ ∋𝔩_{r}. Therefore 𝔩_{r} is unimodular and admissible, hence trivial. This implies that 𝔤 = 𝔰 is semisimple and 𝔩 = 𝔦(𝔰) is its symmetric Iwasawa algebra. □

Corollary 1.11 The quaternionic Kählerian solvmanifolds do admit quotients of finite volume if and only if they are symmetric.

Proof. Since the corresponding solvable metric Lie algebras are admisible (v. [C] Prop. II.29) the result follows from the preceding corollary. □

2 Full Isometry Algebra of Quaternionic Kählerian Solvmanifolds

In this section we will determine the full isometry algebra of quaternionic Kählerian solvmanifolds (L, g) using the algorithm described in section 1. The non-symmetric examples are naturally grouped into three families, which will be treated parallelly: T -, W- and V-spaces. These families contain symmetric examples as well; they correspond to special values of the integer parameters and will be excluded in our discussion.

We will begin by recalling the definition of each of the three families for convenience of the reader. Since the corresponding metric Lie algebras 𝔩 = 𝔞+𝔫 are admissible (v. [C]) and irreducible (v. [Be] 14.45 Thm.) we can apply

Thm. 1.1. The first step is to list all possible decompositions 𝔫 = 𝔫_{1} + ∋^{⊥} 𝔫_{2} of 𝔫 = [𝔩, 𝔩] into sums of root-spaces of 𝔞. The main step is to prove the existence of a suitable SR-decomposition 𝔩 = 𝔩_{s} + 𝔩_{r} such that 𝔫_{s} = 𝔫_{1} and 𝔫_{r} = 𝔫_{2}, where 𝔫 = 𝔫_{1} + 𝔫_{2} is a decomposition from the list such that 𝔫_{1} ≠ 𝔫 is maximal. The last property implies that 𝔩 = 𝔩_{s} + 𝔩_{r} is a maximal suitable SR-decomposition. Thanks to Thm. 1.1 this provides us with an explicit description of the principal part 𝔰+ ∋𝔩_{r} (𝔰 = 𝔦(𝔩_{s})) of the full isometry algebra 𝔤 = (𝔰 ⊕ 𝔡_{0}(𝔩_{r}))+ ∋𝔩_{r}. The remaining step is the straightforward computation of 𝔡_{0}(𝔩_{r}).

At this point the following observation is pertinent. Denote by (L, g) the quaternionic Kählerian solvmanifold associated with a non-symmetric admissible quaternionic Kählerian Lie algebra (𝔩, ⟨, ⟩, 𝔮). Let 𝔤 = 𝔨 + 𝔩 be the full isometry algebra of (L, g). Consider the linear isotropy algebra 𝔨̄ ⊂ 𝔰𝔬(𝔩), i.e. the image of the isotropy algebra 𝔨 under the isotropy representation on 𝔤/𝔨 ≅ 𝔩. Since 𝔩 is quaternionic Kählerian and non-symmetric, we know (s.

[Be] 10.92 Cor.) that 𝔥𝔬𝔩 = 𝔫(𝔮) = 𝔮 + 𝔷(𝔮) (≅ 𝔰𝔭(1) ⊕ 𝔰𝔭(n)), where 𝔫(𝔮) resp. 𝔷(𝔮) denotes the normalizer resp. the centralizer of the quaternionic structure 𝔮 in 𝔰𝔬(𝔩). Therefore

𝔨̄ ⊂ 𝔫(𝔥𝔬𝔩) = 𝔫(𝔫(𝔮)) = 𝔫(𝔮) .

We shall see that the projection of 𝔨̄ on 𝔮 is always surjective. This implies that the twistor space and the canonical SO_{3}-principal bundle over the quaternionic Kählerian solvmanifold are homogeneous under the isometry group of the base.

The following notion will be used frequently.

Definition 2.1 Let φ be a representation of a key algebra 𝔣 = span{G = JH, H} with root μ on a Euclidean vector space (𝔵, ⟨, ⟩) with complex structure

J_{1}. We say that 𝔵 = 𝔵_{+} +^{⊥} 𝔵_{−}, 𝔵_{+} = J_{1}𝔵_{−}, is a weight decomposition if

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

Note that φ is symplectic with respect to ρ_{1} = ⟨J_{1}·, ·⟩ and that φ(JF)^{sym} = J_{1}φ(F)^{sym} for all F ∈ 𝔣 (cf. Example 4).

2.1 T -Spaces

The family 𝔱 = 𝔱(p), p = 0, 1, 2 …, of admissible quaternionic Kählerian Lie algebras is defined by a family of admissible Kählerian Lie algebras 𝔲 = 𝔲(p)

and their unique Q-representations T as explained in Example 5. 𝔲 = 𝔣_{0} ⊕^{⊥} 𝔲_{0} can be described as follows:

𝔲_{0} = 𝔢_{1} ∈ +^{⊥} 𝔣_{2} , 𝔢_{1} = 𝔣_{1} + 𝔵_{1} = 𝔢(p + 1, 1) , 𝔣_{2} = 𝔢(1, \frac{1}{\sqrt{2}}) ;

the key algebras 𝔣_{i} = span{G_{i}, H_{i}}, i = 0, 1, 2, commute and ad_{𝔣_{2}}|𝔵_{1} has a weight decomposition 𝔵_{1} = 𝔵_{+} + 𝔵_{−}. 𝔲 has a unique Q-representation T : 𝔲 → End(𝔲̃). 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}) ,

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} it is given by (X_{±} ∈ 𝔵_{±}):

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} .

(Here we identify covectors with vectors by means of the scalar product ⟨, ⟩.) In particular, T_{𝔣_{2}}|𝔵̃_{1} has weight decomposition 𝔵̃_{1} = 𝔵̃_{+} + 𝔵̃_{−}.

T (0) = SO_{0}(3, 4)/(SO(3) × SO(4)) is the only symmetric space in the family of quaternionic Kähler manifolds T (p) associated to 𝔱(p), p = 0, 1, 2 … We will always assume p ≠ 0.

As usual, let 𝔱 = 𝔞 + 𝔫 denote the decomposition (1).

Lemma 2.1 The table below lists all semidirect orthogonal decompositions 𝔫 = 𝔫_{1} + ∋^{⊥} 𝔫_{2} of the Lie algebra 𝔫 = [𝔱, 𝔱], such that any root space 𝔫_{α}, α ∈ R, (with respect to the Cartan subalgebra 𝔞 of 𝔱 = 𝔞 + 𝔫) is contained in 𝔫_{1} or 𝔫_{2}:

\begin{matrix}𝔫_{1} & 𝔫_{2} \\ 0 & ⟨⟨G_{2}, Q̃_{−}, 𝔵_{−} ⟩⟩ = 𝔫 \\ ≪ G_{2} ≫ & ⟨⟨Q̃_{−}, 𝔵_{−}⟩⟩ \\ ≪ Q̃_{−} ≫ & ⟨⟨G_{2}, 𝔵_{−}⟩⟩ \\ ≪ 𝔵_{−} ≫ & ⟨⟨G_{2}, Q̃_{−}⟩⟩ \\ ≪ G_{2}, Q̃_{−} ≫ & ⟨⟨𝔵_{−}⟩⟩ \\ ≪ G_{2}, 𝔵_{−} ≫ & ⟨⟨Q̃_{−}⟩⟩ \\ ≪ Q̃_{−}, 𝔵_{−} ≫ & ⟨⟨G_{2}⟩⟩ \\ ≪ G_{2}, Q̃_{−}, 𝔵_{−} ≫ & 0 \end{matrix}

≪ ⋯ ≫= 𝔫_{1} respectively ⟨⟨⋯⟩⟩ = 𝔫_{2} means that ⋯ generates the subalgebra 𝔫_{1} respectively the ideal 𝔫_{2}. Since 𝔫_{1} ⊥ 𝔫_{2} it is enough to specify 𝔫_{1} or 𝔫_{2}.

Given a decomposition 𝔫 = 𝔫_{1} + 𝔫_{2} as in the lemma we canonically associate a decomposition 𝔞 = 𝔞_{1} + 𝔞_{2} by defining 𝔞_{2} := {A ∈ 𝔞| ad_{A}𝔫_{1} = 0},

𝔞_{1} ⊥ 𝔞_{2}. Setting 𝔱_{i} = 𝔞_{i} + 𝔫_{i}, i = 1, 2, we obtain a decomposition 𝔱 = 𝔱_{1} + ∋^{⊥} 𝔱_{2}. It is clear that every SR-decomposition of 𝔱 has to show up in this way. The converse is not true: 𝔫_{1} = 𝔫, 𝔫_{2} = 0 does not correspond to an SRdecomposition of 𝔱(p), since we are assuming p ≠ 0.

Remark 7: For 𝔫_{1} =≪ G_{2}, 𝔵_{−} ≫ we obtain 𝔞_{1} + 𝔫_{1} = 𝔲_{0} the principal part of the (totally geodesic) Kählerian subalgebra 𝔲 = 𝔣_{0} + 𝔲_{0}. As Lie algebras we have 𝔲_{0} ≅ 𝔦(𝔰𝔬(p + 2, 2)). Nevertheless, the Riemannian homogeneous space associated with the metric Lie algebra 𝔲_{0} is not the hermitian symmetric space SO_{0}(p + 2, 2)/(SO(p + 2) × SO(2)); simply because the scalar product on 𝔲_{0} is not symmetric. In all the other cases (excluding 𝔫_{1} = 𝔫) 𝔞_{1} + 𝔫_{1} is a symmetric totally geodesic subalgebra of 𝔱. As we shall see, 𝔫_{1} =≪ G_{2}, Q̃_{−} ≫ gives rise to the maximal suitable SR-decomposition of 𝔱. The subalgebra 𝔫_{1} =≪ G_{2}, Q̃_{−} ≫ is maximal with the property that the full isometry algebra of the totally geodesic symmetric submanifold corresponding to 𝔞_{1} + 𝔫_{1} extends to an isometry algebra on the ambient quaternionic Kähler manifold T (p).

Proposition 2.2 The maximal suitable SR-decomposition 𝔱 = 𝔱_{s} + 𝔱_{r} of 𝔱 = 𝔱(p), p ≠ 0, is given as follows

1) 𝔱_{s} = 𝔞_{s} + 𝔫_{s}, 𝔞_{s} = span{H_{0} − H_{1}, H_{2}}, 𝔫_{s} = ≪ G_{2}, Q̃_{−} ≫ = span{G_{2}, Q̃_{0}, Q̃_{+}, Q̃_{−}},

2) 𝔱_{r} = 𝔞_{r} + 𝔫_{r}, 𝔞_{r} = span{H_{0} + H_{1}}, 𝔫_{r} = ⟨⟨𝔵_{−}⟩⟩ = span{G_{0}, G_{1}, P̃_{0}, P̃_{+}, P̃_{−}} + 𝔵_{1} + 𝔵̃_{1},

3) 𝔱_{s} = 𝔦(𝔰), 𝔰 = 𝔰𝔬(𝔳_{3,2}) ≅ 𝔰𝔬(3, 2), where a scalar product (,) of signature (3, 2) on the vector space 𝔳_{3,2} = span{G_{0}, G_{1}, P̃_{0}, P̃_{+}, P̃_{−}} = 𝔷𝔢𝔫𝔱(𝔫_{r}) is defined by the formulas (i = 0, 1, j = 0, +, −)

(G_{0}, G_{1}) = (P̃_{+}, P̃_{−}) = (P̃_{0}, P̃_{0}) = 1 ;
(G_{i}, G_{i}) = (P̃_{±}, P̃_{±}) = (G_{i}, P̃_{j}) = 0 .

We consider 𝔱_{s} as subalgebra of 𝔰 = 𝔰𝔬(𝔳_{3,2}) ≅ ⋀^{2} 𝔳_{3,2} via the embedding

\{ \begin{matrix}H_{0} − H_{1} ↦ & G_{0} ˆ∧G_{1} \\ \sqrt{2}H_{2} ↦ & P̃_{+} ˆ∧P̃_{−} \\ G_{2} ↦ & −P̃_{0} ˆ∧P̃_{+} \\ Q̃_{0} ↦ & P̃_{0} ˆ∧G_{0} \\ Q̃_{+} ↦ & G_{0} ˆ∧P̃_{+} \\ Q̃_{−} ↦ & G_{0} ˆ∧P̃_{−} \end{matrix}

((U ˆ∧V)(W) = (V, W)U − (U, W)V, U, V, W ∈ 𝔳_{3,2}). The extension ρ : 𝔰 → 𝔡𝔢𝔯(𝔱_{r}) of the representation ad : 𝔱_{s} → 𝔡𝔢𝔯(𝔱_{r}) is trivial on span{H_{0} + H_{1}}, standard on 𝔳_{3,2} and acts on 𝔵_{1} +𝔵̃_{1} as a sum of p semi-spinor representations defined by the following formulas:

ρ(P̃_{0} ˆ∧G_{1})|𝔵_{1} = 0 , ρ(P̃_{0} ˆ∧G_{1})|𝔵̃_{1} = \frac{1}{\sqrt{2}}J_{2} ;

ρ(P̃_{0} ˆ∧P̃_{−})|(𝔵_{+} + 𝔵̃_{+}) = \frac{1}{\sqrt{2}}J_{1} , ρ(P̃_{0} ˆ∧P̃_{−})|(𝔵_{−} + 𝔵̃_{−}) = 0 ;

ρ(G_{1} ˆ∧P̃_{−})|𝔵̃_{+} = J_{3} , ρ(G_{1} ˆ∧P̃_{−})|(𝔵_{1} + 𝔵̃_{−}) = 0 ;
ρ(P̃_{+} ˆ∧G_{1})|𝔵̃_{−} = −J_{3} , ρ(P̃_{+} ˆ∧G_{1})|(𝔵_{1} + 𝔵̃_{+}) = 0 .

Proof. The embedding 𝔱_{s} ↪ 𝔰 shows that 𝔱_{s} is a symmetric Iwasawa algebra of 𝔰. It is straightforward to check that the formulas given in the proposition define a representation ρ of 𝔰 on 𝔱_{r} which extends ad : 𝔱_{s} → 𝔡𝔢𝔯(𝔱_{r}) and acts by derivations.

To see that 𝔨^{ρ} := ρ(𝔰) ∩ 𝔡(𝔱_{r}) is maximal compact in ρ(𝔰) it is enough to note that 𝔨_{s} is maximal compact in 𝔰, ρ is faithful and ρ(𝔨_{s}) ⊂ 𝔡(𝔱_{r}). We prove this inclusion by a direct computation:

𝔨_{s} = span{P̃_{0} ˆ∧R_{1}^{+}, P̃_{0} ˆ∧R_{2}^{+}, R_{1}^{+} ˆ∧R_{2}^{+}} ⊕ span{R_{1}^{−} ˆ∧R_{2}^{−}} ≅ 𝔰𝔬(3) ⊕ 𝔰𝔬(2) ,

where R_{1}^{±} = G_{0} ± G_{1} and R_{2}^{±} = P̃_{+} ± P̃_{−}.

ρ(P̃_{0} ˆ∧R_{1}^{+})|(𝔵_{1} + 𝔵̃_{1}) = \frac{1}{\sqrt{2}}J_{2}  (3)
ρ(P̃_{0} ˆ∧R_{2}^{+})|(𝔵_{1} + 𝔵̃_{1}) = \frac{1}{\sqrt{2}}J_{1}  (4)
ρ(R_{1}^{+} ˆ∧R_{2}^{+})|(𝔵_{1} + 𝔵̃_{1}) = J_{3}  (5)

ρ(R_{1}^{−} ˆ∧R_{2}^{−})|(𝔵_{−} + 𝔵̃_{+}) = J_{3} , ρ(R_{1}^{−} ˆ∧R_{2}^{−})|(𝔵_{+} + 𝔵̃_{−}) = −J_{3} .

We see that ρ(𝔨_{s})|(𝔵_{1} + 𝔵̃_{1}) and therefore ρ(𝔨_{s}) consists of skew-symmetric endomorphisms. This shows that 𝔱 = 𝔱_{s} + 𝔱_{r} is a suitable SR-decomposition. It corresponds to the splitting 𝔫 = 𝔫_{1} + 𝔫_{2}, 𝔫_{2} = ⟨⟨𝔵_{−}⟩⟩, in Lemma 2.1. One sees from the list that there is no other splitting 𝔫 = 𝔫′_{1}+𝔫′_{2}, with 𝔫_{1} ⊂ 𝔫′_{1} ≠ 𝔫. This proves the maximality. □

Corollary 2.3 Consider the Cartan decomposition 𝔰 = 𝔨_{s} +𝔭 of 𝔰 = 𝔰𝔬(𝔳_{3,2}) into its skew-symmetric part 𝔨_{s} and its symmetric part 𝔭 with respect to the Euclidean scalar product ⟨, ⟩ on 𝔳_{3,2}. Then ρ(𝔨_{s}) (respectively, ρ(𝔭)) consists of skew-symmetric (respectively, symmetric) endomorphisms.

Proof. We know already that ρ(𝔨_{s}) consists of skew-symmetric endomorphisms. The fact that ρ(𝔭) consists of symmetric endomorphisms is established similarly by computing ρ(𝔭)|(𝔵_{1} + 𝔵̃_{1}) starting from

𝔭 = span{P̃_{0} ˆ∧R_{i}^{−}, R_{i}^{+} ˆ∧R_{j}^{−}; i, j = 1, 2} . □

Corollary 2.4 Denote by 𝔨̄_{s} ⊂ 𝔮 + 𝔷(𝔮) ⊂ 𝔰𝔬(𝔱) the image of 𝔨_{s} under the isotropy representation on 𝔰/𝔨_{s} + 𝔱_{r} ≅ 𝔱. Then the projection of 𝔨̄_{s} on 𝔮 is surjective.

Proof. This follows immediately from the equations (3), (4) and (5) since 𝔵_{1} + 𝔵̃_{1} is a quaternionic subspace of 𝔱. □

To formulate the final result we introduce the following notation. Let V = ℍ denote the quaternions with standard scalar product ⟨q, q⟩ = qq̄ and by R_{i}, R_{j}, R_{k}, right multiplication by i, j, k ∈ ℍ. Then ω_{0} = ⟨R_{k}·, ·⟩ is a symplectic form on the real vector space V ≅ ℝ^{4}. Now we define a metric Lie algebra 𝔱_{r}. It will be proved that it is isomorphic to 𝔱_{r} described in Prop. 2.2. We put

𝔱_{r} = V ⊗ ℝ^{p} +^{⊥} ∧^{2}V

with natural scalar product ⟨, ⟩ induced by (V, ⟨, ⟩) and the standard scalar product on ℝ^{p}. We normalize ⟨, ⟩ on ∧^{2}V by |v ∧ w| = 1 for orthonormal vectors v, w. Denote by ω^{0} = ω_{0}^{−1} the element of ∧^{2}V which is inverse to ω_{0} ∈ ∧^{2}V^{∗}. We can decompose

∧^{2}V = ∧_{0}^{2}V + ℝω^{0} ,

where ∧_{0}^{2}V = ker ω_{0}, ω_{0} ∈ ∧^{2}V^{∗}. We note that |ω^{0}|^{2} = 2 and ω^{0} ⊥ ∧_{0}^{2}V.

Now we define the Lie algebra structure

ad_{ω^{0}}|V ⊗ ℝ^{p} = \frac{1}{2}Id, ad_{ω^{0}}| ∧_{0}^{2} V = Id;

it follows that [∧_{0}^{2}V, V ⊗ ℝ^{p} + ∧_{0}^{2}V] = 0. It remains to define [,] : (V ⊗ ℝ^{p}) × (V ⊗ ℝ^{p}) → ∧_{0}^{2}V. It is given by

[v ⊗ x, w ⊗ y] = (v ∧ w)_{0}⟨x, y⟩ ,

where the subindex 0 denotes the natural projection ∧^{2}V → ∧_{0}^{2}V and ⟨, ⟩ the standard scalar product on ℝ^{p}. Denote by 𝔰𝔭(V, ω_{0}) the symplectic Lie algebra of the symplectic vector space (V, ω_{0}). It acts naturally on 𝔱_{r} by derivations. We define also an action of the Lie group (resp. algebra) O(p) (resp. 𝔰𝔬(p)) on 𝔱_{r} by orthogonal automorphisms (resp. skew-symmetric derivations) as follows. An element ϕ ∈ O(p) (resp. φ ∈ 𝔰𝔬(p)) acts on 𝔱_{r} by

ϕ(v ⊗ x) = v ⊗ ϕ(x) ϕ| ∧^{2} V = Id ,
φ(v ⊗ x) = v ⊗ φ(x) φ| ∧^{2} V = 0 .

So we defined on 𝔱_{r} = V ⊗ ℝ^{p} + ∧_{0}^{2}V + ℝω^{0} the structure of a metric Lie algebra such that the Lie algebra 𝔰𝔭(V, ω_{0}) ⊕ 𝔰𝔬(p) acts on 𝔱_{r} by derivations.

Theorem 2.5 1) The quaternionic Kähler Lie algebra 𝔱(p), p ≥ 1, is iso-

morphic (as metric Lie algebra) to the metric Lie algebra 𝔦(𝔰𝔭(V, ω_{0})) + ∋^{⊥} 𝔱_{r}, where 𝔦(𝔰𝔭(V, ω_{0})) is the symmetric Iwasawa algebra associated with 𝔰 = 𝔰𝔭(V, ω_{0}), s. Example 1.

2) The full isometry algebra 𝔤(T (p)) of the quaternionic Kähler manifold T (p), p ≥ 1, is given by

𝔤(T (p)) = (𝔰𝔭(V, ω_{0}) ⊕ 𝔰𝔬(p))+ ∋𝔱_{r} .

The adjoint representation of this Lie algebra on its ideal 𝔱_{r} defines an embedding 𝔤(T (p)) ↪ 𝔡𝔢𝔯(𝔱_{r}).

Proof. The proof follows from Prop. 2.2. Indeed, the maximal semisimple subalgebra of non-compact type 𝔰 ≅ 𝔰𝔬(3, 2) acts on 𝔵_{1} + 𝔵̃_{1} as a sum of p semi-spinor representations. It is known that the semi-spinor representation of 𝔰𝔬(3, 2) can be identified with the standard representation of 𝔰𝔭(4, ℝ) ≅ 𝔰𝔬(3, 2) on V = ℝ^{4}. Moreover, the induced representation of 𝔰𝔭(4, ℝ) on ∧^{2}V can be decomposed as ∧^{2}V = ∧_{0}^{2}V + ℝω^{0}, where ∧_{0}^{2}V is identified with the standard representation of 𝔰𝔬(3, 2). The natural 𝔰-invariant symplectic form ω_{0} on 𝔵_{1} + 𝔵̃_{1} is given by

ω_{0} = ∑_{i=1}^{p}(X_{−}^{i} ∧ J̃X_{−}^{i} + JX_{−}^{i} ∧ X̃_{−}^{i}) ,

where (X_{−}^{i})_{i=1,...,p} is an orthonormal basis of 𝔵_{−}. Now we can identify as 𝔰-modules 𝔵_{1} + 𝔵̃_{1} ≅ V ⊗ ℝ^{p}, 𝔳_{3,2} ≅ ∧_{0}^{2}V and 𝔞_{r} ≅ ℝω^{0}, where H_{0} − H_{1} = ω^{0}.

It remains to determine the Lie algebra 𝔡_{0}(𝔱_{r}). For future application in section 3 we do more. We determine the corresponding (not necessarily connected) subgroup A^{s}(𝔱_{r}) of I(T (p)) which consists of orthogonal automorphisms of 𝔱_{r} commuting with the action of 𝔰. This will be done in the following lemma. □

Lemma 2.6 A^{s}(𝔱_{r}) = O(p) with the action described before Theorem 2.5.

Proof. It is evident that O(p) ⊂ A^{s}(𝔱_{r}). Conversely, let ϕ ∈ A^{s}(𝔱_{r}). We may identify 𝔱_{r} = V ⊗ ℝ^{p} + ∧_{0}^{2}V + ℝω^{0}. Since ϕ commutes with 𝔰 = 𝔰𝔭(V, ω_{0}) it preserves the decomposition. Moreover, ϕ|V ⊗ ℝ^{p} ∈ O(p), ϕ| ∧_{0}^{2} V = ϵId and ϕω^{0} = ϵ′ω^{0} (ϵ, ϵ′ ∈ {±1}), due to Schurs lemma and orthogonality of ϕ. Commutator relations immediately imply ϵ = ϵ′ = +1. This proves the lemma and Thm. 2.5. □

2.2 W-Spaces

We are going to define W- and V-spaces simultaneously. The following concept is important.

Definition 2.2 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 is commutative:

\begin{aligned}𝔵 & × & 𝔷 & ⟶^{ψ} & 𝔶 \\ & ↓ & σ×τ & & ↓ & υ \\ 𝔵′ & × & 𝔷′ & ⟶^{ψ′} & 𝔶′ \end{aligned}

Accordingly, we say that the data (σ, τ, υ) define an equivalence between ψ and ψ′. If ψ = ψ′ we speak of an autoequivalence. Two isometric mappings ψ : 𝔵 × 𝔷 → 𝔶 and ψ′ : 𝔵 × 𝔷′ → 𝔶′ are said to be isomorphic, if there are isomorphisms τ : 𝔷 → 𝔷′ and υ : 𝔶 → 𝔶′ of Euclidean vector spaces such that the following diagram is commutative:

\begin{aligned}𝔵 & × & 𝔷 & ⟶^{ψ} & 𝔶 \\ & ↓ & Id×τ & & ↓ & υ \\ 𝔵 & × & 𝔷′ & ⟶^{ψ′} & 𝔶′ \end{aligned}

We say that the data (τ, υ) define an isomorphism between ψ and ψ′. If ψ = ψ′ we speak of an automorphism. 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.

Let ψ : 𝔵 × 𝔷 → 𝔶 be a special isometric mapping. The transpose of ψ is the special isometric mapping defined by (X ∈ 𝔵, Y ∈ 𝔶, Z ∈ 𝔷)

⟨ψ^{t}(X, Y), Z⟩ := ⟨Y, ψ(X, Z)⟩ .

Given a second special isometric mapping ψ′ : 𝔵 × 𝔷′ → 𝔶′ (over the same Euclidean vector space 𝔵) the sum ψ + ψ′ is the special isometric mapping ψ + ψ′ : 𝔵 × (𝔷 + 𝔷′) → 𝔶 + 𝔶′ defined by (X ∈ 𝔵, Z ∈ 𝔷, Z′ ∈ 𝔷′)

(ψ + ψ′) (X, Z + Z′) := ψ(X, Z) + ψ′ (X, Z′) .

We recall that special isometric mappings and ℤ_{2}-graded Clifford modules are equivalent notions. The natural correspondence is induced by the following construction. To a special isometric mapping ψ : 𝔵 × 𝔷 → 𝔶 we can associate 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) ,

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

Let 𝔵_{−}, 𝔷_{−} and 𝔶_{−} be Euclidean vector spaces. Every isometric mapping ψ : 𝔵_{−} × 𝔷_{−} → 𝔶_{−} defines an admissible Kählerian Lie algebra 𝔲(ψ) = (𝔣_{0} + 𝔲_{0}, J) by means of the following recipe

1. 𝔲_{0} is a semidirect orthogonal 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 have 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 (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_{+})⟩ .

𝔲(ψ) admits a (unique) Q-representation if 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.

The corresponding quaternionic Kählerian Lie algebras (solvmanifolds) are denoted by 𝔴(p, q) (W(p, q)) and 𝔳(ψ) (V(ψ)) in the cases (i) and (ii) respectively. We recall that the Kählerian Lie subalgebra 𝔲 = 𝔲(ψ) is called principal Kählerian subalgebra. The corresponding subgroup U is a totally geodesic Kählerian submanifold of the quaternionic Kählerian solvmanifold.

The definition of the Q-representation T of 𝔲(ψ) is given as 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}) ,
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 have 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 into the third. With these conventions the expressions 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} ,

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} .

For the convenience of the reader we remark that the bilinear mapping “∘” introduced above is completely determined by the following relations, 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 ∈ {−, +} and we used the standard rule of multiplication of signs: −− = + etc.

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

W(p, 0) ≅ W(0, p) = SO_{0}(p + 4, 4)/(SO(p + 4) × SO(4)) are the only symmetric W-spaces. The multiplication in the division algebras ℝ, ℂ, ℍ and 𝕆 defines special isometric mappings of order 1, 2, 4 and 8 respectively. The corresponding V-spaces are the only symmetric V-spaces. For the remaining of this paper the W- and V-spaces under consideration will always be assumed to be non-symmetric.

First we repeat the analysis of the previous subsection for the W-spaces. V-spaces will be treated in the next subsection.

Consider 𝔴 = 𝔴(p, q) and let 𝔴 = 𝔞 + 𝔫 denote the decomposition (1).

Lemma 2.7 The semidirect orthogonal decompositions 𝔫 = 𝔫_{1} + ∋^{⊥} 𝔫_{2} of the Lie algebra 𝔫 = [𝔴, 𝔴], such that any root space 𝔫_{α}, α ∈ R, (with respect to the Cartan subalgebra 𝔞 of 𝔴 = 𝔞 + 𝔫) is contained in 𝔫_{1} or 𝔫_{2}, are given by (cf. Lemma 2.1):

𝔫_{1} = 0, ≪ G_{2} ≫, ≪ G_{3} ≫, ≪ Q̃_{0} ≫,
≪ 𝔶_{−} ≫, ≪ 𝔷_{−} ≫, ≪ G_{2}, G_{3} ≫, ≪ G_{2}, Q̃_{0} ≫,
≪ G_{2}, 𝔶_{−} ≫, ≪ G_{2}, 𝔷_{−} ≫, ≪ G_{3}, Q̃_{0} ≫,

≪ G_{3}, 𝔷_{−} ≫, ≪ Q̃_{0}, 𝔶_{−} ≫, ≪ Q̃_{0}, 𝔷_{−} ≫,
≪ 𝔶_{−}, 𝔷_{−} ≫, ≪ G_{2}, G_{3}, Q̃_{0} ≫, ≪ G_{2}, Q̃_{0}, 𝔶_{−} ≫,
≪ G_{3}, Q̃_{0}, 𝔷_{−} ≫, ≪ Q̃_{0}, 𝔶_{−}, 𝔷_{−} ≫, ≪ G_{2}, G_{3}, 𝔶_{−}, 𝔷_{−} ≫,
≪ G_{2}, G_{3}, 𝔶_{−}, 𝔷_{−}, P̃_{0} ≫ or 𝔫 .

Proposition 2.8 The maximal suitable SR-decomposition 𝔴 = 𝔴_{s} + 𝔴_{r} of 𝔴 = 𝔴(p, q), p ≠ 0 and q ≠ 0, is as follows

1) 𝔴_{s} = 𝔞_{s} + 𝔫_{s}, 𝔞_{s} = span{H_{0} − H_{1}, H_{2}, H_{3}}, 𝔫_{s} = ≪ G_{2}, G_{3}, Q̃_{0} ≫ = span{G_{2}, G_{3}, P̃_{2}, P̃_{3}, Q̃_{0}, Q̃_{1}},

2) 𝔴_{r} = 𝔞_{r} + 𝔫_{r}, 𝔞_{r} = span{H_{0} + H_{1}}, 𝔫_{r} = ⟨⟨𝔶_{−}, 𝔷_{−}⟩⟩ = span{G_{0}, G_{1}, P̃_{0}, P̃_{1}, Q̃_{2}, Q̃_{3}} + 𝔵_{1} + 𝔵̃_{1}, 𝔵_{1} = 𝔶 + 𝔷,

3) 𝔴_{s} = 𝔦(𝔰), 𝔰 = 𝔰𝔬(𝔳_{3,3}) ≅ 𝔰𝔬(3, 3),

where a scalar product (,) of signature (3, 3) on 𝔳_{3,3} = span{G_{0}, G_{1}, P̃_{0}, P̃_{1}, Q̃_{2}, Q̃_{3}} = 𝔷𝔢𝔫𝔱(𝔫_{r}) is defined by the formulas (i, j = 0, 1, k = 2, 3)

(G_{0}, G_{1}) = (−P̃_{0}, P̃_{1}) = (Q̃_{2}, Q̃_{3}) = 1 ;
(G_{i}, G_{i}) = (P̃_{i}, P̃_{i}) = (Q̃_{k}, Q̃_{k}) = (G_{i}, P̃_{j}) = (G_{i}, Q̃_{k}) = (P̃_{j}, Q̃_{k}) = 0 .

We consider 𝔴_{s} as subalgebra of 𝔰 = 𝔰𝔬(𝔳_{3,3}) ≅ ⋀^{2} 𝔳_{3,3} via the embedding

\begin{matrix}H_{0} − H_{1} ↦ & G_{0} ˆ∧G_{1} \\ H_{2} ↦ & −(P̃_{0} ˆ∧P̃_{1} + Q̃_{2} ˆ∧Q̃_{3})/2 \\ H_{3} ↦ & −(P̃_{0} ˆ∧P̃_{1} − Q̃_{2} ˆ∧Q̃_{3})/2 \\ G_{2} ↦ & P̃_{0} ˆ∧Q̃_{3} \\ G_{3} ↦ & P̃_{0} ˆ∧Q̃_{2} \\ P̃_{2} ↦ & G_{0} ˆ∧Q̃_{3} \\ P̃_{3} ↦ & G_{0} ˆ∧Q̃_{2} \\ Q̃_{0} ↦ & G_{0} ˆ∧P̃_{1} \\ Q̃_{1} ↦ & G_{0} ˆ∧P̃_{0} , \end{matrix}

where the notation ˆ∧ was introduced in Prop. 2.2. The extension ρ : 𝔰 → 𝔡𝔢𝔯(𝔴_{r}) of the representation ad : 𝔴_{s} → 𝔡𝔢𝔯(𝔴_{r}) is trivial on span{H_{0} + H_{1}}, standard on 𝔳_{3,3} and acts on 𝔵_{1} + 𝔵̃_{1} as a sum of p + q semi-spinor representations defined by the following formulas:

ρ(−P̃_{0} ˆ∧G_{1})|(𝔶̃_{−} + 𝔷̃_{−}) = −J_{3} , ρ(−P̃_{0} ˆ∧G_{1})|(𝔵_{1} + 𝔶̃_{+} + 𝔷̃_{+}) = 0 ;
ρ(Q̃_{3} ˆ∧G_{1})|(𝔶̃_{−} + 𝔷̃_{+}) = −J_{2} , ρ(Q̃_{3} ˆ∧G_{1})|(𝔵_{1} + 𝔶̃_{+} + 𝔷̃_{−}) = 0 ;

ρ(Q̃_{3} ˆ∧P̃_{1})|(𝔶_{+} + 𝔷̃_{+}) = −J_{1} , ρ(Q̃_{3} ˆ∧P̃_{1})|(𝔶_{−} + 𝔷 + 𝔷̃_{−} + 𝔶̃) = 0 ;
ρ(G_{1} ˆ∧P̃_{1})|(𝔶̃_{+} + 𝔷̃_{+}) = J_{3} , ρ(G_{1} ˆ∧P̃_{1})|(𝔵_{1} + 𝔶̃_{−} + 𝔷̃_{−}) = 0 ;
ρ(G_{1} ˆ∧Q̃_{2})|(𝔶̃_{+} + 𝔷̃_{−}) = J_{2} , ρ(G_{1} ˆ∧Q̃_{2})|(𝔵_{1} + 𝔶̃_{−} + 𝔷̃_{+}) = 0 ;
ρ(P̃_{1} ˆ∧Q̃_{2})|(𝔷_{+} + 𝔶̃_{+}) = J_{1} , ρ(P̃_{1} ˆ∧Q̃_{2})|(𝔶 + 𝔷_{−} + 𝔶̃_{−} + 𝔷̃) = 0 .

More precisely, 𝔶 + 𝔶̃ (respectively, 𝔷 + 𝔷̃) consists of p (respectively, of q) equivalent semi-spinor representations and the irreducible summands of 𝔶+𝔶̃ are not equivalent to the summands of 𝔷 + 𝔷̃.

Proof. The proof is completely analogous to the proof of Prop. 2.2. We only remark that the semi-spinor representations on 𝔶 + 𝔶̃ and 𝔷 + 𝔷̃ are precisely related by the outer automorphism ξ of 𝔰𝔬(𝔳_{3,3}) given in Prop. 3.1. □

Corollary 2.9 Consider the Cartan decomposition 𝔰 = 𝔨_{s} +𝔭 of 𝔰 = 𝔰𝔬(𝔳_{3,3}) into its skew-symmetric part 𝔨_{s} and its symmetric part 𝔭 with respect to the Euclidean scalar product ⟨, ⟩ on 𝔳_{3,3}. Then ρ(𝔨_{s}) (respectively, ρ(𝔭)) consists of skew-symmetric (respectively, symmetric) endomorphisms.

Corollary 2.10 Denote by 𝔨̄_{s} ⊂ 𝔮 + 𝔷(𝔮) ⊂ 𝔰𝔬(𝔴) the image of 𝔨_{s} under the isotropy representation on 𝔰/𝔨_{s} + 𝔴_{r} ≅ 𝔴. Then the projection of 𝔨̄_{s} on 𝔮 is surjective.

We reformulate the results using the following notation. Let V = (ℝ^{4}, ⟨, ⟩) denote the standard Euclidean vector space. Note that the volume form vol on V defines an indefinite scalar product (,) on ∧^{2}V by

σ ∧ η = (σ, η) vol .

We identify ∧^{2}V ≅ ∧^{2}V^{∗} using (,).

Now we define a metric Lie algebra 𝔴_{r}, which as we will show is isomorphic to 𝔴_{r} from Prop. 2.8. We set

𝔴_{r} = V ⊗ ℝ^{p} +^{⊥} V^{∗} ⊗ ℝ^{q} +^{⊥} ∧^{2}V +^{⊥} ℝh ,

with the scalar product ⟨, ⟩ induced on V ⊗ ℝ^{p} + V^{∗} ⊗ ℝ^{q} + ∧^{2}V by the standard scalar product on V and ℝ; in addition ⟨h, h⟩ = 2.

The Lie brackets are defined by

ad_{h}|(V ⊗ ℝ^{p} + V^{∗} ⊗ ℝ^{q}) = \frac{1}{2}Id , ad_{h}| ∧^{2} V = Id;

this implies [∧^{2}V, V ⊗ℝ^{p}+V^{∗}⊗ℝ^{q}+∧^{2}V] = 0. Moreover [V ⊗ℝ^{p}, V^{∗}⊗ℝ^{q}] = 0 and

[v ⊗ x, w ⊗ y] = v ∧ w ⟨x, y⟩

for v, w ∈ V or V^{∗}. In the latter case we used the identification ∧^{2}V ≅ ∧^{2}V^{∗} via (,). Denote by 𝔰𝔩(V) ≅ 𝔰𝔩(4, ℝ) the Lie algebra of endomorphisms of V which annihilate vol. It naturally acts by derivations on 𝔴_{r} with trivial action on ℝh. We have a natural action of O(p)×O(q) by orthogonal automorphisms on 𝔴_{r}, similar to the case of T -spaces. We denote by 𝔰𝔬(p) ⊕ 𝔰𝔬(q) the corresponding Lie algebra of skew-symmetric derivations.

So we defined on 𝔴_{r} = V ⊗ ℝ^{p} + V^{∗} ⊗ ℝ^{q} + ∧^{2}V + ℝh the structure of a metric Lie algebra such that the Lie algebra 𝔰𝔩(V) ⊕ 𝔰𝔬(p) ⊕ 𝔰𝔬(q) acts on 𝔴_{r} by derivations.

Theorem 2.11 1) The quaternionic Kähler Lie algebra 𝔴(p, q), p, q ≥ 1, is

isomorphic (as metric Lie algebra) to the metric Lie algebra 𝔦(𝔰𝔩(V)) + ∋^{⊥} 𝔴_{r}, where 𝔦(𝔰𝔩(V)) is the symmetric Iwasawa algebra associated with 𝔰 = 𝔰𝔩(V), s. Example 1.

2) The full isometry algebra 𝔤(W(p, q)) of the quaternionic Kähler manifold W(p, q), p, q ≥ 1, is given by

𝔤(W(p, q)) = (𝔰𝔩(V) ⊕ 𝔰𝔬(p) ⊕ 𝔰𝔬(q))+ ∋𝔴_{r} .

The adjoint representation of this Lie algebra on its ideal 𝔴_{r} defines an embedding 𝔤(W(p, q)) ↪ 𝔡𝔢𝔯(𝔴_{r}).

Proof. The proof reduces to a reformulation of Prop. 2.8 if we remark that the semi-spinor representation of 𝔰𝔬(3, 3) is the standard representation of 𝔰𝔩(V) ≅ 𝔰𝔬(3, 3) and the induced representation of 𝔰𝔩(V) on ∧^{2}V is exactly the standard representation of 𝔰𝔬(3, 3). To identify the notations of Prop. 2.8 and Thm. 2.11 we have to identify

𝔶 + 𝔶̃ ≅ V ⊗ ℝ^{p}, 𝔷 + 𝔷̃ ≅ V^{∗} ⊗ ℝ^{q}
𝔳_{3,3} ≅ ∧^{2}V, 𝔞_{r} ≅ ℝh, h = H_{0} − H_{1} .

To finish the proof we must check that 𝔰𝔬(p) ⊕ 𝔰𝔬(q) = 𝔡_{0}(𝔴_{r}). This follows from the next lemma. □

Lemma 2.12 A^{s}(𝔴_{r}) = O(p) × O(q), where A^{s}(𝔴_{r}) is the group of orthogonal automorphisms of 𝔴_{r} which commute with the action of 𝔰 = 𝔰𝔩(V). The Lie algebra of A^{s}(𝔴_{r}) is 𝔡_{0}(𝔴_{r}) = 𝔰𝔬(p) ⊕ 𝔰𝔬(q).

Proof. The proof is the same as for T -spaces (cf. Lemma 2.6) since V, V^{∗}, ∧^{2}V and ℝh are non-equivalent irreducible 𝔰𝔩(V)-modules. □

2.3 V-Spaces

The definition of the non-symmetric V-spaces was given in subsection 2.2. Let 𝔳 = 𝔞 + 𝔫 denote the decomposition (1) for the quaternionic Kähler Lie algebra 𝔳 = 𝔳(ψ) associated with a special isometric mapping ψ.

Lemma 2.13 The semidirect orthogonal decompositions 𝔫 = 𝔫_{1} + ∋^{⊥} 𝔫_{2} of the Lie algebra 𝔫 = [𝔳, 𝔳], such that any root space 𝔫_{α}, α ∈ R, (with respect to the Cartan subalgebra 𝔞 of 𝔳 = 𝔞 + 𝔫) is contained in 𝔫_{1} or 𝔫_{2}, are given by (cf. Lemma 2.1):

𝔫_{1} = 0, ≪ G_{3} ≫, ≪ Q̃_{0} ≫, ≪ 𝔷_{−} ≫,
≪ 𝔵_{−} ≫, ≪ G_{3}, Q̃_{0} ≫, ≪ G_{3}, 𝔷_{−} ≫, ≪ G_{3}, 𝔵_{−} ≫,
≪ 𝔵_{−}, 𝔷_{−} ≫, ≪ Q̃_{0}, 𝔷_{−} ≫, ≪ Q̃_{0}, 𝔵_{−} ≫,
≪ G_{3}, Q̃_{0}, 𝔷_{−} ≫, ≪ G_{3}, Q̃_{0}, 𝔵_{−} ≫,
≪ Q̃_{0}, 𝔵_{−}, 𝔷_{−} ≫, ≪ G_{3}, 𝔵_{−}, 𝔷_{−} ≫ or 𝔫 .

Proposition 2.14 The maximal suitable SR-decomposition 𝔳 = 𝔳_{s} + 𝔳_{r} of 𝔳 = 𝔳(ψ), 𝔳 non-symmetric, is given by

1) 𝔳_{s} = 𝔞_{s} + 𝔫_{s}, 𝔞_{s} = span{H_{0} − H_{1}, H_{2}, H_{3}}, 𝔫_{s} = ≪ G_{3}, Q̃_{0}, 𝔵_{−} ≫ = span{G_{2}, G_{3}, P̃_{2}, P̃_{3}, Q̃_{0}, Q̃_{1}} + 𝔵 + 𝔵̃_{−},

2) 𝔳_{r} = 𝔞_{r} + 𝔫_{r}, 𝔞_{r} = span{H_{0} + H_{1}}, 𝔫_{r} = ⟨⟨𝔷_{−}⟩⟩ = span{G_{0}, G_{1}, P̃_{0}, P̃_{1}, Q̃_{2}, Q̃_{3}} + 𝔵_{1} + 𝔵̃_{1}, 𝔵_{1} = 𝔶 + 𝔷,

3) 𝔳_{s} = 𝔦(𝔰), 𝔰 = 𝔰𝔬(𝔳_{3,3+k}) ≅ 𝔰𝔬(3, 3 + k),

where a scalar product (,) of signature (3, 3 + k) is defined on 𝔳_{3,3+k} = 𝔳_{3,3} + 𝔵̃_{+} = 𝔷𝔢𝔫𝔱(𝔫_{r}) by the requirement that 𝔳_{3,3} and 𝔵̃_{+} are (,)-orthogonal, that the scalar product coincides with −⟨, ⟩ on 𝔵̃_{+} and is the same as in Prop. 2.8 on 𝔳_{3,3} = span{G_{0}, G_{1}, P̃_{0}, P̃_{1}, Q̃_{2}, Q̃_{3}}. We consider 𝔳_{s} as subalgebra of 𝔰 = 𝔰𝔬(𝔳_{3,3+k}) ≅ ⋀^{2} 𝔳_{3,3+k} via the embedding (X_{±} ∈ 𝔵_{±}):

\begin{matrix}X_{+} ↦ & P̃_{0} ˆ∧X̃_{+} \\ X_{−} ↦ & −Q̃_{3} ˆ∧J̃X_{−} \\ X̃_{−} ↦ & −G_{0} ˆ∧J̃X_{−} , \end{matrix}

s. Prop. 2.2 for the notation ˆ∧. The embedding of the subalgebra 𝔳_{s} ∩ (𝔵 + 𝔵̃_{−})^{⊥} ≅ 𝔴_{s} is given by the same formulas as in Prop. 2.8.

The extension ρ : 𝔰 → 𝔡𝔢𝔯(𝔳_{r}) of the representation ad : 𝔳_{s} → 𝔡𝔢𝔯(𝔴_{r}) is trivial on span{H_{0} + H_{1}}, standard on 𝔳_{3,3+k} and its action on 𝔵_{1} + 𝔵̃_{1} is as follows. The action of 𝔰𝔬(3, 3) on 𝔵_{1} + 𝔵̃_{1} is given by the same formulas as in Prop. 2.8. For all X_{−} ∈ 𝔵_{−} we have:

ρ(G_{1} ˆ∧J̃X_{−})|(𝔶̃_{+} + 𝔷̃_{−}) = J_{2}X_{−}∘ ,
ρ(G_{1} ˆ∧J̃X_{−})|(𝔶̃_{−} + 𝔷̃_{+}) = −J_{2}X_{−}∘ ,
ρ(G_{1} ˆ∧J̃X_{−})|𝔵_{1} = 0 ;
ρ(P̃_{1} ˆ∧J̃X_{−})|(𝔶_{+} + 𝔷̃_{+}) = −J_{1}X_{−}∘ ,
ρ(P̃_{1} ˆ∧J̃X_{−})|(𝔷_{+} + 𝔶̃_{+}) = J_{1}X_{−}∘ ,
ρ(P̃_{1} ˆ∧J̃X_{−})|(𝔶_{−} + 𝔷_{−} + 𝔶̃_{−} + 𝔷̃_{−}) = 0 ;
ρ(Q̃_{2} ˆ∧J̃X_{−})|(𝔶_{−} + 𝔷_{+} + 𝔶̃_{+} + 𝔷̃_{−}) = X_{−}∘ ,
ρ(Q̃_{2} ˆ∧J̃X_{−})|(𝔶_{+} + 𝔷_{−} + 𝔶̃_{−} + 𝔷̃_{+}) = 0 ;

ρ(J̃X_{−} ˆ∧J̃X′_{−})|(𝔵_{1} + 𝔵̃_{1}) = \frac{1}{2}(X_{−} ∘ X′_{−} − X′_{−} ∘ X_{−})∘ ,

where X_{−}∘ denotes the 𝔮-linear endomorphism of 𝔵_{1} + 𝔵̃_{1} obtained from the map (s. p. 22) U ↦ X_{−} ∘ U, U ∈ 𝔶 + 𝔷, by J_{2}-linear extension and J_{α}X_{−}∘ is the composition of the endomorphisms J_{α}|(𝔵_{1} + 𝔵̃_{1}) and X_{−}∘.

Corollary 2.15 Consider the Cartan decomposition 𝔰 = 𝔨_{s} + 𝔭 of the Lie algebra 𝔰 = 𝔰𝔬(𝔳_{3,3+k}) into its skew-symmetric part 𝔨_{s} and its symmetric part 𝔭 with respect to the Euclidean scalar product ⟨, ⟩ on 𝔳_{3,3+k}. Then ρ(𝔨_{s}) (respectively, ρ(𝔭)) consists of skew-symmetric (respectively, symmetric) endomorphisms.

Corollary 2.16 Denote by 𝔨̄_{s} ⊂ 𝔮 + 𝔷(𝔮) ⊂ 𝔰𝔬(𝔳) the image of 𝔨_{s} under the isotropy representation on 𝔰/𝔨_{s} + 𝔳_{r} ≅ 𝔳. Then the projection of 𝔨̄_{s} on 𝔮 is surjective.

We recall that the isometry classes of V-spaces are in one-to-one correspondence with the equivalence classes of special isometric mappings. Furthermore, every special isometric mapping can be decomposed into a sum of irreducible ones. Two special isometric mappings of order k ≢ 0 (4) are equivalent if and only if they have the same number l = 1, 2, 3 … of irreducible summands. Every special isometric mapping ψ of order k ≡ 0 (4) admits a decomposition ψ = pψ^{+} + qψ^{−}, where ψ^{±} are two non-isomorphic special isometric mappings and p, q ∈ {0, 1, 2 …}, l = p + q ≥ 1; pψ^{+} + qψ^{−} and p′ψ^{+} + q′ψ^{−} are equivalent if and only if {p, q} = {p′, q′}.

For the final description of the full isometry algebra 𝔤(V(ψ)) of V(ψ) we introduce further notation. Denote by V = ℝ^{3,3+k} the vector representation of 𝔰 = 𝔰𝔬(3, 3 + k) and by S a (real) spinorial representation of lowest dimension, i.e. spinor representation if it is irreducible and semi-spinor otherwise. In other words, S is spinor if k ≡ 1, 2, 3, 5 (8) and semi-spinor if k ≡ 0, 4, 6, 7 (8). For k ≡ 0 (4) we denote the two non-equivalent semi-spinor representations by S^{+} and S^{−}. In order to describe the principal part 𝔰+ ∋𝔳_{r} of the full isometry algebra 𝔤(V(ψ)) = (𝔰 ⊕ 𝔡_{0}(𝔳_{r}))+ ∋𝔳_{r} of V(ψ) in more invariant terms we shall make use of the following facts concerning the spinorial representation S.

Proposition 2.17 1) There exists a Spin(V)-invariant isomorphism S ≅ S^{∗} if k ≢ 0 (4) and S^{−} ≅ (S^{+})^{∗} if k ≡ 0 (4).

2) ∧^{2}S contains a unique Spin(V)-submodule isomorphic to V.

3) Assume k ≡ 0 (4) and denote the unique vector submodules of ∧^{2}S^{+} and ∧^{2}S^{−} by V and V′ resp. Then V and V′ are canonically isomorphic.

Remark 8: The explicit description of Spin(V)-invariant structures on the spinor representation such as invariant bilinear forms is part of a general program which will be presented in a forthcomming paper [A-C]. As result of this program we obtain the explicit construction of embeddings of the vector representation into the (exterior or symmetric) square of the spinor or semi-spinor representation. This is used to construct Lie algebra and Lie superalgebra extensions of the (generalized) Poincaré algebra.

Proof. 1) Follows from the existence of a non-trivial Spin(V)-invariant bilinear form on S for k ≢ 0 (4) and on the spinor module S^{+} + S^{−} for k ≡ 0 (4); S^{±} being isotropic in the last case (s. [A-C]).

2) The explicit construction of the embedding V ↪ ∧^{2}S is given in [A-C], cf. Remark 9 below. The uniqueness can be extracted from [O-V] Table 1.

3) By 1) and 2) V′ ⊂ ∧^{2}S^{−} ≅ (∧^{2}S^{+})^{∗} is canonically identified with V^{∗}. Since V^{∗} is in turn canonically isomorphic to V via (,) this gives a canonical isomorphism V ≅ V′. □

We remark that only 3) and the existence statement in 2) will be used in the sequel. These can be easily derived from Prop. 2.14 without refering to [A-C]. We have not done so for expository reasons.

Remark 9: The explicit description of the embedding V ↪ ∧^{2}S can be given in terms of Clifford multiplication μ and an appropriate (symmetric or skew-symmetric) Spin(V)-invariant bilinear form β. For example, if k ≡ 1, 2, 3, 5 (8) it is given by

v ↦ β(μ(v)·, ·), v ∈ V ,

where μ : V ⊗ S → S and β is a Spin(V)-invariant skew-symmetric bilinear form on S.

With the notations above we associate to each V-space V(ψ) a Lie algebra 𝔤 = 𝔰𝔬(3, 3 + k)+ ∋𝔳_{r}. We will prove later that 𝔤 is exactly the principal part 𝔰+ ∋𝔳_{r} of the full isometry algebra 𝔤(V(ψ)).

First we define an 𝔰𝔬(3, 3 + k)-module 𝔳_{r} by

𝔳_{r} = S ⊗ ℝ^{l} + V + ℝh , if k ≢ 0 (4);  (6)
𝔳_{r} = S^{+} ⊗ ℝ^{p} + S^{−} ⊗ ℝ^{q} + V + ℝh , if k ≡ 0 (4);  (7)

where the modules ℝh, ℝ^{l}, ℝ^{p} and ℝ^{q} are trivial.

Now we define the Lie algebra structure on 𝔳_{r}

ad_{h}|S ⊗ ℝ^{l} = \frac{1}{2}Id , if k ≢ 0 (4),

ad_{h}|(S^{+} ⊗ ℝ^{p} + S^{−} ⊗ ℝ^{q}) = \frac{1}{2}Id , if k ≡ 0 (4),

ad_{h}|V = Id ,

it follows that [V, S ⊗ ℝ^{l} + V] = 0 (resp. [V, S^{+} ⊗ ℝ^{p} + S^{−} ⊗ ℝ^{q} + V] = 0). The Lie bracket on S ⊗ ℝ^{l} (k ≢ 0 (4)) is defined by

[v ⊗ x, w ⊗ y] = pr(v ∧ w)⟨x, y⟩ , v, w ∈ S ,

where pr is the natural projection of ∧^{2}S onto its unique vector submodule (s. Prop. 2.17 2)). In case k ≡ 0 (4) the same formula defines the Lie bracket on S^{+} ⊗ℝ^{p} and on S^{−} ⊗ℝ^{q}, using the identification of the vector submodules in ∧^{2}S^{+} and ∧^{2}S^{−} given in Prop. 2.17 3). Finally, we put [S^{+}⊗ℝ^{p}, S^{−}⊗ℝ^{q}] = 0.

It is clear that the representation of 𝔰𝔬(3, 3 + k) on 𝔳_{r} defined by the decompositions (6) and (7) acts by derivations of the Lie algebra structure just defined.

Theorem 2.18 1) The quaternionic Kähler Lie algebra 𝔳(ψ) of the nonsymmetric quaternionic Kähler V-space V(ψ) is isomorphic to the Lie algebra 𝔦(𝔰𝔬(3, 3 + k))+ ∋𝔳_{r}, where 𝔦(𝔰𝔬(3, 3 + k)) denotes the Iwasawa algebra of 𝔰 = 𝔰𝔬(3, 3 + k).

2) The full isometry algebra 𝔤(V(ψ)) of V(ψ) is given by

𝔤(V(ψ)) = (𝔰𝔬(3, 3 + k) ⊕ 𝔡_{0}(𝔳_{r}))+ ∋𝔳_{r} ,

where the Lie algebra 𝔡_{0}(𝔳_{r}) = Lie A^{s}(𝔳_{r}) is given in the next lemma.

Proof. Consider the 𝔰-Module (𝔰 = 𝔰𝔬(3, 3 + k)) 𝔳_{r} = 𝔵_{1} + 𝔵̃_{1} + 𝔳_{3,3+k} + 𝔞_{r} described in Prop. 2.14. Using the formulas given there we can identify

𝔵_{1} + 𝔵̃_{1} ≅ S ⊗ ℝ^{l} , if k ≢ 0 (4),

𝔵_{1} + 𝔵̃_{1} ≅ S^{+} ⊗ ℝ^{p} + S^{−} ⊗ ℝ^{q} , if k ≡ 0 (4),
𝔳_{3,3+k} ≅ V , 𝔞_{r} ≅ ℝh .

It is easy to see that this isomorphism of 𝔰-modules is also a Lie algebra isomorphism, if we identify h = H_{0} − H_{1}. □ Lemma 2.19 A^{s}(𝔳_{r}) is isomorphic to

• O(l) if k ≡ 1, 7 (8),

• U(l) if k ≡ 2, 6 (8),

• Sp(l) if k ≡ 3, 5 (8),

• Sp(p) × Sp(q) if k ≡ 4 (8),

• O(p) × O(q) if k ≡ 0 (8).

Proof. Let ϕ ∈ A^{s}(𝔳_{r}). We can consider ϕ as orthogonal automorphism of the metric Lie algebra 𝔳(ψ) acting trivially on 𝔳_{s}. It was shown in [C] (s. Prop. II.23) that orthogonal automorphism of 𝔳(ψ) correspond exactly to autoequivalences of ψ in the sense of Def. 2.2. Since ϕ acts as identity on 𝔳_{s} ⊃ 𝔵_{−} it is an automorphism of ψ in the sense of Def. 2.2. Using the natural correspondence between special isometric mappings of order k, ℤ_{2}- graded Cℓ_{k}-modules and (ungraded) Cℓ_{k−1}-modules, now the proof follows from next lemma and the fact that Cℓ_{k−1} is a

• simple real matrix algebra if k ≡ 1, 7 (8),

• simple complex matrix algebra if k ≡ 2, 6 (8),

• simple quaternionic matrix algebra if k ≡ 3, 5 (8),

• sum of two simple quaternionic matrix algebras if k ≡ 4 (8),

• sum of two simple real matrix algebras if k ≡ 0 (8). □

Lemma 2.20 1) Let 𝕂(n) denote the simple matrix algebra over 𝕂 = ℝ, ℂ or ℍ and V = l 𝕂^{n} = 𝕂^{n}⊗ℝ^{l} a sum of l copies of the standard representation considered as real vector space with action of 𝕂(n) ⊂ End(V). Then

Z_{GL(V)}(𝕂(n)) = 𝕂^{∗} ⊗ GL(l, ℝ) = GL(l, 𝕂)

and

Z_{O(V)}(𝕂(n)) = \{ \begin{matrix}O(l) if & 𝕂 = ℝ \\ U(l) if & 𝕂 = ℂ \\ Sp(l) if & 𝕂 = ℍ, \end{matrix}

where O(V) is the orthogonal group with respect to some positive definite scalar product on V.

2) Let A = 𝕂(n) ⊕ 𝕂(n)′ be the direct sum of two copies of 𝕂(n) and denote by 𝕂^{n} and 𝕂^{n}′ the two non-equivalent irreducible representations of A, with trivial action of 𝕂(n)′ and 𝕂(n) respectively. Consider V = p𝕂^{n} ⊕ q𝕂^{n}′ = (𝕂^{n} ⊗ ℝ^{p}) ⊕ (𝕂^{n}′ ⊗ ℝ^{q}) as real vector space and A ⊂ End(V). Then

Z_{GL(V)}(A) = GL(p, 𝕂) × GL(q, 𝕂)

and

Z_{O(V)}(A) = \{ \begin{matrix}O(p) × O(q) if & 𝕂 = ℝ \\ U(p) × U(q) if & 𝕂 = ℂ \\ Sp(p) × Sp(q) if & 𝕂 = ℍ. \end{matrix}

Recall that the subgroup U(ψ) ⊂ V(ψ) corresponding to the principal Kählerian subalgebra 𝔲(ψ) is a totally geodesic Kählerian submanifold. Now we describe the Lie algebra 𝔤_{𝔲} of the group of isometries of V(ψ) which preserve U(ψ). Denote by 𝔰𝔬(2, 2 + k) the subalgebra of 𝔰𝔬(3, 3 + k) = 𝔰𝔬(𝔳_{3,3+k}) which acts trivially on the two-dimensional subspace 𝔳_{1,1} = span{G_{0}, G_{1}} ⊂ 𝔳_{3,3+k} and by 𝔳_{2,2+k} the orthogonal complement of 𝔳_{1,1} in 𝔳_{3,3+k}.

Lemma 2.21 1) The 𝔰𝔬(2, 2+k)-module S decomposes as direct sum of two irreducible submodules S_{𝔲}, S̃_{𝔲} of the same dimension. Moreover, we may assume that

S_{𝔲} ⊗ ℝ^{l} = 𝔵_{1} and S̃_{𝔲} ⊗ ℝ^{l} = 𝔵̃_{1} ,

in case k ≢ 0 (4). In the case k ≡ 0 (4) we have S = S^{±} = S𝔲^{±} ⊕ S̃𝔲^{±} and may assume

S𝔲^{+} ⊗ ℝ^{p} + S𝔲^{−} ⊗ ℝ^{q} = 𝔵_{1} and S̃𝔲^{+} ⊗ ℝ^{p} + S̃𝔲^{−} ⊗ ℝ^{q} = 𝔵̃_{1} .

2) The natural projection pr : ∧^{2}S → V = 𝔳_{3,3+k} maps ∧^{2}S_{𝔲} and ∧^{2}S̃_{𝔲} onto trivial 𝔰𝔬(2, 2 + k)-submodules of V and S_{𝔲} ⊗ S̃_{𝔲} onto the 𝔰𝔬(2, 2 + k)-vector submodule 𝔳_{2,2+k}. More precisely, from our identifications it follows that

pr(∧^{2}S_{𝔲}) = ℝG_{1} , pr(∧^{2}S̃_{𝔲}) = ℝG_{0} .

3) For k ≢ 0 (4)

𝔲_{r} = S_{𝔲} ⊗ ℝ^{l} + pr(∧^{2}S_{𝔲}) + pr(∧^{2}S̃_{𝔲}) + ℝh

is a subalgebra of 𝔳_{r}. For k ≡ 0 (4) the same is true for

𝔲_{r} = S𝔲^{+} ⊗ ℝ^{p} + S𝔲^{−} ⊗ ℝ^{q} + pr(∧^{2}S_{𝔲}) + pr(∧^{2}S̃_{𝔲}) + ℝh

(pr(∧^{2}S_{𝔲}) and pr(∧^{2}S̃_{𝔲}) do not dependent on the choice S_{𝔲} = S𝔲^{±}). Moreover,

𝔲_{r} = 𝔵_{1} + span{G_{0}, G_{1}} + 𝔞_{r} = 𝔲 ∩ 𝔳_{r} .

Proof. The lemma follows from the formulas in Prop. 2.14. □

Theorem 2.22 1) The maximal suitable SR-decomposition of the non-sym-metric quaternionic Kähler Lie algebra 𝔳(ψ) induces a (not maximal) suitable SR-decomposition of the principal Kähler subalgebra 𝔲 = 𝔲(ψ). More precisely, 𝔲 = 𝔲_{s} + 𝔲_{r}, where 𝔲_{s} is the Iwasawa subalgebra of 𝔰𝔬(2, 2 + k), 𝔲_{r} = 𝔲 ∩ 𝔳_{r} ⊕ ℝ(H_{0} − H_{1}) and 𝔲 ∩ 𝔳_{s} = 𝔲_{s} ⊕ ℝ(H_{0} − H_{1}).

2) The Lie algebra 𝔤_{𝔲} is given by

𝔤_{𝔲} = (𝔰𝔬(2, 2 + k) ⊕ 𝔡_{0}(𝔳_{r})^{𝔲_{r}})+ ∋𝔲_{r} ,

where 𝔡_{0}(𝔳_{r})^{𝔲_{r}} = {φ ∈ 𝔡_{0}(𝔳_{r})| φ𝔲_{r} ⊂ 𝔲_{r}}.

Proof. One can check immediately using Prop. 2.14 that

𝔲 ∩ 𝔳_{s} = 𝔞_{s} + span{G_{2}, G_{3}} + 𝔵 ,
𝔲 ∩ 𝔳_{r} = 𝔞_{r} + span{G_{0}, G_{1}} + 𝔵_{1} .

This implies 1). To prove 2) we remark that 𝔤_{𝔲} = 𝔨_{𝔲} + 𝔲, where 𝔨_{𝔲} = 𝔨 ∩ 𝔲 is a compact subalgebra, hence 𝔨_{𝔲} ∩ 𝔳_{r} = 0 and 𝔲_{r} = 𝔤_{𝔲} ∩ 𝔳_{r}. This implies that 𝔲_{r} is an ideal in 𝔤_{𝔲}, because 𝔳_{r} is an ideal in 𝔤(V). This shows that 𝔤_{𝔲} ⊂ 𝔫_{𝔤(V)}(𝔲_{r}). On the other hand, one can check that

𝔫_{𝔤(V)}(𝔲_{r}) = (𝔰𝔬(2, 2 + k) ⊕ 𝔡_{0}(𝔳_{r})^{𝔲_{r}})+ ∋𝔲_{r} ⊂ 𝔤_{𝔲} . □

3 Full Isometry Group of Quaternionic Kählerian Solvmanifolds

3.1 Automorphism Group of 𝔰 = 𝔰𝔬(3, 3 + k)

In section 2 we proved that the maximal semisimple subalgebra 𝔰 of noncompact type in the full isometry algebra 𝔤(L) of the quaternionic Kähler solvmanifolds (L, g) is isomorphic to 𝔰𝔬(3, 3 + k), where k = −1, 0, 1, … is the order of the isometric mapping defining (L, g). We will apply Cor. 1.9 to construct the full (not necessarily connected) isometry group I(L, g). For this we need a description of the group Aut(𝔰) of all automorphisms of the Lie algebra 𝔰 = 𝔰𝔬(3, 3 + k). It is given in the following proposition.

Proposition 3.1 1) Aut(𝔰) = Int(𝔰) ∪ ξInt(𝔰) for 𝔰 = 𝔰𝔬(3, 3 + k), if 0, 2 ≠ k ≥ −1;

2) Aut(𝔰) = Int(𝔰) ∪ ξInt(𝔰) ∪ ηInt(𝔰) ∪ ξηInt(𝔰) for 𝔰 = 𝔰𝔬(3, 3) (k = 0), where ξ = Ad_{D}, η = Ad_{C} and

D = ( \begin{matrix}1_{k+5} & 0 \\ 0 & −1 \end{matrix} ) , C = ( \begin{matrix}0 & 1_{3} \\ 1_{3} & 0 \end{matrix} ) .

Proof. The proof will split into several lemmas. We denote by 𝔨_{s} = 𝔰𝔬(3) ⊕ 𝔰𝔬(3 + k) the maximal compact subalgebra of 𝔰 = 𝔰𝔬(3, 3 + k) and by 𝔰ℂ = 𝔰𝔬(6 + k, ℂ) the complexification of 𝔰. We will consider 𝔰 and 𝔰ℂ as linear Lie algebras of the vector spaces V = ℝ^{6+k} and V ℂ = ℂ^{6+k} respectively.

Lemma 3.2 [He]

Aut(𝔰ℂ) = \{ \begin{matrix}Int(𝔰ℂ) if k is odd \\ Int(𝔰ℂ) ∪ ξInt(𝔰ℂ) if k ≠ 2 is even, \end{matrix}

where ξ = Ad_{D}, D = diag(1_{k+5}, −1). The automorphism ξ preserves 𝔰 and modulo Int(𝔰) is equivalent to Ad_{E} : X ↦ −X^{t}, E = diag(1_{3}, −1_{3+k}).

Denote by Aut_{e}(𝔨_{s}) the group of all automorphisms of 𝔨_{s} = 𝔰𝔬(3)⊕𝔰𝔬(3+ k) which can be extended to an automorphism of 𝔰.

Lemma 3.3 1) Aut(𝔨_{s}) = Int(𝔨_{s}) for k even ≠ 0, 2,

2) Aut(𝔨_{s}) = Int(𝔨_{s}) ∪ ξInt(𝔨_{s}) for k odd ≠ 1,

3) Aut_{e}(𝔨_{s}) = Int(𝔨_{s}) ∪ ξInt(𝔨_{s}) for k = 1,

4) Aut(𝔨_{s}) = Int(𝔨_{s}) ∪ ηInt(𝔨_{s}) for k = 0,

where ξ = Ad_{D} and η = Ad_{C} with matrices D and C given in Prop. 3.1.

Corollary 3.4 For k ≠ 1, Aut(𝔨_{s}) = Aut_{e}(𝔨_{s}).

Proof. The proof of 1), 2), 4) follows from [He]. To prove 3) it is sufficient to show that an automorphism φ of 𝔨_{s} = 𝔰𝔬(3) ⊕ 𝔰𝔬(4) can be extended to an automorphism of 𝔰 = 𝔰𝔬(3, 4) if and only if it preserves the ideal 𝔰𝔬(3) (hence, also 𝔰𝔬(4)). Since any automorphism of 𝔰ℂ = 𝔰𝔬(7, ℂ) is inner, φ = Ad_{A} for some A ∈ SO(7, ℂ). Hence, the decomposition of the vector space V ℂ = ℂ^{7} into irreducible components with respect to the Lie algebras 𝔰𝔬(3) and φ(𝔰𝔬(3)) = A𝔰𝔬(3)A^{−1} are isomorphic. This implies that φ(𝔰𝔬(3)) = 𝔰𝔬(3), since 𝔰𝔬(3) acts trivially on a 4-dimensional subspace of V ℂ and both 𝔰𝔬(3)- ideals of 𝔰𝔬(4) act trivially only on a 3-dimensional subspace of V ℂ. □

Lemma 3.5 The only non-trivial automorphism of 𝔰 which acts trivially on 𝔨_{s} is Ad_{E}, E = diag(1_{3}, −1_{3+k}).

Proof. By Lemma 3.2 we may assume that the automorphism has the form Ad_{A}|𝔰 for some A ∈ SO(6 + k, ℂ); where Ad_{A} preserves 𝔰 and acts trivially on 𝔨_{s}. The last condition means that [A, 𝔨_{s}] = 0. The linear Lie algebra 𝔨_{s} = 𝔰𝔬(3) ⊕ 𝔰𝔬(3 + k) ⊂ 𝔤𝔩(V ℂ) has two non-equivalent invariant irreducible subspaces. By Schur’s Lemma A = diag(λ1_{3}, μ1_{3+k}), λ, μ ∈ ℂ. Since A ∈ SO(6 + k, ℂ), we have λ^{2} = μ^{2} = 1. Hence, A = ±E and Ad_{A} = Ad_{E}. □

The proof of Proposition 3.1 now follows from Lemmas 3.3 and 3.5

3.2 T -Spaces

Theorem 3.6 The (full) isometry group of the quaternionic Kähler manifolds T (p), p ≥ 1, is given by

I(T (p)) = G^{s} ∪ φG^{s} ↪ Aut(𝔱_{r}) ,

G^{s} = (S×A^{s}(𝔱_{r}))⋉T_{r}, where S = Sp(4, ℝ) = Spin_{0}(3, 2) and T_{r} are the linear subgroups of Aut(𝔱_{r}) generated by the linear Lie algebras 𝔰 = 𝔰𝔭(4, ℝ), 𝔱_{r} ⊂ 𝔡𝔢𝔯(𝔱_{r}) (s. Thm. 2.5) and A^{s}(𝔱_{r}) = O(p) is given in Lemma 2.6. The automorphism φ ∈ Aut(𝔱_{r}) is defined in terms of the decomposition 𝔱_{r} = V ⊗ ℝ^{p} + ∧_{0}^{2}V + ℝω^{0} as follows:

φ|V ⊗ ℝ^{p} = B ⊗ Id, B = diag(1, 1, −1, −1) ,
φ(v ∧ w)_{0} = (φv ∧ φw)_{0} and φω^{0} = ω^{0} .

Proof. According to Cor. 1.9 and Prop. 3.1 it is sufficient to prove that the automorphism φ of 𝔱_{r} normalizes 𝔰 = 𝔰𝔬(3, 2) = 𝔰𝔭(4, ℝ) and induces an outer automorphism on 𝔰. Since Bω_{0} = −ω_{0}, conjugation with B will be an outer automorphism of 𝔰𝔭(4, ℝ) which is extended by φ to an automorphism of 𝔰+ ∋𝔱_{r}. □

3.3 W-Spaces

Theorem 3.7 The (full) isometry group of the quaternionic Kähler manifolds W(p, q), p, q ≥ 1, is given by

I(W(p, q)) = G^{s} ∪ χG^{s} ↪ Aut(𝔴_{r}) if p ≠ q;
I(W(p, p)) = G^{s} ∪ χG^{s} ∪ φG^{s} ∪ (χφ)G^{s} ↪ Aut(𝔴_{r}) ;

G^{s} = (S × A^{s}(𝔴_{r}))⋉W_{r}, where S = SL(4) = Spin_{0}(3, 3) and W_{r} are the linear subgroups of Aut(𝔴_{r}) generated by the linear Lie algebras 𝔰 = 𝔰𝔩(4, ℝ), 𝔴_{r} ⊂ 𝔡𝔢𝔯(𝔴_{r}) (s. Thm. 2.11) and A^{s}(𝔴_{r}) = O(p) × O(q) is described in Lemma 2.12. In terms of the decomposition 𝔴_{r} = V ⊗ ℝ^{p} + V^{∗} ⊗ ℝ^{q} + ∧^{2}V + ℝh, χ is induced by the automorphism diag(−1, 1, 1, 1) of V putting χh = h. For p = q the automorphism φ of 𝔴_{r} is induced by the canonical isomorphisms V → V^{∗} and V^{∗} → V via ⟨, ⟩ putting φh = h (recall that we have a canonical 𝔰𝔩(V)-invariant identification ∧^{2}V ≅ ∧^{2}V^{∗}; note that φ| ∧^{2} V = ∗ (Hodge-star)).

Proof. χ is an orthogonal automorphism of 𝔴_{r} and conjugation by the matrix diag(−1, 1, 1, 1) in 𝔰 = 𝔰𝔩(4, ℝ) is an outer automorphism of 𝔰𝔩(4, ℝ) = 𝔰𝔬(3, 3). Moreover, this automorphism interchanges the simple summands of the maximal compact subalgebra 𝔰𝔬(4) = 𝔰𝔬(3) ⊕ 𝔰𝔬(3) and modulo inner automorphisms is equivalent to η in Prop. 3.1 2). According to Cor. 1.9 we have proved G^{s} ∪ χG^{s} ⊂ I(W(p, q)).

Let us now consider the outer automorphism φ : X ↦ −X^{t} of 𝔰 = 𝔰𝔩(4, ℝ).

Lemma 3.8 φ can be extended to an automorphism of 𝔰+ ∋𝔴_{r} if and only if p = q. The extension (p = q) is orthogonal on 𝔴_{r}.

Proof. Assume φ can be extended. Then we have

Aφv = −φA^{t}v, A ∈ 𝔰𝔩(4, ℝ), v ∈ V ⊗ e ≅ V ,

which shows that the 𝔰𝔩(4, ℝ)-module φ(V ⊗ e) ≅ V^{∗}. This implies that the multiplicitiy p of V in 𝔴_{r} equals the multiplicitiy q of V^{∗}. If p = q then φ is canonically extended by φ given in Theorem 3.7. □

Using Cor. 1.9 we immediately derive Theorem 3.7 from Prop. 3.1 2) and the lemma. □

3.4 V-Spaces

Theorem 3.9 The (full) isometry group of the non-symmetric quaternionic Kähler manifolds V(ψ) is given by

I(V(ψ)) = G^{s} ↪ Aut(𝔳_{r}) , if k ≠ 2 is even ,
I(V(ψ)) = G^{s} ∪ φG^{s} ↪ Aut(𝔳_{r}) , if k is odd .

The automorphism φ ∈ Aut(𝔳_{r}) is defined by

φ|(𝔷 + 𝔷̃ + 𝔵̃_{+}) = −Id , φ|(𝔶 + 𝔶̃ + 𝔳_{3,3} + 𝔞_{r}) = Id

and G^{s} = (S × A^{s}(𝔳_{r}))⋉V_{r}, where S = Spin_{0}(3, 3 + k), V_{r} are the linear subgroups generated by the linear algebras 𝔰 = 𝔰𝔬(3, 3 + k), 𝔳_{r} ⊂ 𝔡𝔢𝔯(𝔳_{r}) and A^{s}(𝔳_{r}) was determined in Lemma 2.19.

Proof. Consider first the case when k is odd. The outer automorphism Ad_{D_{o}}, D_{o}|𝔳_{3,3} = Id, D_{o}|𝔵̃_{+} = −Id, of 𝔰𝔬(𝔳_{3,3+k}) ≅ 𝔰𝔬(3, 3 + k) is extended by φ to an automorphism of 𝔰+ ∋𝔳_{r}, which is orthogonal on 𝔳_{r}. By Cor. 1.9 and Prop. 3.1 this implies the theorem in case k is odd.

If k is even we consider the outer automorphism θ = Ad_{D_{e}}, D_{e}Q̃_{2} = −Q̃_{3}, D_{e}Q̃_{3} = −Q̃_{2}, D_{e}|(span{G_{0}, G_{1}, P̃_{0}, P̃_{1}} + 𝔵̃_{+}) = Id, of 𝔰𝔬(𝔳_{3,3+k}) ≅ 𝔰𝔬(3, 3 + k). By the formulas for the embedding 𝔳_{s} ↪ 𝔰 (s. Prop. 2.14) we know that θ preserves the Cartan subalgebra 𝔞_{s} of 𝔳_{s} and θ|𝔞_{s} = Id. Assume θ has been extended to θ ∈ Aut(𝔰+ ∋𝔳_{r}). Since θ|𝔞_{s} = Id, it follows e.g. that θ𝔶_{−} = 𝔶_{−}, because 𝔶_{−} is the root space of 𝔞_{s} with root −(H_{0} − H_{1} + 2H_{3})/4. Now we check Lie brackets between 𝔰 = 𝔰𝔬(𝔳_{3,3+k}) = ∧^{2}𝔳_{3,3+k} and 𝔳_{r} ⊃ 𝔶_{−}using the formulas in Prop. 2.14:

[θ(−Q̃_{3} ˆ∧J̃X_{−}), θY_{−}] = [Q̃_{2} ˆ∧J̃X_{−}, θY_{−}]
= ρ(Q̃_{2} ˆ∧J̃X_{−})θY_{−} = X_{−} ∘ θY_{−} ≠ 0 ,

if X_{−} ∈ 𝔵_{−} − {0} and Y_{−} ∈ 𝔶_{−} − {0}. On the other hand:

θ[−Q̃_{3} ˆ∧J̃X_{−}, Y_{−}] = θρ(−Q̃_{3} ˆ∧J̃X_{−})Y_{−} = θ[X_{−}, Y_{−}] = 0 .

This shows that θ cannot be extended to an automorphism of 𝔰+ ∋𝔳_{r} and proves the theorem, thanks to Cor. 1.9 and Prop. 3.1. □

3.5 Conclusion

A remarkable consequence of our analysis is the following.

Theorem 3.10 The (connected) isometry group of the quaternionic Kählerian solvmanifolds acts transitively on the twistor space and on the SO_{3}- principal bundle associated with the quaternionic structure.

Proof. The result follows from Corollaries 2.4, 2.10 and 2.16. □

References

[A 1] D.V. Alekseevskiĭ: Conjugacy of polar factorizations of Lie groups, Math. USSR. Sbornik 13, No. 1 (1971), 12–25.

[A 2] D.V. Alekseevskiĭ: Classification of quaternionic spaces with a transitive solvable group of motions, Math. USSR Izvestija 9, No. 2 (1975), 297–339.

[A-C] D.V. Alekseevsky, V. Cortés: Classification of N-(super)- extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p, q), Commun. Math. Phys. 183 (1997), 477–510.

[Ar] S.I. Araki: On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34.

[A-W 2] R. Azencott, E. Wilson: Homogeneous manifolds with negative curvature II, Mem. Amer. Math. Soc. 8 (1976).

[Be] A.L. Besse: Einstein Manifolds, Springer, Berlin, 1987.

[Ce] S. Cecotti: Homogeneous Kähler Manifolds and T-Algebras in N=2 Supergravity and Superstrings, Commun. Math. Phys. 124 (1989), 23–55.

[C-F-G] S. Cecotti, S. Ferrara, L. Girardello: Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories, Int. J. Mod. Phys. A4 (1989) 2475–2529.

[C] V. Cortés: Alekseevskian Spaces, Diff. Geom. Appl. 6 (1996), no. 2, 129-168.

[dW-VP 0] B. de Wit, A. Van Proeyen: Symmetries of dual-quaternionic manifolds, Phys. Lett. B252 (1990), 221–229.

[dW-VP 1] B. de Wit, A. Van Proeyen: Special Geometry, Cubic Polynomials and Homogeneous Quaternionic Spaces, Commun. Math. Phys. 149 (1992), 307–333.

[dW-VP 2] B. de Wit, A. Van Proeyen: Broken sigma-model isometries in very special geometry, Phys. Lett. B293 (1992), 94–99.

[dW-VP 3] B. de Wit, A. Van Proeyen: Isometries of special manifolds, to be published in the Proceedings of the Meeting on Quaternionic Structures in Mathematics and Physics, Trieste, September 1994.

[dW-V-VP] B. de Wit, F. Vanderseypen, A. Van Proeyen: Symmetry structure of special geometries, Nucl. Phys. B400 (1993), 463-521.

[G-PS-V] S.G. Gindikin, I.I. Pyateckiĭ-Shapiro, E.B. Vinberg: Homogeneous Kähler manifolds in Geometry of Homogeneous Bounded Domains (C.I.M.E., 3^{∘} Ciclo, Urbino, 1967), Edizioni Cremonese, Rome, 1968, 3–87.

[G-W] C.S. Gordon, N. Wilson: Isometry Groups of Riemannian Solvmanifolds, Trans. Amer. Math. Soc., 307 (1988), 245-269.

[He] S. Helgason: Differential geometry, Lie groups and symmetric spaces, Academic Press, Orlando, 1978.

[K-N 1] S. Kobayashi, K. Nomizu: Foundations of differential geometry, Vol. I, Interscience, Wiley, New York, 1963.

[O-V] A.L. Onishchik, E.B. Vinberg: Lie Groups and Algebraic Groups, Springer, Berlin, Heidelberg, 1990.

[PS] I.I. Pyateckiĭ-Shapiro: Automorphic Functions and the Geometry of Classical Domains, Gordon and Breach, New York, 1969.

[Wolt] T. Wolter: Homogene Mannigfaltigkeiten nichtpositiver Krümmung, Dissertation, Zürich, 1989.