HOMOGENEOUS SPECIAL GEOMETRY

VICENTE CORTÉS

Abstract. Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Methods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transitive reductive algebraic group actions on pseudo Riemannian hypersurfaces. The theory is applied to the case of cubic hypersurfaces, which is the case most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special Kähler manifolds.

Introduction

The concept of special geometry was developed in the context of string theory and supergravity. It is a leading principle established by Witten et al (s. e.g. [B-W]) that in order to understand and construct Lagrangeans L for supersymmetric field theories one has to interpret the invariance of the action ∫ Ldμ under the gauge super group as special geometric property of a Riemannian metric; namely the metric g defined by the coefficients g_{ij} of the kinetic terms of the scalar fields ϕ^{1}, …, ϕ^{n} in

L = ∑_{μ=1}^{D} ∑_{i,j=1}^{n} g_{ij}(ϕ^{1}, …, ϕ^{n})∂_{μ}ϕ^{i}∂^{μ}ϕ^{j} + ⋯

In a more narrow sense (cf. e.g. [dW-VP0], [dW-VP] and [St]) special Kähler geometry is the special geometry associated to N = 2 supergravity-Yang-Mills theories in D = 4 spacetime dimensions. It has attracted a lot of attention due to its relation to Mirror Symmetry, s. [Y]. E.g. the Weil-Petersson metric of the moduli space of Calabi-Yau 3-folds is special Kähler, s. Example 1 on p. 8. Similarly, special real geometry [dW-VP] is defined as the geometry associated to N = 2 supergravity-Yang-Mills theories in dimension D = 5. Dimensional reduction of supergravity theories [Cr] from dimension D = 5 to D = 4 induces a correspondence called the r-map [dW-VP] relating special real to special Kähler geometry.

In the first part of the paper, which is a purely mathematical presentation of special geometry, we discuss two natural generalizations of the r-map.

Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University). Research at MSRI is supported in part by grant DMS-9022140.

The basic mathematical object considered is an affine hypersurface H ⊂ ℝ^{n} defined by a homogeneous polynomial h, which induces on H a canonical pseudo Riemannian metric, s. 1.1. By the first generalization of the r-map we canonically associate to H a pseudo Kähler tube domain U, s. 1.2. The second generalization of the rmap associates to H a complex Lagrangean cone C ⊂ T^{∗}ℂ^{n+1}, s. 1.3. If H is a cubic hypersurface, then the pseudo Kähler tube domain U is isomorphic to the projectivized cone P (C) endowed with a canonical pseudo Kähler metric induced by the embedding C ⊂ T^{∗}ℂ^{n+1}, s. Theorem 1.12. If H is moreover a Riemannian cubic hypersurface, then U ≅ P (C) is precisely the special Kähler manifold associated to the special real manifold H via the r-map.

Based on the first construction, we develop a method for the classification of homogeneous Riemannian hypersurfaces of arbitrary degree d ≥ 2, s. 2.1. It is easily applied to the cases 2 ≤ d ≤ 3. As a result, s. Theorem 2.8, the classification of homogeneous cubic Riemannian hypersurfaces and their associated special Kähler (and also quaternionic Kähler) manifolds is obtained, providing a short and conceptual proof for the results of [dW-VP]. Our method uses the basic theory of normal J-algebras (s. Theorem 2.3), avoiding the local cubic tensor calculus of [dW-VP]. A series of homogeneous Riemannian hypersurfaces of arbitrary degree d ≥ 2 is constructed at the end of 2.1.

Finally, a method for the classification of transitive linear reductive algebraic group actions on pseudo Riemannian hypersurfaces of degree d ≥ 2 is presented and carried through for 2 ≤ d ≤ 3, s. 2.2. We remark that by Corollary 1.3 any transitive linear action of a group G ⊂ GL(n, ℝ) on a pseudo Riemannian hypersurface H ⊂ ℝ^{n} induces a transitive affine action of the group Ĝ = ℝ^{n}⋊(ℝ^{+} × G) ⊂ Aff(ℂ^{n}) on the corresponding pseudo Kähler tube domain U ⊂ ℂ^{n} by holomorphic isometries. If H is cubic, U is special pseudo Kähler by Theorem 1.12 and we obtain a transitive affine action on the special pseudo Kähler manifold U by holomorphic isometries.

Acknowledgements The author is very grateful to S.-S. Chern, R. Osserman and MSRI for hospitality and support. He would also like to thank D. Alekseevsky, O. Baues, J.-M. Hwang, M. Kontsevich, A. Van Proeyen, E.B. Vinberg and J.A. Wolf for discussions related to the subject of this paper.

Contents

Introduction i 1. Basic constructions of special geometry 1 1.1. The canonical metric of a hypersurface 1 1.2. The pseudo Kählerian tube domain associated to a pseudo Riemannian

hypersurface of degree d 2

1.3. Special pseudo Kähler geometry and the r-map 3 2. Homogeneous case 9 2.1. Classification of homogeneous Riemannian hypersurfaces and corresponding

Siegel domains 9

2.2. Classification of transitive reductive group actions on pseudo Riemannian

hypersurfaces 23

2.3. Appendix: Special Kähler submanifolds of Alekseevsky’s quaternionic

Kähler manifolds and their pseudo Riemannian analogues 29

References 32

1. Basic constructions of special geometry

1.1. The canonical metric of a hypersurface. Let h be a real or complex polynomial in n variables x^{1}, …, x^{n} which is homogeneous of degree d. Consider the level set

H_{c}(h) = {X = (x^{1}, …, x^{n}) ∈ 𝕂^{n}|h(X) = c}

where c ∈ 𝕂 = ℝ or ℂ. If c ≠ 0, we can write H_{c}(h) = H_{1}(h/c).

Definition 1.1. A hypersurface of degree d is a smooth, open and connected subset H ⊂ H_{1}(h). The basic polynomial h is homogeneous of degree d and not a power of a polynomial of lower degree.

Remark 1: Similar constructions to the ones which will be discussed in the following can be presented for projective hypersurfaces H ⊂ {𝕂X ∈ P (𝕂^{n})|h(X) = 0}.

Now we define the canonical metric g = g(h) of a hypersurface H ⊂ H_{1}(h) of degree d ≥ 2. Let H ∈ ∨^{d}(𝕂^{n})^{∗} be the symmetric d-linear form obtained by polarizing h, i.e. h(X) = H(X, …, X). Then we put

g_{X_{0}}(X, X) := −(d − 1)H(X_{0}, …, X_{0}, X, X) ,
X ∈ T_{X_{0}}H = {X ∈ 𝕂^{n}|H(X_{0}, … X_{0}, X) = 0} .

Formula:

g = −\frac{1}{d}∂^{2}h = −\frac{1}{d}∂^{2} log h on H .  (1)

Remark 2: By the preceding formula, we can define the canonical metric without assuming that h is a homogeneous polynomial. However, we will use the homogeneity of h in the next sections, therefore we have assumed it from the very beginning.

Definition 1.2. A hypersurface H of degree d is called nondegenerate if its canonical metric is nondegenerate.

In the real case (𝕂 = ℝ), the canonical metric g defines on any nondegenerate hypersurface H a canonical structure of pseudo Riemannian hypersurface. In the complex case the canonical metric defines the structure of complex Riemannian hypersurface on any nondegenerate hypersurface.

Remark 3: Special real manifolds in the sense of de Wit and Van Proeyen [dW-VP] are precisely Riemannian cubic hypersurfaces.

1.2. The pseudo Kählerian tube domain associated to a pseudo Riemannian hypersurface of degree d. Let H ⊂ H_{1}(h) ⊂ ℝ^{n} be a pseudo Riemannian hypersurface of degree d with canonical metric g of signature (k, l). We will construct a totally geodesic isometric embedding ι : (H, g) ↪ (U, g^{c}) of H into a pseudo Kähler manifold (U, g^{c}) of complex dimension n and complex signature (k+1, l). We consider the positive cone V := ℝ^{+} · H over H and the tube domain U = ℝ^{n} + iV ⊂ ℂ^{n} with complex coordinates Z = X + iY. We define a canonical pseudo Kähler metric g^{c} on U by the Kähler potential

K(Z) = −\frac{4}{d} log(h(Y)) .

Proposition 1.1. The map ι : (H, g) ↪ (U, g^{c}) given by Y ↦ iY is a totally geodesic isometric embedding of the pseudo Riemannian hypersurface (H, g) of signature (k, l) into the pseudo Kählerian tube domain (U, g^{c}) of complex signature (k+1, l). The cone iV ⊂ U is totally geodesic and isometric to the Riemannian product (ℝ, can) × (H, g) via

ℝ × H ∋ (t, Y) ↦ ie^{t}Y ∈ iV .

Moreover, (U, g^{c}) admits the following global isometries preserving the cone iV:

(i) scaling Z ↦ λZ by λ ∈ ℝ^{+}, (ii) translations Z ↦ Z + X_{0} by real vectors X_{0} ∈ ℝ^{n}, (iii) reflection X + iY ↦ −X + iY and (iv) “inversion” X + iY ↦ X + i\frac{Y}{h(Y)^{2/d}} with respect to iH.

Proof: By definition we have

g_{Z}^{c} = ∑_{i,j=1}^{n} \frac{∂^{2}}{∂z^{i}∂z̄^{j}} K(Z)dz^{i} ⊗ dz̄^{j} =

−\frac{1}{d} ∑_{i,j=1}^{n} \frac{∂^{2}}{∂y^{i}∂y^{j}} log h(Y)(dx^{i} ⊗ dx^{j} + dy^{i} ⊗ dy^{j}) ,

where z^{j} = x^{j} + iy^{j} are the standard coordinates on ℂ^{n} ⊃ ℝ^{n} + iV = U, and Z = X + iY = (x^{1}, …, x^{n}) + i(y^{1}, …, y^{n}). It is clear from this expression, that (ii) and (iii) define isometries of U and from the formula (1) it is obvious that ι : (H, g) → (iH, g^{c}|iH) is an isometry. Using the formula

∂^{2} log h = \frac{∂^{2}h}{h} − \frac{(∂h)^{2}}{h^{2}}  (2)

and the homogeneity of h, it is easy to check that (i) defines an isometric ℝ^{+} action on (U, g^{c}) preserving the cone iV. The induced Killing vector field on the cone is just the radial vector field R(iY) = iY. It has unit length and is orthogonal to the hypersurface iH. From this it follows that iV is a Riemannian product. In particular, the inclusions iH ⊂ iV ⊂ U are totally geodesic, the cone iV being totally geodesic as fixed point set of the reflection. Finally, the fact that (iv) is an isometry of U follows from the fact that iY ↦ iY/h(Y)^{2/d} is an isometry of iV. It corresponds to the isometry (t, Y) ↦ (−t, Y) of ℝ × H ≅ iV. □

Now we show that our construction behaves nicely with respect to group actions.

Proposition 1.2. Let G ⊂ GL(n, ℝ) be a group preserving the pseudo Riemannian hypersurface H ⊂ H_{1}(h) ⊂ ℝ^{n}. Then G acts on (H, g) by isometries with respect to the canonical metric g. Moreover, this action on iH ⊂ U is uniquely extended to a (complex) linear action on ℂ^{n} ⊃ U preserving U, namely by the inclusion G ⊂ GL(n, ℝ) ⊂ GL(n, ℂ). The extended action on U is isometric (and holomorphic) with respect to the canonical metric g^{c} on the tube domain U.

Proof: Assume that A ∈ GL(n, ℝ) preserves H ⊂ ℝ^{n}. This means that A^{∗}h = h on H and hence on the open set V = ℝ^{+} · H, implying A^{∗}h = h. In particular, the transformation A preserves the canonical metric g. The action of A ∈ GL(n, ℝ) ⊂ GL(n, ℂ) on ℂ^{n} is given by X + iY ↦ AX + iAY and it clearly preserves the tube domain U and its canonical metric g^{c}. □

From Propositions 1.1 and 1.2 we obtain the following corollary.

Corollary 1.3. If the hypersurface H is homogeneous under a subgroup

G ⊂ GL(n, ℝ) ⊂ GL(n, ℂ),

then the associated tube domain (U, g^{c}) is homogeneous as pseudo Kähler manifold under the subgroup

Ĝ = ℝ^{n}⋊(ℝ^{+} × G) ⊂ Aff(ℂ^{n}) = ℂ^{n}⋊GL(n, ℂ) .

1.3. Special pseudo Kähler geometry and the r-map. First we define special pseudo Kähler geometry in terms of canonical metrics on Lagrangean cones. Second we derive the coordinate expressions (well known in the physical literature) for these metrics in terms of a local holomorphic function homogeneous of degree 2. Then we define a straightforward generalization of the physical r-map. By this we associate a Lagrangean cone C to every real hypersurface H of degree d. If H is a cubic pseudo Riemannian hypersurface, then the projectivized cone P (C) with its special metric, s. Definition 1.5, is isometric to the tube domain (U, g^{c}) associated to H. This shows, in particular, that (U, g^{c}) is a special pseudo Kähler manifold.

Consider the following fundamental algebraic data:

1) A complex symplectic vector space (V, ω)

2) A compatible real structure, i.e. a ℂ-antilinear involution τ : V → V satisfying

ω(τX, τY) = \overline{ω(X, Y)} , ∀X, Y ∈ V .

Given these data we define a sesquilinear form γ on V by

γ(X, Y) := iω(X, τY) .

Lemma 1.4. The form γ is a Hermitian form of signature (n + 1, n + 1), where dimℂ V = 2n + 2.

Proof: We first check that γ is Hermitian:

\overline{γ(X, Y)} = −i \overline{ω(X, τY)} = −iω(τX, Y) = iω(Y, τX) = γ(Y, X) .

Now we show that the Hermitian form γ has signature (n + 1, n + 1). The restriction of the symplectic form ω to the fixed point set V^{τ} of the real structure τ : V → V is a real symplectic form on the real vector space V^{τ}, hence we can choose a basis p_{α}, q_{β}, α, β = 1, …, n + 1, of V^{τ} such that

ω(p_{α}, q_{β}) = δ_{αβ} , ω(p_{α}, p_{β}) = ω(q_{α}, q_{β}) = 0 .

Then p_{α}, iq_{β}, α, β = 1, …, n + 1, is a Witt basis for γ, i.e.

γ(p_{α}, iq_{β}) = δ_{αβ} , γ(p_{α}, p_{β}) = ω(iq_{α}, iq_{β}) = 0 .

This shows that γ has signature (n + 1, n + 1). □

Remark that up to isomorphism we can assume that the fundamental data (V, ω, τ) are the following: V = T^{∗}ℂ^{n+1} = T^{∗}ℝ^{n+1}+iT^{∗}ℝ^{n+1}, ω the complex bilinear extension of the standard symplectic form on T^{∗}ℝ^{n+1} and τ complex conjugation with respect to the real form V^{τ} = T^{∗}ℝ^{n+1}.

We recall that a complex vector subspace L of a complex symplectic vector space (V, ω) is called Lagrangean if it is maximally isotropic.

Definition 1.3. Given fundamental algebraic data (V, ω, τ) a Lagrangean subspace L ⊂ V is called nondegenerate if γ|L is nondegenerate. A connected submanifold C ⊂ V is called (nondegenerate) Lagrangean cone if ℝ^{+} · C = C and if L = T_{v}C is a (nondegenerate) Lagrangean subspace of V for all v ∈ C.

Remark that a Lagrangean subspace L ⊂ V is nondegenerate if and only if L∩τL = 0. From Definition 1.3 it follows that a nondegenerate Lagrangean cone C is a pseudo Kählerian submanifold of the pseudo Kähler manifold (V, γ).

Definition 1.4. The induced metric g^{C} = γ|C is called the canonical metric of the Lagrangean cone C ⊂ (V, ω, τ).

Now we define a canonical pseudo Kähler metric g^{P (C)} on the projective image P (C) ⊂ P (V) of a nondegenerate Lagrangean cone C. For this we assume that γ(u, u) ≠ 0 for all u ∈ C. If this additional condition is satisfied we shall say that the cone C is properly nondegenerate. Remark that a Riemannian cone (C, g^{C}) is automatically properly nondegenerate. Denote by π : V → P (V) the canonical projection. We define

g_{πu}^{P (C)}(dπv, dπv) = \frac{γ(v, v)}{γ(u, u)} − |\frac{γ(u, v)}{γ(u, u)}|^{2}

for u ∈ C ⊂ V, v ∈ T_{u}C ⊂ V.

Definition 1.5. The Hermitian metric g^{P (C)} is called the special metric of the projectivized Lagrangean cone P (C). A pseudo Kähler manifold is called special if it is locally isometric to (P (C), g^{P (C)}) for some Lagrangean cone C ⊂ (V, ω, τ).

Proposition 1.5. Assume that C is a properly nondegenerate Lagrangean cone. Then the special metric g^{P (C)} on P (C) is a pseudo Kähler metric with Kähler form

Im(g^{P (C)}) = 2πc_{1}(U, γ),

where c_{1}(U, γ) is the Chern form of the universal bundle p : U → P (C) with Hermitian metric induced by γ. If the canonical metric g^{C} has Riemannian signature, then the special metric g^{P (C)} is a (Riemannian) Kähler metric.

Proof: Let ζ be a holomorphic section of U. Then

2πc_{1}(U, γ) = −i∂ ¯∂log γ(ζ, ζ) =

−i\frac{∂ ¯∂γ(ζ, ζ)}{γ(ζ, ζ)} + i\frac{∂γ(ζ, ζ) ∧ ¯∂γ(ζ, ζ)}{γ(ζ, ζ)^{2}} =

\frac{Im(γ)}{γ(ζ, ζ)} + i\frac{γ(·, ζ) ∧ γ(ζ, ·)}{γ(ζ, ζ)^{2}} = Im(g_{p(ζ)}^{P (C)})

(here we have used the convention a ∧ b̄ = (a ⊗ b̄ − b̄ ⊗ a)/2). □

Now we describe Lagrangean cones C and the special metric g^{P (C)} on the projectivization P (C) by a basic function F, thereby relating our presentation to the usual construction of special Kähler geometry in the physical literature. We consider the standard model V = T^{∗}ℂ^{n+1}. Let F (Z), Z = (z^{0}, …, z^{n}) ∈ ℂ^{n+1}, be a locally defined holomorphic function on ℂ^{n+1} which is homogeneous of degree 2, i.e. F (λZ) = λ^{2}F (Z), λ ∈ ℂ − {0}, where this equation makes sense. We call F a basic function. For example, we may take F = p/q to be the quotient of two homogeneous polynomials p, q, deg p = deg q + 2. We are interested in the image C_{F} ⊂ T^{∗}ℂ^{n+1} of the differential dF : Z ↦ dF |_{Z}.

Proposition 1.6. Let F be a basic function. Then the connected components of C_{F} − {0} are Lagrangean cones. Conversely, every Lagrangean cone which locally projects isomorphically onto ℂ^{n+1} is locally of this form for some basic function F.

Proof: It is a well known fact in symplectic geometry, that a Lagrangean submanifold of T^{∗}ℂ^{n+1} which locally projects isomorphically onto ℂ^{n+1} is locally the image C_{F} of a differential dF. Now C_{F} is a cone if and only if dF is homogeneous of degree one, i.e. if and only if F is homogeneous of degree two. □

Now let C = C_{F} be a Lagrangean cone which is the image of the differential dF of a basic function F. We want to express the metrics g^{C} and g^{P (C)} in terms of F. Denote by Z = (z^{0}, …, z^{n}) and P = (p_{0}, …, p_{n}) be the canonical coordinates for T^{∗}ℂ^{n+1}.

Proposition 1.7. The canonical metric g^{C} of the Lagrangean cone C = C_{F} at the point dF |_{Z} ∈ C_{F} is given by:

g^{C} = −2 ∑_{j,k=0}^{n} Re (i\frac{∂^{2}F (Z)}{∂z^{j}∂z^{k}}) dz^{j} ⊗ dz̄^{k} = 2 ∑_{j,k=0}^{n} Im (\frac{∂^{2}F (Z)}{∂z^{j}∂z^{k}}) dz^{j} ⊗ dz̄^{k} .

Proof: The standard symplectic form ω on T^{∗}ℂ^{n+1} is

ω = ∑_{j=0}^{n}(dz^{j} ⊗ dp_{j} − dp_{j} ⊗ dz^{j}) .

The corresponding Hermitian metric γ = iω(·,¯·) of signature (n+1, n+1) on T^{∗}ℂ^{n+1} is given by

γ = i ∑_{j=0}^{n}(dz^{j} ⊗ dp̄_{j} − dp_{j} ⊗ dz̄^{j}) .

On C = C_{F} we have p_{j} = ∂F/∂z^{j}, hence

g^{C} = γ|C = i ∑_{j=0}^{n}(dz^{j} ⊗ d\frac{∂F̄}{∂z̄^{j}} − d\frac{∂F}{∂z^{j}} ⊗ dz̄^{j})

=^{(∗)} i ∑_{j,k=0}^{n} (\frac{∂^{2}F̄}{∂z̄^{j}∂z̄^{k}} − \frac{∂^{2}F}{∂z^{j}∂z^{k}} ) dz^{j} ⊗ dz̄^{k} = 2 ∑_{j,k=0}^{n} Im (\frac{∂^{2}F}{∂z^{j}∂z^{k}} ) dz^{j} ⊗ dz̄^{k} ,

where at (∗) we have used that F is holomorphic. □

Since we are assuming that C = C_{F}, the cone C ⊂ T^{∗}ℂ^{n+1} is mapped isomorphically onto an open subset of ℂ^{n+1} under the canonical projection T^{∗}ℂ^{n+1} → ℂ^{n+1}. Therefore, we can use inhomogeneous coordinates of P (ℂ^{n+1}) as coordinates on P (C), e.g. ζ^{j} = z^{j}/z^{0}, j = 1, …, n. Recall that in order for our definition of the special metric g^{P (C)} to make sense we assume that C is properly non degenerate, i.e. γ(u, u) ≠ 0 for all u ∈ C. Without restriction of generality we may assume γ(u, u) > 0 for all u ∈ C.

Corollary 1.8. The special metric g^{P (C)} is given with respect to inhomogeneous coordinates on Pℂ^{n} by the pseudo Kähler potential

K^{F} (ζ^{1}, …, ζ^{n}) = log γ(ζ, ζ) = log (2 ∑_{j,k=0}^{n} Im (\frac{∂^{2}F (ζ)}{∂z^{j}∂z^{k}}) ζ^{j}ζ̄^{k}) ,

where ζ = (ζ^{0} = 1, ζ^{1}, …, ζ^{n}).

Proof: The first equation follows from Proposition 1.5, since (ζ^{1}, …, ζ^{n}) ↦ dF |_{ζ} ∈ C ⊂ T^{∗}ℂ^{n+1} is a local section of the universal bundle over the projectivized cone P (C). Remark that (ζ^{1}, …, ζ^{n}) are the inhomogeneous coordinates of the point πdF |_{ζ} ∈ P (C) ⊂ P (T^{∗}ℂ^{n+1}). The second equation follows from Proposition 1.7. Remark that the functions \frac{∂^{2}F}{∂z^{j}∂z^{k}} are homogeneous of degree zero and hence well defined on Pℂ^{n} as they must be. □

By Proposition 1.6 to every local holomorphic function F homogeneous of degree two we can associate a Lagrangean cone C_{F} and the special metric g^{F} := g^{P (C_{F})} on the projectivization P (C_{F}). The r-map maps by definition any homogeneous cubic polynomial h(x^{1}, … x^{n}) on ℝ^{n} to the (possibly degenerate) special metric g^{F_{h}} with

F_{h}(Z) = \frac{h(z^{1}, …, z^{n})}{z^{0}} , Z = (z^{0}, …, z^{n}) .

We can easily extend the r-map to homogeneous polynomials h of degree d by

h ↦ F_{h}(Z) = \frac{h(z^{1}, …, z^{n})}{(z^{0})^{d−2}} .

Now we want to compare the special metric g^{F_{h}} on the projectivized Lagrangean cone P (C_{F_{h}}) to the canonical metric g^{c} on the tube domain U = ℝ^{n} + iV, V = ℝ^{+} · H, associated to a hypersurface H ⊂ H_{1}(h). Consider the biholomorphism

φ : ℂ^{n} ⊃ U ∋ Z ↦ πdF_{h}|_{(1,Z)} ∈ P (C_{F_{h}}) ⊂ P (T^{∗}ℂ^{n+1}) ,

where π : T^{∗}ℂ^{n+1} → P (T^{∗}ℂ^{n+1}) is the canonical projection.

Proposition 1.9. The metric g^{s} = φ^{∗}g^{F_{h}} on the tube domain U associated to the homogeneous polynomial h of degree d has the Kähler potential

K^{s}(Z) = (φ^{∗}K^{F_{h}})(Z) = −\frac{4}{d} log(2Im (−H(Z, …, Z)(d − 2) + H(Z, …, Z, Z̄)d)) ,

where H is the polarization of h. In particular, g^{F_{h}} is nondegenerate if and only if ∂ ¯∂K^{s} is nondegenerate.

Proof: We have to show that

(∂^{2}F_{h})(ζ, ζ̄) = −H(Z, …, Z)(d − 2) + H(Z, …, Z, Z̄)d ,

where ∂^{2}F_{h} = (\frac{∂^{2}F_{h}}{∂z^{i}∂z^{j}})_{i,j=0,... ,n} is the complex Hessian form at the point ζ = (1, Z). We compute:

(∂^{2}F_{h})(ζ, ζ̄) = h(Z)(d − 2)(d − 1) − ∂hZ̄(d − 2)
−∂hZ(d − 2) + (∂^{2}h)(Z, Z̄)
= H(Z, …, Z)(d − 2)(d − 1) − H(Z, …, Z, Z̄)(d − 2)d
−H(Z, …, Z)(d − 2)d + H(Z, …, Z, Z̄)(d − 1)d
= −H(Z, …, Z)(d − 2) + H(Z, …, Z, Z̄)d . □

Proposition 1.10. The metrics g^{c} and g^{s} on the tube domain associated to a homogeneous cubic polynomial h coincide.

Proof: The following lemma shows that the Kähler potentials defining g^{s} and g^{c} coincide up to an additive constant. □

Lemma 1.11. Let H ∈ ∨^{3}(ℝ^{n})^{∗} be the polarization of the homogeneous cubic polynomial h and Z = X + iY ∈ ℂ^{n}, then

Im (−H(Z, Z, Z) + 3H(Z, Z, Z̄)) = 4h(Y) .

The following theorem is a consequence of Propositions 1.1, 1.9 and 1.10.

Theorem 1.12. The metrics g^{c} = g^{s} = φ^{∗}g^{F_{h}} and g^{F_{h}} associated to a pseudo Riemannian cubic hypersurface (H, g) ⊂ H_{1}(h) with canonical metric g of signature (k, l) are special pseudo Kählerian of complex signature (k + 1, l). In particular, the r-map maps special real manifolds (i.e. Riemannian cubic hypersurfaces) to special Kähler manifolds.

Example 1: Let X be a Kähler manifold with holonomy algebra 𝔥𝔬𝔩 = 𝔰𝔲(3), i.e. a general Calabi-Yau 3-fold. Consider the complex vector space V = H^{3}(X, ℂ) with standard real structure τ, V^{τ} = H^{3}(X, ℝ). The “intersection” form ω

ω(ξ, η) = ∫_{X} ξ ∧ η , ξ, η ∈ V ,

is a complex skew symmetric bilinear form on V compatible with τ, i.e.

\overline{ω(ξ, η)} = ω(τξ, τη) , ξ, η ∈ V .

By the Hodge-Riemann bilinear relations for primitive cohomology, s. e.g. [We] Ch. V Sec. 6, the form ω is non degenerate and thus a complex symplectic form. Indeed, the third cohomology is primitive, i.e. Ω ∧ ξ = 0 ∈ H^{5}(X, ℂ) for all ξ ∈ H^{3}(X, ℂ), where Ω is the Kähler class of X. This follows from the formula h_{0}^{p,q} = h^{p,q} −h^{p−1,q−1}, which relates the primitive Dolbeault numbers h_{0}^{p,q} to the usual ones h^{p,q} and from the fact that the only holomorphic form on X (up to scaling) is the volume form, due to our holonomy assumption.

Summing up, to every general Calabi-Yau 3-fold we have associated the fundamental algebraic data (V, ω, τ), cf. definition on p. 4.

Again by the Hodge-Riemann bilinear relations, the complementary subspaces W = H^{3,0}(X)+H^{1,2}(X) and \overline{W} := τW are Lagrangean (with respect to ω), the Hermitian form γ = iω(·, τ·) is positively defined on W × \overline{W} and H^{3,0}(X) and H^{1,2}(X) are γ-orthogonal.

Now we consider the moduli space of X, i.e deformations of its complex structure. It is known (s. [Ti], [To], cf. [Kn]) that there exists a local universal deformation δ : X → S, X_{s} := δ^{−1}(s), X_{s_{0}} = X, of X. The base (S, s_{0}) is the Kuranishi moduli space, which can be identified with a connected open subset of H^{2,1}(X).

Since X is Kähler the Dolbeault numbers h^{p,q}(X_{s}) are independent of the deformation parameter s, by well known results of Kodaira and Spencer’s deformation theory (combine [We] Ch. V Cor. 6.6 and [M-K] Thm. 4.6). After identification of the fibres of δ : X → S by a diffeomorphism X ≅ S × X we can define the period map

P er : S ∋ s ↦ H^{3,0}(X_{s}) ∈ P (H^{3}(X, ℂ)) = P (V) .

The differential dP er_{s} maps T_{s}S ≅ H^{2,1}(X) isomorphically onto dπ_{p}(H^{3,0}(X_{s}) + H^{2,1}(X_{s})) = dπ_{p}H^{2,1}(X_{s}), where π : V → P (V) is the canonical projection and 0 ≠ p ∈ P er(s) = H^{3,0}(X_{s}). Remark that the subspace W_{s} = H^{3,0}(X_{s}) + H^{2,1}(X_{s}) ⊂ V is still Lagrangean, because a local deformation of a Kähler manifold is again Kähler, s. [M-K] Thm. 4.6. This shows that the lines P er(s) ⊂ V, s ∈ S, form a Lagrangean cone C = ∪_{s}P er(s) ⊂ V.

The special metric g^{P (C)} on P (C) = P er(S) is negatively defined and hence −g^{P (C)} is a genuine Kähler metric. In fact, γ is negatively defined on H^{2,1}(X_{s}) = P er(s)^{⊥} ∩ T_{p}C, p ∈ P er(s)−{0}; here ⊥ denotes the γ-orthogonal complement. The (negatively defined) special “Kähler” metric g^{P (C)} is known as Weil-Petersson metric. Remark 4: The Weil-Petersson metric on the Kuranishi moduli space for general Calabi-Yau m-folds of arbitrary dimension m can be defined similarly. The reasons why we have concentrated on the case m = 3 are the following: first P er(S) is always a totally isotropic cone with respect to the intersection form, but in general not maximally isotropic in H^{m}(X, ℂ), and second the intersection form is symmetric if m is even.

2. Homogeneous case

2.1. Classification of homogeneous Riemannian hypersurfaces and corresponding Siegel domains. Let H ⊂ H_{1}(h) ⊂ ℝ^{n} be a pseudo Riemannian hypersurface of degree d ≥ 2 with basic polynomial h and canonical metric g. We define the real algebraic linear group

Aut(h) = {φ ∈ GL(n, ℝ)|φ^{∗}h = h} ,

which acts naturally on H_{1}(h).

Definition 2.1. A pseudo Riemannian hypersurface H ⊂ H_{1}(h) of degree d ≥ 2 is said to be homogeneous if the connected component Aut_{0}(h) acts transitively on H.

From Definition 2.1 and the definition of the canonical metric g (p. 1) it follows that a homogeneous pseudo Riemannian hypersurface (H, g) of degree d ≥ 2 admits a transitive group of isometries, namely Aut_{0}(h) ⊂ Isom(H, g). By Corollary 1.3 to any homogeneous (pseudo) Riemannian hypersurface of degree d we have canonically associated a homogeneous (pseudo) Kähler manifold, which is special (pseudo) Kähler if d = 3. This motivates the interest in the classification problem for homogeneous such hypersurfaces, which we study in this section.

The first step is to reduce the classification of homogeneous pseudo Riemannian hypersurfaces (H, g) of degree d ≥ 2 to the case of hypersurfaces admitting a transitive triangular subgroup L ⊂ Aut_{0}(h). By a decomposition theorem for real algebraic groups due to Vinberg [V] we prove now that this reduction is possible if the canonical metric g is Riemannian.

Theorem 2.1. Let (H, g) be a homogeneous Riemannian hypersurface of degree d ≥ 2.Then the homogeneous pseudo Kählerian tube domain U = ℝ^{n} + iV associated to H ⊂ H_{1}(h) ⊂ ℝ^{n} (cf. Corollary 1.3) is a Siegel domain of type I, i.e. V is convex and does not contain any line. Moreover, V is a connected component of ℝ^{n} −H_{0}(h). Finally, Aut_{0}(h) admits the polar decomposition

Aut_{0}(h) = K · L ,

where K is the stabilizer of a point v ∈ H and is a maximal compact connected subgroup and L is a maximal triangular subgroup acting simply transitively on H. In particular, H is contractible.

Proof: The stated properties of the cone V follow, by well known arguments of Koszul [K], from the existence of the (ℝ^{+} ×Aut_{0}(h))-invariant closed 1-form α = −d log h on the homogeneous cone V = ℝ^{+}H ⊂ ℝ^{\} with positively defined Euclidean covariant derivative ∂α > 0. Recall that the Hessian bilinear form ∂α = −∂^{2} log h defines up to a positive factor the canonical (ℝ^{+} ×Aut_{0}(h))-invariant Riemannian metric g^{V} ˆ=g^{c}|iV of the cone (V, g^{V}) ≅ (iV, g^{c}|iV) ⊂ (U, g^{c}), cf. Propositions 1.1 and 1.2.

The polar decomposition follows from Vinberg’s theorem [V] and the convexity of the cone. In fact, since Aut(h) is defined by a polynomial equation, it is a real algebraic linear group. Hence, by [V] we have Aut_{0}(h) = K · L, where K is maximal compact and connected and L is maximal triangular.

From the convexity of the cone V it follows that K fixes a point v ∈ H (centre of gravity of a compact orbit), hence K is contained in the isotropy group K^{1} of Aut_{0}(h) at v. Since Aut_{0}(h) acts by isometries of the Riemannian metric g, the isotropy group K^{1} is compact and hence K = K^{1}. □

The second step is to reformulate our classification problem in terms of metric Lie algebras.

Definition 2.2. A metric Lie group (L, g) is a Lie group L together with a leftinvariant (pseudo) Riemannian metric g. Its metric Lie algebra (𝔩, ⟨·, ·⟩) consists of the Lie algebra 𝔩 = Lie L together with the scalar product ⟨·, ·⟩ = g_{e}, e ∈ L the identity. A (pseudo) Kähler Lie group (L, g, J̃) is a metric Lie group (L, g) together with a parallel, left-invariant and orthogonal complex structure J̃. Its (pseudo) Kähler Lie algebra (𝔩, ⟨·, ·⟩, J) consists of its metric Lie algebra (𝔩, ⟨·, ·⟩) together with the orthogonal complex structure J := J̃_{e} on 𝔩. We say that (𝔩, ⟨·, ·⟩) (resp. (𝔩, ⟨·, ·⟩, J)) is metric (resp. pseudo Kähler) Lie algebra for the pseudo Riemannian (resp. pseudo Kähler) manifold M if it is the metric (resp. pseudo Kähler) Lie algebra of a metric (resp. pseudo Kähler) Lie group which as pseudo Riemannian (resp. as pseudo Kähler) manifold is isomorphic to M. A Kähler Lie algebra (𝔩, ⟨·, ·⟩, J) is called a normal J-algebra if 𝔩 is splittable solvable and if its Kähler form ⟨·, J·⟩ is the differential of a 1-form ω on 𝔩, i.e. if ω([X, Y]) = −⟨X, JY ⟩ for all X, Y ∈ 𝔩.

It is known [PS] that any normal J-algebra is Kähler Lie algebra for a bounded homogeneous domain, the Kähler metric not necessarily being the Bergmann metric. Example 2: The basic examples of normal J-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. Given 𝔣 and a Euclidean vector space 𝔵 with orthogonal complex structure, 𝔢 = 𝔣 + 𝔵 carries a canonical Euclidean scalar product ⟨·, ·⟩ and complex structure J. The structure of 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 (i.e. orthogonal isomorphism of Lie algebras) by n = dimℂ 𝔵 and μ. If we wish to specify these parameters, we shall write 𝔢 = 𝔢(n + 1, μ). The elementary Kählerian Lie algebra 𝔢(n + 1, μ) is Kähler Lie algebra for the complex hyperbolic space Hℂ^{n+1} with suitably normalized holomorphic sectional curvature. Remark that as Lie algebra 𝔢(n+1, μ) is isomorphic to the Iwasawa Lie algebra of the semisimple Lie algebra 𝔰𝔲(1, n + 1).

Let (H, g) be a homogeneous Riemannian hypersurface of degree d ≥ 2 with basic polynomial h and B_{0} ⊂ Aut_{0}(h) a simply transitive triangular subgroup, which exists by Theorem 2.1. By Corollary 1.3 the Lie group B := ℝ^{+}×B_{0} ⊂ GL(n, ℝ) (resp. U_{0} := ℝ^{n}⋊B ⊂ Aff(ℂ^{n})) acts simply transitively and isometrically (resp. and isometrically and biholomorphically) on the cone (V, g^{V}) ≅ (iV, g^{c}|iV) (resp. on the Siegel domain (U = ℝ^{n} + iV, g^{c})). The orbit map

φ : U_{0} ∋ u ↦ u(ip) ∈ U , p ∈ H ,

induces diffeomorphisms B_{0} ≅ B_{0}(ip) = iH ≅ H, B ≅ B(ip) = iV ≅ V and U_{0} ≅ U_{0}(ip) = U defining the structure of metric Lie group on B_{0}, B and U_{0} by the condition that φ be an isometry. Moreover, the pull back of the complex structure of the tube domain via φ is a left-invariant, parallel orthogonal complex structure on U_{0}. Hence we have on U_{0} the structure of Kähler Lie group. Its Kähler Lie algebra (𝔲_{0}, ⟨·, ·⟩, J) contains the (metric) Lie algebras 𝔟_{0} = Lie B_{0} and 𝔟 = Lie B as metric subalgebras.

Proposition 2.2. The Kähler Lie algebra (𝔲_{0}, ⟨·, ·⟩, J) associated to a homogeneous Riemannian hypersurface H ⊂ H_{1}(h) of degree d ≥ 2 is a normal J-algebra and admits the orthogonal (semidirect) decompositions:

𝔲_{0} = 𝔟+ ∋J𝔟 , 𝔟 = ℝB_{0} ⊕ 𝔟_{0} ,

where J𝔟 is an Abelian ideal and the vector B_{0} is in the centre of 𝔟 and satisfies the equations

ad_{B_{0}}JX = ad_{X}JB_{0} = JX for all X ∈ 𝔟 .

The polynomial h_{0} = h ∘ dφ_{e}|J𝔟 is homogeneous of degree d and invariant under the adjoint action of B_{0} on the ideal J𝔟 ⊂ 𝔲_{0}. Moreover, the differential dφ_{e} of the orbit map φ maps the B_{0}-orbit H^{0} = B_{0}(JB_{0}) ⊂ H_{1}(h_{0}) ⊂ J𝔟 of the vector JB_{0} diffeomorphically onto the hypersurface H ⊂ H_{1}(h) ⊂ ℝ^{n} ⊂ T_{ip}U, ip = φ(e). It is an isometry with respect to the canonical metrics on H^{0} and H.

Proof: The decompositions of metric Lie algebras are induced by the decompositions U_{0} = B⋉ℝ^{n} and B = ℝ^{+} × B_{0} of metric Lie groups.

The definition of the Kähler form Ω of the tube domain U by the Kähler potential K(Z) = −\frac{4}{d} log h(Y) on p. 2 shows that Ω is exact in the complex of U_{0}-invariant differential forms on U. In fact, ω = \overline{∂} log h(Y) is an U_{0}-invariant 1-form on the tube domain U and dω = ∂\overline{∂} log h(Y) is proportional to Ω. Since, by Theorem 2.1, 𝔲_{0} is splittable solvable, this shows that (𝔲_{0}, ⟨·, ·⟩, J) is a normal J-algebra.

The last statements follow from the fact that the representations B_{0} → GL(J𝔟) and B_{0} ↪ GL(ℝ^{n}) are isomorphic via

dφ_{e}|J𝔟 : 𝔟 + J𝔟 = 𝔲_{0} ⊃ J𝔟 ⟶^{∼} ℝ^{n} ⊂ T_{ip}U = ℝ^{n} + iℝ^{n} . □

Definition 2.3. Let 𝔵, 𝔶 and 𝔷 be pseudo Euclidean vector spaces. A bilinear map ψ : 𝔵 × 𝔶 → 𝔷 is said to be isometric, if

⟨ψ(X, Y), ψ(X, Y)⟩ = ⟨X, X⟩⟨Y, Y ⟩

for all X ∈ 𝔵 and Y ∈ 𝔶. The number k = dim 𝔵 is the order of the isometric map. An isometric map ψ : 𝔵 × 𝔶 → 𝔷 is called special if dim 𝔶 = dim 𝔷.

The transpose ψ^{t} : 𝔵 × 𝔷 → 𝔶 of an isometric map ψ : 𝔵 × 𝔶 → 𝔷 is defined by

⟨ψ^{t}(X, Z), Y ⟩ := ⟨Z, ψ(X, Y)⟩ , X ∈ 𝔵 , Y ∈ 𝔶 and Z ∈ 𝔷 .

Remark that the transpose ψ^{t} of an isometric map ψ is isometric if and only if ψ (and hence ψ^{t}) is a special isometric map.

The following important structure result can be extracted from [PS], cf. [G-PS-V].

Theorem 2.3. (S.G. Gindikin, I.I. Pyateckiĭ-Shapiro, E.B. Vinberg) Any normal J-algebra (𝔲_{0}, ⟨·, ·⟩, J) has an orthogonal semidirect decomposition

𝔲_{0} = 𝔢_{1} + 𝔢_{2} + ⋯ + 𝔢_{l}  (3)

into elementary Kählerian subalgebras 𝔢_{j} = 𝔣_{j} + 𝔵_{j} with root μ_{j}, j = 1, 2, …, l. More precisely, [𝔣_{i}, 𝔢_{j}] = 0 and [𝔵_{i}, 𝔢_{j}] ⊂ 𝔵_{i} if i < j.

The normal J-algebra (𝔲_{0}, ⟨·, ·⟩, J) is Kähler Lie algebra for a Siegel domain of type I if and only if we have the following orthogonal decompositions

𝔲_{0} = 𝔟+ ∋J𝔟 , 𝔟 = ℝB_{0} ⊕ 𝔟_{0} ,  (4)

where J𝔟 is an Abelian ideal and B_{0} = ∑_{i=1}^{l} \frac{1}{μ_{i}} H_{i} is in the centre of 𝔟. Under this assumption, we have orthogonal decompositions

𝔵_{i} = ∑_{j=i+1}^{l} 𝔵_{ij} , 𝔵_{ij} = 𝔵_{ij}^{−} + 𝔵_{ij}^{+} , 𝔵_{ij}^{+} = J𝔵_{ij}^{−} , i = 1, …, l − 1 , 𝔵_{l} = 0 ,

such that

𝔟 = 𝔞 + ∑_{j>i} 𝔵_{ij}^{−} , where 𝔞 = span{H_{i}|i = 1, …, l}

and we have the commutator relations

[𝔣_{k}, 𝔵_{ij}] = 0 if i < j, i < k and j ≠ k;
ad_{H_{j}}|𝔵_{ij}^{±} = ±\frac{μ_{j}}{2} Id ,
ad_{G_{j}}|𝔵_{ij}^{+} = 0 ,
ad_{G_{j}}|𝔵_{ij}^{−} = −μ_{j}J if i < j;
[𝔵_{st}, 𝔵_{ij}] = 0 if s < t, i < j, i < s and s ≠ j ≠ t;
[𝔵_{jk}^{±}, 𝔵_{ij}^{−}] ⊂ 𝔵_{ik}^{±} ,
[𝔵_{jk}, 𝔵_{ij}^{+}] = 0 if i < j < k;
[𝔵_{kj}^{±}, 𝔵_{ij}^{∓}] ⊂ 𝔵_{ik}^{+} ,
[𝔵_{kj}^{±}, 𝔵_{ij}^{±}] = 0 if i < k < j .

The Lie bracket [·, ·] : 𝔵_{jk}^{±} × 𝔵_{ij}^{−} → 𝔵_{ik}^{±} for i < j < k is given by an isometric map ψ_{ijk} : 𝔵_{jk}^{−} × 𝔵_{ij}^{−} → 𝔵_{ik}^{−} as follows:

[X, Y] = \frac{1}{\sqrt{2}}ψ_{ijk}(X, Y) ,
[JX, Y] = J[X, Y] , X ∈ 𝔵_{jk}^{−} , Y ∈ 𝔵_{ij}^{−} .

The Lie bracket [·, ·] : 𝔵_{kj}^{±} × 𝔵_{ij}^{∓} → 𝔵_{ik}^{+} for i < k < j is given by

⟨[X, Y], Z⟩ = −\frac{1}{\sqrt{2}}⟨JY, ψ_{ikj}(X, JZ)⟩ ,

[X, Y] = [JX, JY] , X ∈ 𝔵_{kj}^{−} , Y ∈ 𝔵_{ij}^{+} , Z ∈ 𝔵_{ik}^{+} .

The number l of elementary Kählerian subalgebras in the decomposition (3) is called the rank of the normal J-algebra 𝔲_{0}. The normal J-algebras for Siegel domains of type I will be called normal J-algebras of type I.

Lemma 2.4. Let (𝔲_{0}, ⟨·, ·⟩, J) be a normal J-algebra for a Siegel domain of type I. Then the orbit H = B_{0}(JB_{0}) of the vector JB_{0} ∈ J𝔟 under the adjoint action of B_{0} ⊂ U_{0} on the ideal J𝔟 ⊂ 𝔲_{0} = 𝔟 + J𝔟 is a smooth hypersurface.

Proof: It is sufficient to prove that the map

𝔟_{0} ∋ X ↦ ad_{X}JB_{0} ∈ J𝔟

has maximal rank. This follows from the equation ad_{X}JB_{0} = JX (X ∈ 𝔟), which characterizes the vector B_{0} ∈ 𝔟. □

By Proposition 2.2 and Lemma 2.4 the classification of homogeneous Riemannian hypersurfaces of degree d ≥ 2 reduces to the following problem, which can be studied using Theorem 2.3. Problem 1: Classify all normal J-algebras (𝔲_{0}, ⟨·, ·⟩, J) for Siegel domains of type I (i.e. which admit the decompositions (4)) such that the hypersurface H = B_{0}(JB_{0}) is contained in the level set H_{1}(h) ⊂ J𝔟 of a homogeneous polynomial h of degree d ≥ 2 on J𝔟 and the canonical metric of H is Riemannian.

Remark that if d = 3, then to any solution (𝔲_{0}, ⟨·, ·⟩, J) to Problem 1 we can associate a homogeneous special Kähler manifold, namely the Kählerian Siegel domain of type I associated to the homogeneous Riemannian cubic hypersurface H.

For this reason we will give the complete solution to Problem 1 for 2 ≤ d ≤ 3. Examples of homogeneous Riemannian hypersurfaces of arbitrary degree d ≥ 2 and the corresponding homogeneous Kählerian Siegel domains of type I will be presented at the end of this section. An interesting class of homogeneous pseudo Riemannian cubic hypersurfaces and the corresponding homogeneous special pseudo Kähler and pseudo quaternionic Kähler manifolds will be discussed in 2.3.

Before studying the general case, we consider the normal J-algebras of the form 𝔲_{0} = 𝔣_{1} ⊕ 𝔣_{2} ⊕ ⋯ ⊕ 𝔣_{l}, i.e. all the elementary Kählerian subalgebras are key algebras. We have the orthogonal decomposition 𝔲_{0} = 𝔟 + J𝔟 with

𝔟 = 𝔞 = span\{H_{i}|i = 1, …, l\} = ℝB_{0} ⊕^{⊥} 𝔟_{0} , B_{0} = ∑_{i=1}^{l} \frac{1}{μ_{i}} H_{i} .

Lemma 2.5. If 𝔲_{0} = 𝔣_{1} ⊕ 𝔣_{2} ⊕ ⋯ ⊕ 𝔣_{l} is an orthogonal direct sum of key algebras, then a B_{0}-invariant homogeneous function f is defined near the point JB_{0} ∈ J𝔟 by

f(η) = ∏_{j=1}^{l} a_{j}^{∏_{k≠j} μ_{k}^{2}} , η = ∑_{j=1}^{l} a_{j}G_{j} ∈ J𝔟 = span{G_{j}|j = 1, …, l} .

Any such (local) function is proportional to a real power of f.

Proof: To prove that f is B_{0}-invariant it is sufficient to check that f is annihilated by the adjoint action of μ_{1}H_{1} − μ_{2}H_{2} ∈ 𝔟_{0}. The last statement follows from the fact that a homogeneous B_{0}-invariant function defined on a neighborhood of JB_{0} in J𝔟 is uniquely determined by its degree ∈ ℝ and its constant value on the hypersurface H = B_{0}(JB_{0}). □

Proposition 2.6. Under the assumption of Lemma 2.5 there exists a non constant B_{0}-invariant homogeneous polynomial h on J𝔟 if and only if

(\frac{μ_{i}}{μ_{j}} )^{2} ∈ ℚ for all i, j = 1, …, l .

Under this condition we can, up to homothety of metric Lie algebras, assume that μ_{j}^{2} = p_{j}/q_{j} is a reduced fraction for all j and p_{1} = q_{1} = 1. Then the unique (up to scaling) non constant B_{0}-invariant homogeneous polynomial of lowest degree d is

h(η) = ∏_{j=1}^{l} a_{j}^{(q_{j} ∏_{k≠j} p_{k})/N} , N = gcd{q_{j} ∏_{k≠j} p_{k}|j = 1, …, l} ,

where “gcd” stands for “greatest common divisor”. The degree d of h is

d = \frac{1}{N} ∑_{j=1}^{l}(q_{j} ∏_{k≠j} p_{k}) .

Proof: A real power of f is a non constant polynomial if and only if there exists a λ ∈ ℝ^{∗} such that

∏_{k≠j} μ_{k}^{2} ∈ λℚ for all j = 1, …, l

or equivalently such that for all i, j = 1, …, l we have

\frac{μ_{i}^{2}}{μ_{j}^{2}} = \frac{∏_{k≠j} μ_{k}^{2}}{∏_{k≠i} μ_{k}^{2}} ∈ ℚ .

By scaling the scalar product of the normal J-algebra (𝔲_{0}, ⟨·, ·⟩, J) we can assume that μ_{1} = 1 and hence μ_{2}^{2}, … μ_{l}^{2} ∈ ℚ. Now the remaining statements are immediate. □ Lemma 2.7. The solutions (𝔲_{0}, ⟨·, ·⟩, J) to Problem 1 (p. 14) which admit a direct orthogonal decomposition 𝔲_{0} = 𝔣_{1}+𝔣_{2}+⋯+𝔣_{l} into key algebras and for which 2 ≤ d ≤ 3 are up to scaling listed in the following table, where h(η), η = ∑_{j=1}^{l} a_{j}G_{j}, is a basic polynomial for the (flat) homogeneous Riemannian hypersurface H = B_{0}(JB_{0}) ⊂

H_{1}(h) ⊂ J𝔟 of degree d. \begin{matrix}d & l & (μ_{i}^{2} , i = 1, …, l) & h(η) \\ 2 & 2 & (1,1) & a_{1}a_{2} \\ 3 & 2 & (1,2) & a_{1}^{2}a_{2} \\ 3 & 3 & (1,1,1) & a_{1}a_{2}a_{3} \end{matrix}

Proof: It is straightforward, using Proposition 2.6, to determine the solutions to Problem1 for which 𝔲_{0} = 𝔣_{1} + 𝔣_{2} + ⋯ + 𝔣_{l} and 2 ≤ d ≤ 3. If we normalize μ_{1} = 1, then d = 2 implies l = 2, μ_{2} = 1 and h(η) = a_{1}a_{2} and d = 3 implies either l = 3, μ_{2} = μ_{3} = 1 and h(η) = a_{1}a_{2}a_{3} or l = 2 and μ_{2} ∈ \{\sqrt{2}, 1/\sqrt{2}\}. In the last two cases the corresponding polynomials are h(η) = a_{1}^{2}a_{2} and a_{1}a_{2}^{2} respectively. Up to scaling the scalar product of 𝔲_{0}, it is sufficient to consider the case μ_{2} = \sqrt{2} and h(η) = a_{1}^{2}a_{2}. □ Remark 5: The Riemannian hypersurfaces defined by the two polynomials a_{1}^{2}a_{2} and a_{1}a_{2}^{2} are isometric via the linear transformation a_{1}G_{1} + a_{2}G_{2} ↦ a_{2}G_{1} + a_{1}G_{2}. In particular, the Kählerian tube domains associated to these hypersurfaces are isomorphic.

Now we consider the general case of a normal J-algebra for a Siegel domain of type I. We use the notations and decompositions introduced above, cf. Theorem 2.3; in particular,

𝔲_{0} = 𝔢_{1} + 𝔢_{2} + ⋯ + 𝔢_{l} , 𝔢_{j} = 𝔢(n_{j} + 1, μ_{j}) , j = 1, …, l , n_{l} = 0 .

Consider the decomposition J𝔟 = A_{1,0} + A_{0,1}, where

A_{1,0} := J𝔞 = span{G_{j}|j = 1, …, l} , A_{0,1} := ∑_{j>i} 𝔵_{ij}^{+} .

It defines an 𝔞-invariant decomposition for the homogeneous polynomials of degree d on J𝔟:

∨^{d}(J𝔟)^{∗} = ∑_{p+q=d} A^{p,q} := π ((A^{1,0})^{⊗p} ⊗ (A^{0,1})^{⊗q}) ,

where A^{1,0} ≅ A_{1,0}^{∗} and A^{0,1} ≅ A_{0,1}^{∗} are the subspaces of (J𝔟)^{∗} which annihilate A_{0,1} and A_{1,0} respectively and π : ⊗^{d}(J𝔟)^{∗} → ∨^{d}(J𝔟)^{∗} denotes the natural projection from the tensor product to the symmetric tensor product. We can decompose any homogeneous polynomial h ∈ ∨^{d}(J𝔟)^{∗} into its pure components:

h = ∑_{p+q=d} h^{p,q} , h^{p,q} ∈ A^{p,q} .

Put 𝔞_{0} = 𝔟_{0} ∩ 𝔞 = {A ∈ 𝔞|⟨A, B_{0}⟩ = 0}. Remark that if h is a 𝔟_{0}-invariant homogeneous polynomial, then h and its pure components h^{p,q} are 𝔞_{0}-invariant. Next we will apply these general considerations to the case of degree 2 ≤ d ≤ 3.

To simplify the terminology we will use the following definition.

Definition 2.4. We say that a (pseudo) Kähler Lie algebra (𝔲_{0}, ⟨·, ·⟩, J) for a homogeneous (pseudo) Kähler manifold U, s. Definition 2.2, is of degree d if, up to scaling the metric, U is isometric to the (pseudo) Kählerian tube domain associated to a homogeneous (pseudo) Riemannian hypersurface of degree d.

Theorem 2.8. The quadratic normal J-algebras are (up to scaling) precisely the normal J-algebras of type I (s. Theorem 2.3) of the form

𝔲_{0} = 𝔢_{1} + 𝔣_{2} , μ_{1} = μ_{2} = 1 .

The 𝔟_{0}-invariant homogeneous quadratic polynomial on J𝔟 = J𝔞 + 𝔵_{12}^{+},

J𝔞 = span{G_{1}, G_{2}},

is given by

h(η) = a_{1}a_{2} − \frac{1}{2}⟨X, X⟩ , η = a_{1}G_{1} + a_{2}G_{2} + X , X ∈ 𝔵_{12}^{+} .

All cubic normal J-algebras have rank l = 2 or 3. The cubic normal J-algebras of rank 2 are (up to scaling) precisely the normal J-algebras of type I of the form

𝔲_{0} = 𝔢_{1} + 𝔣_{2} , μ_{1} = 1 , μ_{2} = \frac{1}{\sqrt{2}} ,

h(η) = a_{1}a_{2}^{2} − \frac{1}{\sqrt{2}}a_{2}⟨X, X⟩ .

The cubic normal J-algebras of rank 3 are (up to scaling) precisely the normal Jalgebras of type I of the form

𝔲_{0} = 𝔲_{0}(ψ) = 𝔢_{1} + 𝔢_{2} + 𝔣_{3} , μ_{1} = μ_{2} = μ_{3} = 1 ,

determined by an isometric map ψ := ψ_{123} : 𝔵_{23}^{−} × 𝔵_{12}^{−} ↦ 𝔵_{13}^{−}, which has to be special or of order zero, s. Definition 2.3. The 𝔟_{0}-invariant homogeneous cubic polynomial on J𝔟 = J𝔞 + 𝔵_{23}^{+} + 𝔵_{13}^{+} + 𝔵_{12}^{+} is

h(η) = a_{1}a_{2}a_{3} − \frac{1}{2} ∑_{α=1}^{3} a_{α}⟨X_{βγ}, X_{βγ}⟩ + \frac{1}{\sqrt{2}}⟨ψ(JX_{23}, JX_{12}), JX_{13}⟩ ,

where η = ∑_{α=1}^{3}(a_{α}G_{α} + X_{βγ}), X_{βγ} ∈ 𝔵_{βγ}, β < γ and {α, β, γ} = {1, 2, 3}.

Corollary 2.9. The quadratic normal J-algebras are the normal J-algebras for the Hermitian symmetric spaces SO_{0}(2, 2 + p)/(SO(2) × SO(2 + p)). The cubic normal J-algebras are precisely the normal J-algebras for the special Kähler submanifolds of the Alekseevsky spaces, s. 2.3.

Proof(of the theorem): Let (𝔲_{0}, ⟨·, ·⟩, J) be a quadratic normal J-algebra and h the basic polynomial of the quadratic Riemannian hypersurface H = B_{0}(JB_{0}) ⊂ J𝔟. We have the decomposition

h = h^{2,0} + h^{1,1} + h^{0,2}

into 𝔞_{0}-invariant homogeneous quadratic polynomials. The polynomial h^{2,0} is nonzero because otherwise the canonical metric of H defined by h would be zero on the subspace J𝔞_{0} ⊂ J𝔟_{0} = T_{JB_{0}}H, which is impossible for a Riemannian metric. Hence h^{2,0} defines a 𝔞_{0}-invariant nonzero homogeneous quadratic polynomial on J𝔞. The canonical metric defined by h^{2,0} on a hypersurface in J𝔞 through the point JB_{0} ∈ J𝔞 is Riemannian. By Lemma 2.7 this can only occur if l = 2, i.e. 𝔲_{0} = 𝔢_{1} +𝔣_{2}, μ_{1} = μ_{2} = 1 (up to scaling) and h^{2,0}(η) = a_{1}a_{2} for η = a_{1}G_{1} + a_{2}G_{2}.

In the cubic case we have

h = h^{3,0} + h^{2,1} + h^{1,2} + h^{0,3}

and h^{3,0} is a nonzero 𝔞_{0}-invariant homogeneous cubic polynomial. As above, we conclude in this case that l = 2 or l = 3. If l = 2 we have (up to scaling) μ_{1} = 1 and either μ_{2} = \sqrt{2} and h^{3,0}(η) = a_{1}^{2}a_{2} or μ_{2} = \frac{1}{\sqrt{2}} and h^{3,0}(η) = a_{1}a_{2}^{2} for η = a_{1}G_{1}+a_{2}G_{2}. If l = 3 we have (up to scaling) μ_{1} = μ_{2} = μ_{3} = 1 and h^{3,0}(η) = a_{1}a_{2}a_{3} for η = a_{1}G_{1} + a_{2}G_{2} + a_{3}G_{3}.

Using the decomposition of ∨^{3}(J𝔟)^{∗} introduced above we show next that in the first case the polynomial h^{3,0}(η) = a_{1}^{2}a_{2} cannot be extended to a 𝔟_{0}-invariant homogeneous cubic polynomial on J𝔟 unless 𝔲_{0} = 𝔢_{1}∈ +𝔣_{2} = 𝔣_{1} ⊕ 𝔣_{2} is a direct sum of key algebras.

Assume that h = h^{3,0} + h^{2,1} + h^{1,2} + h^{0,3} is such an extension, denote by π_{p,q} : ∨^{3}(J𝔟)^{∗} → A^{p,q}, p+q = 3, the natural projection and by Y ↦ ad_{Y}^{∗} the representation of 𝔟_{0} on ∨^{3}(J𝔟)^{∗} induced by the adjoint representation of 𝔟_{0} on the ideal J𝔟 ⊂ 𝔲_{0}. Let Y ∈ 𝔵_{12}^{−} ⊂ 𝔟_{0}. Then the equation ad_{Y}^{∗} h = 0 implies the equations

π_{2,1}ad_{Y}^{∗} h = ad_{Y}^{∗} h^{3,0} + π_{2,1}ad_{Y}^{∗} h^{1,2} = 0 ,
π_{0,3}ad_{Y}^{∗} h = π_{0,3}ad_{Y}^{∗} h^{1,2} = 0

for the polynomial

h^{1,2}(η) = c_{1}a_{1}q_{1}(X, X) + c_{2}a_{2}q_{2}(X, X) , η = a_{1}G_{1} + a_{2}G_{2} + X ,

where q_{1} and q_{2} are quadratic forms on 𝔵_{12}^{+} ∋ X and c_{1} and c_{2} are real constants. Since ad_{Y}^{∗} h^{3,0}(η) = −2a_{1}a_{2}⟨JY, X⟩, the first equation is satisfied for all Y ∈ 𝔵_{12}^{−} ≠ 0 if and only if h^{1,2}(η) = −\frac{1}{\sqrt{2}}a_{1}⟨X, X⟩. However, this implies

π_{0,3}ad_{Y}^{∗} h^{1,2}(η) = −\frac{1}{\sqrt{2}}⟨JY, X⟩⟨X, X⟩ ,

so the second equation is not satisfied. This shows that the case μ_{2} = \sqrt{2} is impossible if 𝔵_{12}^{−} ≠ 0.

To conclude the proof in the case l = 2 we have to check that if μ_{2} = 1 (resp. if μ_{2} = \frac{1}{\sqrt{2}}) h^{2,0}(η) = a_{1}a_{2} (resp. h^{3,0}(η) = a_{1}a_{2}^{2}) is extended by h given in the theorem to a homogeneous 𝔟_{0}-invariant quadratic (resp. cubic) polynomial on J𝔟, that the canonical metric of H = B_{0}(JB_{0}) ⊂ H_{1}(h) ⊂ J𝔟 is Riemannian and, more precisely, that that (𝔲_{0}, ⟨·, ·⟩, J) is Kähler Lie algebra for the Kählerian Siegel domain associated to H. This is a special case of Proposition 2.10.

Now we show in the case l = 3 that if h = h^{3,0} + h^{2,1} + h^{1,2} + h^{0,3} is a 𝔟_{0}-invariant polynomial on J𝔟 and h^{3,0}(η) = a_{1}a_{2}a_{3}, then h must be the polynomial given above.

It is easy to check that the only 𝔞_{0}-invariant element of A^{2,1} is zero, the 𝔞_{0}-invariant elements of A^{1,2} are spanned by the three polynomials

a_{α}⟨X_{βγ}, X_{βγ}⟩ ,

and finally, the 𝔞_{0}-invariant elements f of A^{0,3} are given by trilinear functions

t : 𝔵_{23} × 𝔵_{13} × 𝔵_{12} → ℝ , i.e. f(X_{23} + X_{13} + X_{12}) = t(X_{23}, X_{13}, X_{12}) ,

where

X_{βγ} ∈ 𝔵_{βγ} , {α, β, γ} = {1, 2, 3} , β < γ .

This implies

h^{2,1} = 0 , h^{1,2}(η) = ∑_{α=1}^{3} c_{α}a_{α}⟨X_{βγ}, X_{βγ}⟩ , h^{0,3}(η) = t(X_{23}, X_{13}, X_{12}) ,

where c_{α} are real constants, t is a trilinear function as above and

η = ∑_{α=1}^{3}(a_{α}G_{α} + X_{βγ}) , X_{βγ} ∈ 𝔵_{βγ} , {α, β, γ} = {1, 2, 3} , β < γ .

Now we let X ∈ ∑_{i<j} 𝔵_{ij}^{−} and consider the equations

0 = π_{2,1}ad_{X}^{∗}h = ad_{X}^{∗}h^{3,0} + π_{2,1}ad_{X}^{∗}h^{1,2}
0 = π_{1,2}ad_{X}^{∗}h = π_{1,2}ad_{X}^{∗}h^{1,2} + π_{1,2}ad_{X}^{∗}h^{0,3}
0 = π_{0,3}ad_{X}^{∗}h = π_{0,3}ad_{X}^{∗}h^{1,2} + π_{0,3}ad_{X}^{∗}h^{0,3} .

The first equation is satisfied if and only if c_{1} = c_{2} = c_{3} = −\frac{1}{2}. Then the second equation is equivalent to

t(X_{23}, X_{13}, X_{12}) = \frac{1}{\sqrt{2}}⟨ψ(JX_{23}, JX_{12}), JX_{13}⟩ .

Now the third equation is satisfied only if 𝔵_{23}^{−} = 0 or if ψ is a special isometric map. In fact, if e.g. X ∈ 𝔵_{23}^{−} then

[X, ∑_{α=1}^{3}(a_{α}G_{α} + X_{βγ})] = a_{3}JX + ⟨JX, X_{23}⟩G_{2} + \frac{1}{\sqrt{2}}Jψ^{t}(X, JX_{13})

and the third equation reads

0 = \frac{1}{2}⟨JX, X_{23}⟩⟨X_{13}, X_{13}⟩ + \frac{1}{2}⟨ψ(JX_{23}, ψ^{t}(X, JX_{13}), JX_{13}⟩
= \frac{1}{2}⟨JX, X_{23}⟩⟨X_{13}, X_{13}⟩ + \frac{1}{2}⟨ψ^{t}(X, JX_{13}), ψ^{t}(JX_{23}, JX_{13})⟩ .

If 𝔵_{23}^{−} ≠ 0 we can choose X = JX_{23} ≠ 0 and the last equation shows that ψ^{t}(X, ·) : 𝔵_{13}^{−} → 𝔵_{12}^{−} is injective, hence ψ is a special isometric map.

It only remains to check that the scalar product of the normal J-algebra 𝔲_{0} is (up to scaling) induced by the canonical metric of the tube domain associated to the hypersurface H = B_{0}(JB_{0}) ⊂ H_{1}(h) ⊂ J𝔟. This is a straightforward computation, cf. Proposition 2.10. Remark that the pure component h^{0,3} involving the special isometric map ψ plays no role in this calculation, since its Hessian vanishes at the point JB_{0}. □

Now we give examples of normal J-algebras of arbitrary degree d = 2, 3, 4, . . .. For every degree d ≥ 2 we construct a series (𝔲_{0}(p, d − 1), ⟨·, ·⟩, J), p ∈ ℕ_{0}, of normal J-algebras of degree d and rank 2. The subalgebra 𝔟_{0} for these series is up to scaling isomorphic to the Iwasawa algebra of 𝔰𝔬(1, p + 1) with scalar product induced by the Riemannian metric of hyperbolic (p + 1)-space Hℝ^{p+1} = SO_{0}(1, p + 1)/SO(p + 1).

From now on we write 𝔲_{0} instead of (𝔲_{0}, ⟨·, ·⟩, J). The scalar product and complex structure are understood and always denoted by ⟨·, ·⟩ and J respectively.

For every s = 1, 2, … consider the Kählerian Lie algebra

𝔲_{0} = 𝔲_{0}(p, s) = 𝔢(p + 1, 1)∈ +𝔢(1, \frac{1}{\sqrt{s}}) = (𝔣_{1} + 𝔵_{1}) + 𝔣_{2} ,

where the semidirect orthogonal sum of the key algebra 𝔣_{2} with root μ = \frac{1}{\sqrt{s}} and the ideal 𝔣_{1} + 𝔵_{1} is defined by the condition that ad_{𝔣_{2}}|𝔵_{1} has weight decomposition 𝔵_{1} = 𝔵_{12}^{−} + 𝔵_{12}^{+}, p = dim 𝔵_{12}^{−}.

Proposition 2.10. 𝔲_{0} = 𝔲_{0}(p, s) = 𝔟 + J𝔟, 𝔟 = ℝB_{0} ⊕ 𝔟_{0}, s. Theorem 2.3, is a normal J-algebra of degree d = s + 1 with 𝔟_{0}-invariant polynomial

h(η) = a_{1}(μa_{2})^{s} − \frac{1}{2}(μa_{2})^{s−1}⟨X, X⟩ , μ = \frac{1}{\sqrt{s}} ,

where η = a_{1}G_{1} + a_{2}G_{2} + X, X ∈ 𝔵_{12}^{+}. Its subalgebra 𝔟_{0} is isomorphic to the Iwasawa algebra of 𝔰𝔬(1, p+1). In particular, the Iwasawa subgroup of SO_{0}(1, p+1) acts simply transitively on a Riemannian hypersurface H of degree d and of constant negative curvature. The Kählerian Siegel domain U associated to the Riemannian hypersurface H = B_{0}(JB_{0}) ⊂ J𝔟 of degree d is symmetric only if H is quadratic and in this case U is isometric to the Hermitian symmetric space SO_{0}(2, p + 2)/(SO(2) × SO(p + 2)). If H is cubic (cf. Theorem 2.8), then U is special Kähler; in fact , it is the special Kähler submanifold of the Alekseevvsky space T (p), s. 2.3.

Proof: First we observe that 𝔲_{0} = 𝔟 + J𝔟, where 𝔟 = span{H_{1}, H_{2}} + 𝔵_{12}^{−} has centre

B_{0} = H_{1} + \frac{1}{μ}H_{2} if p ≠ 0. Remark that the metric Lie algebra 𝔟_{0} ⊂ 𝔟 = ℝB_{0} ⊕^{⊥} 𝔟_{0} is isomorphic to the Iwasawa algebra of 𝔰𝔬(1, p + 1) with scalar product induced (up to scaling) by the hyperbolic metric of Hℝ^{p+1} = SO_{0}(1, p + 1)/SO(p + 1).

Consider now the faithful linear representation ρ = ad| : 𝔟 → 𝔤𝔩(J𝔟). We have ρ(B_{0}) = Id and ρ(B)JB_{0} = JB for all B ∈ 𝔟. The corresponding representation R : B → GL(J𝔟) of the simply connected Lie group B = ℝ^{+} × B_{0} with Lie algebra 𝔟 = ℝB_{0} + 𝔟_{0} has the open orbit R(B)JB_{0} and H = B_{0}(JB_{0}) = R(B_{0})JB_{0} is a codimension one orbit of the simply connected Lie group B_{0} with Lie algebra 𝔟_{0}.

We will prove that H is defined by a homogeneous polynomial h of degree d and that the simply connected metric Lie group U_{0} with metric Lie algebra 𝔲_{0} is up to scaling isomorphic to the Kählerian Siegel domain U associated to the hypersurface H.

For η = a_{1}G_{1} + a_{2}G_{2} + X ∈ J𝔟, X ∈ 𝔵_{12}^{+}, we define

h(η) = a_{1}(μa_{2})^{s} − \frac{1}{2}(μa_{2})^{s−1}⟨X, X⟩ ,

where ⟨·, ·⟩ denotes the Euclidean scalar product of the metric Lie algebra 𝔲_{0}. It is an easy computation to check that h is invariant under the linear action of 𝔟_{0} on ∨^{d}(J𝔟)^{∗} induced by ρ|𝔟_{0} : 𝔟_{0} → 𝔤𝔩(J𝔟). Hence H = R(B_{0})JB_{0} is contained in H_{1}(h). Let us now consider the orbit map

φ : B_{0} → H , b ↦ R(b)JB_{0} .

We check that the Euclidean scalar product ⟨·, ·⟩ on 𝔟_{0} = T_{e}B_{0} is

⟨·, ·⟩ = \frac{1}{d}(φ^{∗}g)_{e} ,  (5)

where g = −\frac{1}{d}∂^{2}h is the canonical metric of the hypersurface H defined by the basic polynomial h. We compute

(φ^{∗}g)_{e}(H_{1} − μH_{2}, H_{1} − μH_{2}) = g_{JB_{0}}(G_{1} − μG_{2}, G_{1} − μG_{2})

= g(G_{1}, G_{1}) − 2μg(G_{1}, G_{2}) + μ^{2}g(G_{2}, G_{2})

= 0 + \frac{1}{d}(2μs(μ\frac{1}{μ})^{s−1}μ − μ^{2}s(s − 1)(μ\frac{1}{μ})^{s−2}μ^{2})

= \frac{1}{d}(2μ^{2}s − μ^{4}s(s − 1)) = \frac{1}{d}(2 − \frac{s − 1}{s})

= \frac{1}{d}(1 + μ^{2}) = \frac{1}{d}⟨H_{1} − μH_{2}, H_{1} − μH_{2}⟩ and

(φ^{∗}g)_{e}(X_{−}, X_{−}) = g_{JB_{0}}(JX_{−}, JX_{−})

= \frac{1}{d}(μ\frac{1}{μ})^{s−1}⟨JX_{−}, JX_{−}⟩ = \frac{1}{d}⟨X_{−}, X_{−}⟩ ,

where X_{−} ∈ 𝔵_{12}^{−}. This proves equation (5).

Now we extend the linear representation ρ : 𝔟 → 𝔤𝔩(J𝔟) ⊂ 𝔤𝔩((J𝔟)⊗ℂ) to an affine representation ρ : 𝔲_{0} → 𝔞𝔣𝔣((J𝔟)⊗ℂ) of the real Lie algebra 𝔲_{0} on the complex vector space (J𝔟) ⊗ ℂ = J𝔟 + iJ𝔟. For B ∈ 𝔟 we define

ρ(JB) = JB ∈ J𝔟 ⊂ (J𝔟) ⊗ ℂ ⊂ (J𝔟) ⊗ ℂ + 𝔤𝔩((J𝔟) ⊗ ℂ) = 𝔞𝔣𝔣((J𝔟) ⊗ ℂ) .

The corresponding extension of R : B → GL(J𝔟) ⊂ GL((J𝔟) ⊗ ℂ) will be also denoted by the same letter:

R : U_{0} → Aff((J𝔟) ⊗ ℂ) .

The tube domain U = J𝔟 + iV, V = ℝ^{+}H ⊂ J𝔟, is precisely the orbit of the point iJB_{0} ∈ iV ⊂ U under the affine representation R of the group U_{0} on (J𝔟) ⊗ ℂ, that is: U = R(U_{0})(iJB_{0}).

Now we prove that the scalar product ⟨·, ·⟩ of the metric Lie algebra 𝔲_{0} = T_{e}U_{0} is

⟨·, ·⟩ = \frac{1}{d}(ϕ^{∗}g^{c})_{e} ,

where g^{c} is the canonical metric of the tube domain U and

ϕ : U_{0} → U , u ↦ R(u)(iJB_{0})

is the orbit map.

First we remark that that

(dϕ)_{e} : 𝔲_{0} = 𝔟 + J𝔟 → T_{iJB_{0}}U = J𝔟 + iJ𝔟

maps the orthogonal subspaces 𝔟 and J𝔟 of 𝔲_{0} onto the orthogonal subspaces iJ𝔟 and J𝔟 of T_{iJB_{0}}U respectively:

(dϕ)_{e}|𝔟 : 𝔟 → iJ𝔟 , B ↦ iJB
(dϕ)_{e}|J𝔟 = Id : J𝔟 → J𝔟 .

Moreover, J and multiplication by −i are orthogonal endomorphisms of 𝔲_{0} and T_{iJB_{0}}U respectively which make the following diagram commutative:

𝔲_{0} ⟶^{(dϕ)_{e}} T_{iJB_{0}}U
↓ J  ↓ −i
𝔲_{0} ⟶^{(dϕ)_{e}} T_{iJB_{0}}U

Now since J interchanges the subspaces 𝔟 and J𝔟 of 𝔲_{0} and −i interchanges the subspaces iJ𝔟 and J𝔟 of T_{iJB_{0}}U, it is sufficient to check that the map (dϕ)_{e}|𝔟 : 𝔟 → iJ𝔟 ⊂ T_{iJB_{0}}U becomes a linear isometry after scaling it by \sqrt{d}. We have checked this already on 𝔟_{0} ⊂ 𝔟 = ℝB_{0} + 𝔟_{0}, s. (5). The vector (dϕ)_{e}B_{0} = iJB_{0} is precisely the radial vector at the point iJB_{0} ∈ iV, so it has unit length with respect to the canonical metric g^{c}, thanks to Proposition 1.1. On the other hand, ⟨B_{0}, B_{0}⟩ = 1 + \frac{1}{μ^{2}} = 1 + s = d. This proves that (ϕ^{∗}g^{c})_{e} = \frac{1}{d}⟨·, ·⟩. All remaining statements are easily checked. □

2.2. Classification of transitive reductive group actions on pseudo Riemannian hypersurfaces. Now we explain how it is possible to use results from invariant theory for reductive algebraic groups to classify pseudo Riemannian hypersurfaces of degree d admitting a transitive reductive algebraic group of linear transformations. We will give a complete classification for (quadratic and) cubic hypersurfaces, which is the case relevant to homogeneous special geometry.

Let G be a real algebraic reductive group and V an (algebraic) G-module. Assume that the connected component G_{0} acts transitively on the hypersurface H ⊂ H_{1}(h) ⊂ V, where h is the basic polynomial, s. Def. 1.1. Then G preserves h, i.e. h is a Ginvariant. In fact, any G-invariant is of the form ch^{k}, c ∈ ℂ, k ∈ ℕ_{0}.

Consider now the one dimensional extension G^{1} = ℝ^{∗} × G and on V the canonical structure of G^{1}-module, where ℝ^{∗} acts by the standard scalar multiplication on V. The G^{1}-module V has an open orbit, it is a prehomogeneous vector space (P.V.) in the terminology of [S-K].

In [S-K] T. Kimura and M. Sato have classified all irreducible P.V.s for complex algebraic reductive groups. If we consider the (G^{1})ℂ-module V ℂ and assume that it is irreducible, then it must appear in the classification [S-K]. The (G^{1})ℂ-module V ℂ is irreducible if and only if the Gℂ-module V ℂ is irreducible. We will assume first that this condition is satisfied and treat the reducible case later. Without restriction of generality we also assume that Gℂ acts almost faithfully, i.e. with only discrete kernel. Then V ℂ is an irreducible almost faithful P.V. of (G^{1})ℂ and by a theorem of Cartan (Thm. 1 of [S-K]) G must be semisimple. Remark that since h is a Ginvariant, it must be a relative G^{1}- and (G^{1})ℂ-invariant, i.e. h is preserved up to scaling. A P.V. was called regular in [S-K] if there exists a relative invariant with not identically vanishing Hessian determinant. Remark that a one dimensional P.V. is always regular.

Lemma 2.11. If H = G_{0}v ⊂ H_{1}(h) ⊂ V is a pseudo Riemanian hypersurface in the G-module V ∋ v, then V ℂ is a regular P.V. of (G^{1})ℂ.

Proof: If ∂^{2}h|T_{v}H is nondegenerate, then ∂^{2}h|_{v} is nondegenerate, since h is a homogeneous polynomial of degree d ≥ 2. □

An analogeous discussion applies to complex Riemannian hypersurfaces H ⊂ V ≅ ℂ^{n} admitting a transitive irreducible linear action of a complex algebraic reductive group G. We can easily deduce the classification of such group actions from the classification of P.V.s in [S-K]. We will consider two representations R : G → GL(V) and R′ : G′ → GL(V′) as equivalent if there is an isomorphism GL(V) →^{∼} GL(V′) mapping R(G) onto R(G′).

Theorem 2.12. The following list gives, up to equivalence, all irreducible G-modules V of a connected complex algebraic reductive group G which induce a transitive action on a complex Riemannian cubic hypersurface H ⊂ H_{1}(h) ⊂ V. The quadruples (V^{n}, G, h, K) below contain the G-module V of (complex) dimension n, the basic cubic polynomial h (unique up to multiplicative constant) and the isotropy group K of a point v ∈ H as abstract group.

1) (V^{9} = U⊗ℂ^{3}, H×SL(3, ℂ), det, H), where U = ℂ^{3} is the standard 3-dimensional module of H = SL(3, ℂ), H = SO(3, ℂ) or H = {e} and ℂ^{3} is the standard SL(3, ℂ)- module.

2) (V^{6} = ∨^{2}ℂ^{3}, SL(3, ℂ), det, SO(3, ℂ)),

3) (V^{15} = ∧^{2}ℂ^{6}, SL(6, ℂ), P ff, Sp(3, ℂ)), where P ff is the Pfaffian of a skew symmetric 6 × 6-matrix:

P ff(A) = ∑_{σ∈S_{6}} sgn(σ)a_{σ(1)σ(2)}a_{σ(3)σ(4)}a_{σ(5)σ(6)} , A = (a_{ij}) , i, j = 1, …, 6 .

(Here we use the notation Sp(n, ℂ) for the symplectic group of ℂ^{2n} as in [S-K].)

4) (V^{27} = Herm_{3}(𝕆) ⊗ ℂ = Herm_{3}(C) ⊗ ℂ, E_{6}, det, F_{4}), where Herm_{3}(𝕆) (resp. Herm_{3}(C)) denotes the Hermitian 3 × 3 matrices over the octonions (resp. over the real split Cayley algebra C), which is an irreducible module of the real form E_{6}^{(−26)} (resp. E_{6}^{(6)}) of E_{6}.

Now we classify all real G-modules V of real algebraic reductive groups G such that the connected component G_{0} acts transitively on a cubic pseudo Riemannian hypersurface H ⊂ H_{1}(h) ⊂ V under the condition that the Gℂ-module V ℂ is irreducible. If the last condition is satisfied, we shall say that the module (V, G) is totally irreducible. It is clear that under these conditions the complexified module (V ℂ, Gℂ) must be one of the list in Theorem 2.12. In other words, we must study the real forms of the complex modules classified in Theorem 2.12.

Let now V be a complex G-module of a connected complex algebraic reductive group G.

Definition 2.5. A real structure for the module (V, G) is given by

1) a (real algebraic) antiholomorphic automorphism τ : G → G of the complex algebraic group G and

2) a complex antilinear involution τ : V → V of the complex vector space V (denoted by the same letter) such that

τ(av) = τ(a)τ(v) , a ∈ G , v ∈ V .

The pair (V^{τ}, G^{τ}) is called a real form for (V, G), where the superscript τ stands for fixed point set of τ.

Now we assume that H = Gv ⊂ H_{1}(h) is a complex Riemannian hypersurface for some v ∈ V. Remark that then Gv is a complex Riemannian hypersurface for all v ∈ V − S, where the singular set S is Zariski closed. Given a real structure τ for (V, G) we may assume that h is real, i.e. \overline{h(v)} = h(τv) for all v ∈ V. In fact, multiplying h by a suitable complex constant we may assume that h^{τ} := Re h|_{V^{τ}} ≠ 0. Then the complex extension hℂ^{τ} is a G-invariant polynomial on V of the same degree as h and hence hℂ^{τ} = ch for some c ∈ ℂ^{∗}. Finally, we can assume that H = Gv for some v ∈ V^{τ}. Indeed, the singular set S is a Zariski closed proper subset of V and hence V^{τ} ⊄ S. Under these assumptions we have the following lemma.

Lemma 2.13. The real structure τ preserves the isotropy group K of G at v and the real form K^{τ} of K is precisely the isotropy group of v in G^{τ}. Moreover, the orbit (G^{τ})_{0}v ≅ (G^{τ})_{0}/(K^{τ} ∩ (G^{τ})_{0}) of the connected component (G^{τ})_{0} of G^{τ} coincides with the connected component (H^{τ})_{0} of v in the fixed point set H^{τ}, H = Gv. Finally, (H^{τ})_{0} is pseudo Riemannian if and only if H is complex Riemannian.

Proof: It is clear that K^{τ} = K ∩ G^{τ} is the isotropy group of v in G^{τ}; in particular G^{τ}v ≅ G^{τ}/K^{τ} and (G^{τ})_{0}v ≅ (G^{τ})_{0}/(K^{τ} ∩ (G^{τ})_{0}). From v ∈ V^{τ} it follows that τK = K, τ(Gv) = Gv and G^{τ}v ⊂ (Gv)^{τ}, so K^{τ} is a real form of K and

dimℂ Gv ≥ dimℝ(Gv)^{τ} ≥ dimℝ G^{τ}v
= dimℝ G^{τ}/K^{τ} = dimℂ G/K = dimℂ Gv .

This implies dim(Gv)^{τ} = dim G^{τ}v and (G^{τ})_{0}v = ((Gv)^{τ})_{0}. The last statement of the lemma is trivial since we are assuming that the basic polynomial h is real. □

Theorem 2.14. The totally irreducible modules of real algebraic reductive groups inducing a transitive action of the group’s identity component on a cubic pseudo Riemannian hypersurface are (up to equivalence) the real forms (V^{τ}, G^{τ}) of the complex modules (V, G) classified in Theorem 2.12. We list the triples (V^{τ}, G^{τ}, K^{τ}), where K^{τ} is the isotropy group of v ∈ V^{τ}; H = Gv ⊂ V the underlying complex Riemannian hypersurface. The corresponding homogeneous pseudo Riemannian cubic hypersurfaces are all locally symmetric. (The real cubic hypersurfaces with positively or negatively defined canonical metric correspond to the triples (V^{τ}, G^{τ}, K^{τ}) with compact isotropy group K^{τ}.) 1)(U^{0} ⊗ ℝ^{3}, H^{0} × SL(3, ℝ), H^{0}), where U^{0} = ℝ^{3} is the standard 3-dimensional module of H^{0} = SL(3, ℝ), SO(1, 2), SO(3) or {e}. 2)(∨^{2}ℝ^{3}, SL(3, ℝ), SO(3)),

3) (∧^{2}ℝ^{6}, SL(6, ℝ), Sp(3, ℝ)), ((∧ℂ^{2}ℍ^{3})^{τ}, SL(3, ℍ), Sp(3)), where the real structure τ on ∧ℂ^{2}ℍ^{3} ⊂ ℍ^{3} ⊗ℂ ℍ^{3} is the square of the quaternionic structure on the complex vector space ℍ^{3}.

4) (Herm_{3}(𝕆), E_{6}^{(−26)}, F_{4}^{(−52)}), (Herm_{3}(C), E_{6}^{(6)}, F_{4}^{(4)}), for the notation s. Theorem 2.12, 4).

Proof: The fact that the above list gives all real forms for the complex modules of Theorem 2.12 can be checked using Tits’ tables [Tt]. □

Now we study reducible G-modules of connected complex algebraic reductive groups G inducing a transitive G-action on a nondegenerate hypersurface. We use the following theorem as lemma.

Lemma 2.15. [S-K] Let (V, G) be a prehomogeous vector space of a connected complex algebraic reductive group G. Then the following conditions are equivalent.

(i) (V, G) is regular. (ii) The generic isotropy group G_{v}, v ∈ V, is reductive. (iii) The singular set S = V − Gv is a hypersurface.

Theorem 2.16. Let (V, G) be a faithful module of a simply connected and complex algebraic reductive group G acting transitively on a nondegenerate cubic hypersurface H ⊂ V with basic polynomial h. Consider its canonical extension (V, G^{1}) to a regular prehomogeneous vector space V of the group G^{1} = ℂ^{∗} × G. Then (V, G^{1}) is the sum of at most 3 irreducible regular P.V.s. Moreover, only the three following possibilities can occur.

(1) The module (V, G^{1}) and the basic polynomial h are irreducible and G is semisimple.

(2) The G^{1}-module V is the direct sum V = V_{1} ⊕ V_{2} of two irreducible P.V.s and dim V_{1} = 1. The polynomial h is the product h = lq of a linear function l on V_{1} and a quadratic G-invariant q on V_{2}. Finally, G ≅ ℂ^{∗} × G′, where G′ is semisimple or trivial.

(3) The G^{1}-module V is the direct sum of three one-dimensional P.V.s and G ≅ ℂ^{∗} × ℂ^{∗}.

Proof: First we remark that the G^{1}-module V is the direct sum V = ⊕_{j=1}^{r}V_{j} of irreducible submodules, because G^{1} is a reductive algebraic group. Moreover, the irreducible summands V_{j} are again prehomogeneous vector spaces.

From the assumption that G acts transitively on a nondegenerate hypersurface it follows that (V, G^{1}) is a regular P.V. and by Lemma 2.15 the generic isotropy group G_{v} is reductive. This implies that the generic isotropy group for the irreducible summands V_{j} is also reductive and hence the V_{j} are regular P.V.s. In particular, each of them admits a non-constant relative invariant h_{j}.

We have already remarked that (V, G^{1}) being irreducible and faithful implies the semisimplicity of G. Now we prove that if (V, G^{1}) is irreducible, then the basic polynomial h is also irreducible. In fact, assume that h = fg is the product of two non-constant polynomials. Then f and g are relative G^{1}-invariants and one of them must be linear. However, a module V with a linear relative invariant l splits as V = V_{1} ⊕ V_{2}, where dim V_{1} = 1 and V_{2} = ker l.

Any simply connected complex algebraic reductive group G is the direct product G = Z × G′ of its centre Z, which is an algebraic torus, and a semisimple group G′.

If the G^{1}-module V = ⊕_{j=1}^{r}V_{j} is the sum of r irreducible submodules V_{j}, then it has r algebraically independent relative G^{1}-invariants h_{1}, …, h_{r}, which are necessarily G′-invariant. Remark that at a generic point v ∈ V the differentials dh_{1}, …, dh_{r} are linearly independent. Since G acts transitively on a hypersurface in V, the group G′ has an orbit of codimension ≤ dim Z + 1. This implies r ≤ dim Z + 1. On the other hand, since ℂ^{∗} × Z acts faithfully on V, we have that 1 + dim Z ≤ r, hence

r = dim Z + 1 .

Now we show that r ≤ 3 and hence dim Z = r − 1 ≤ 2. A homogeneous polynomial f = f_{1} ⋯ f_{r} on V which is a product of homogeneous polynomials f_{j} of degree d_{j} on V_{j} will be called a monomial of degree (d_{1}, …, d_{r}), with respect to the G-invariant decomposition V = ⊕_{j=1}^{r}V_{j}. Any homogeneous polynomial f on V of degree d can be decomposed into monomials f_{δ} of degree δ = (d_{1}, …, d_{r}) ∈ ℕ_{0}^{r}, |δ| = d_{1}+⋯+d_{r} = d:

f = ∑_{|δ|=d} f_{δ} .

If f is G-invariant, then the f_{δ} are G-invariant. It follows that the basic cubic polynomial h is a monomial h = h_{1} ⋯ h_{r} and the degrees d_{j} ≥ 0 of the homogeneous polynomials h_{j} must add up to three: d_{1} + ⋯ + d_{r} = 3. Now we use the fact that h has nondegenerate Hessian form ∂^{2}h to conclude that d_{j} ≥ 1 for all j = 1, …, r. This implies r ≤ 3, as claimed above.

If r = 1, V and hence h is irreducible. If r = 2, h = h_{1}h_{2}, where we can assume that h_{1} = l is a linear polynomial on V_{1} and h_{2} = q is a quadratic polynomial on V_{2}. From the irreducibility of V_{2} (and nondegeneracy of ∂^{2}h) it follows that either q is irreducible (and a nondegenerate quadratic form) or it is the square of a linear polynomial and dim V_{2} = 1. The second case implies G′ = {e} by the faithfullness of the G-action. Finally if r = 3, h = h_{1}h_{2}h_{3} is the product of three relative G^{1}-invariant polynomials h_{j} on V_{j}. In particular, dim V_{j} = 1 and G′ = {e}. □.

Theorem 2.16 reduces the classification of homogeneous pseudo Riemannian cubic hypersurfaces of reductive real algebraic groups G to three cases. The first case is that of totally irreducible G-modules and was treated in Theorem 2.14. The third case corresponds to the real forms of the flat complex Riemannian cubic hypersurface in ℂ^{3} with basic polynomial h(z^{1}, z^{2}, z^{3}) = z^{1}z^{2}z^{3} and transitive action of the algebraic torus ℂ^{∗} × ℂ^{∗}. The second case reduces to the classification of homogeneous pseudo Riemannian quadratic hypersurfaces, which we give now.

Every homogeneous quadratic polynomial q on 𝕂^{n} such that ∂^{2}q is nondegenerate is of the form

q(X) = ⟨X, X⟩ , X ∈ 𝕂^{n} ,

for some nondegenerate symmetric 𝕂-bilinear form ⟨·, ·⟩ on 𝕂^{n}. If 𝕂 = ℂ, we call H = H_{1}(q) a complex sphere. If 𝕂 = ℝ, the connected components H of H_{1}(q) are called pseudo spheres. The canonical metric g of H at X_{0} ∈ H is precisely

g_{X_{0}}(X, X) = −⟨X, X⟩ , X ∈ T_{X_{0}}H = {X ∈ 𝕂^{n}|⟨X_{0}, X⟩ = 0} .

In the real case this shows that (H, g) has signature (l, k−1) if ⟨·, ·⟩ has signature (k, l). In particular, the canonical metric of the hyperboloid H_{1}(q) ⊂ ℝ^{1,n−1} is positively defined and that of the sphere H_{1}(q) ⊂ ℝ^{n,0} is negatively defined; where q(X) = x_{1}^{2} + ⋯ x_{k}^{2} − x_{k+1}^{2} − ⋯ − x_{n}^{2} for ℝ^{k,l}, k + l = n.

By Witt’s Theorem, SO_{0}(k, l) acts transitively on the (connected) pseudo spheres H ⊂ ℝ^{k,l}. The identity component of the isotropy group of SO_{0}(k, l) at (1, 0, …, 0) is SO_{0}(k − 1, l). Moreover, the canonical metric defines on H the structure of pseudo Riemannian symmetric space. The symmetry σ_{X_{0}} : H → H at X_{0} ∈ H is given by:

σ_{X_{0}}(X) = − (X − \frac{⟨X, X_{0}⟩}{⟨X_{0}, X_{0}⟩}X_{0}) + \frac{⟨X, X_{0}⟩}{⟨X_{0}, X_{0}⟩}X_{0} .

We sum up our discussion.

Proposition 2.17. Any pseudo Riemannian quadratic hypersurface (always with the canonical metric) is a pseudo sphere H ⊂ ℝ^{k,l} and, in particular, a pseudo Riemannian symmetric space. Its full connected group of linear automorphisms is SO_{0}(k, l), which acts irreducibly on ℝ^{k,l}.

To round up our discussion, we give the classification of transitive linear algebraic actions of reductive groups on complex and on pseudo Riemannian quadratic hypersurfaces. As before, the complex case can be easily extracted from [S-K]:

Theorem 2.18. Any G-module V of a complex algebraic reductive group which induces a transitive action on a nondegenerate quadratic hypersurface is (up to equivalence) one in the following list. All of them are irreducible and hence their basic quadratic polynomial q is the (up to scaling) unique G-invariant quadratic form on V. The corresponding quadratic hypersurfaces are complex spheres. The last entry K in the triples (V^{n}, G, K) is, as before, the generic isotropy group as abstract group and n = dim V.

1) (V^{4} = U × ℂ^{2}, H × SL(2, ℂ), H), U = ℂ^{2}, H = SL(2, ℂ) or = {e},

2) (V^{4} = ∨^{2}ℂ^{2}, SL(2, ℂ), SO(2, ℂ)),

3) (V^{4n} = ℂ^{2n} ⊗ ℂ^{2}, Sp(n, ℂ) × SL(2, ℂ), Sp(n − 1, ℂ) × Sp(1, ℂ)), n ≥ 2. In this case the basic polynomial q can be described as follows. Consider ℂ^{2n} ⊗ ℂ^{2} as vector space of complex (2n) × 2-matrices A = (a_{ij}), then

q(A) = P ff(A^{t}JA) , J = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) ,

is the Pfaffian of the skew symmetric matrix A^{t}JA.

4) (V^{n} = ℂ^{n}, SO(n, ℂ), SO(n − 1, ℂ)),

5) (V^{8} = spinor module, Spin(7, ℂ), G_{2}), 6) (V^{16} = spinor module, Spin(9, ℂ), Spin(7, ℂ)),

7) (V^{7}with highest weight Λ_{2}, G_{2}, SL(3, ℂ)).

Theorem 2.19. The real algebraic G-modules V of reductive groups G with transitive action of G_{0} on a quadratic pseudo Riemannian hypersurface H ⊂ V are obtained from the real forms of the complex modules in Theorem 2.18 and are listed below. The hypersurface H ⊂ H_{1}(q) ⊂ (V, q) is a pseudo sphere in the pseudo Euclidean vector space (V, q), where q is a real basic G-invariant quadratic polynomial of the irreducible G-module V.

1) (ℝ^{2} × ℝ^{2}, H^{0} × SL(2, ℝ)), where H^{0} = SL(2, ℝ) or = {e}, (ℝ^{4}, SO(4)), cf. 4),

2) (∨^{2}ℝ^{2}, SL(2, ℝ)),

3) (ℝ^{2n} ⊗ ℝ^{2}, Sp(n, ℝ) × SL(2, ℝ)), (ℍ^{n}, Sp(n − l, l) × Sp(1)), 0 ≤ l ≤ n ≥ 2,

4) (ℝ^{k,l}, SO(k, l)), k + l = n,

5) spinor module of Spin(7, 0) and Spin(3, 4) and semi spinor modules of Spin(0, 7) and Spin(4, 3),

6) spinor module of Spin(8, 1), Spin(4, 5) and Spin(0, 9) and semi spinor modules of Spin(1, 8), Spin(5, 4) and Spin(9, 0),

7) (Im 𝕆, G_{2}^{(−14)} = Aut 𝕆), (Im C, G_{2}^{(2)} = Aut C), where Im 𝕆 (resp. Im C) denotes the imaginary part and Aut 𝕆 (resp. Aut C) the full irreducible automorphism group of the octonions (resp. of the real split Cayley algebra).

Proof: The real forms are obtained again using tables, s. [Tt] and [L-M]. □

2.3. Appendix: Special Kähler submanifolds of Alekseevsky’s quaternionic Kähler manifolds and their pseudo Riemannian analogues. First of all we explain how one can obtain pseudo Kählerian versions of the cubic (and also of the quadratic) normal J-algebras classified in Theorem 2.8. Consider e.g. the cubic normal J-algebras (𝔲_{0}(p), ⟨·, ·⟩, J) of rank 2. If p ≠ 0, we can define a new pseudo Kähler Lie algebra (𝔲_{0}(p)′, ⟨·, ·⟩′, J) changing the scalar product on the subspace 𝔵_{1} ⊂ 𝔲_{0}(p) only by a sign. The formulas defining the Lie bracket on 𝔲_{0}(p)′ are the same as for 𝔲_{0}(p) only the scalar product occuring in these formulas is substituted by ⟨·, ·⟩′.

Recall (Theorem 2.3) that to every isometric map ψ : 𝔵_{23}^{−} × 𝔵_{12}^{−} → 𝔵_{13}^{−} of Euclidean vector spaces 𝔵_{23}^{−}, 𝔵_{12}^{−} and 𝔵_{13}^{−} we can associate a normal J-algebra 𝔲_{0}(ψ) of type I. If moreover 𝔵_{23}^{−} = 0 or if dim 𝔵_{12}^{−} = dim 𝔵_{13}^{−}, then 𝔲_{0}(ψ) is a cubic normal J-algebra, s. Theorem 2.8. Similarly, to every isometric map ψ : 𝔵_{23}^{−}×𝔵_{12}^{−} → 𝔵_{13}^{−} of pseudo Euclidean vector spaces 𝔵_{23}^{−}, 𝔵_{12}^{−} and 𝔵_{13}^{−} we can associate a pseudo Kählerian Lie algebra 𝔲_{0}(ψ) and we have the following proposition.

Proposition 2.20. The pseudo Kähler Lie algebra (𝔲_{0}(p)′, ⟨·, ·⟩′, J) is cubic. Let ψ : 𝔵_{23}^{−} × 𝔵_{12}^{−} → 𝔵_{13}^{−} be an isometric map of pseudo Euclidean vector spaces 𝔵_{23}^{−}, 𝔵_{12}^{−} and 𝔵_{13}^{−} and assume that ψ is special or of order 0, s. Definition 2.3. Then the pseudo Kählerian Lie algebra (𝔲_{0}(ψ), ⟨·, ·⟩, J) associated to ψ is cubic. In particular, (𝔲_{0}(p)′, ⟨·, ·⟩′, J) and (𝔲_{0}(ψ), ⟨·, ·⟩, J) are pseudo Kähler Lie algebras for special pseudo Kähler tube domains.

Proof: The proof is analogeous to the proof of Theorem 2.8. □

Next we recall some basic facts about Alekssevsky’s quaternionic Kähler manifolds, s. [A], [C] and [A-C] for details.

Definition 2.6. An Alekseevsky space is a quaternionic Kähler manifold M admitting a simply transitive (non-Abelian) splittable solvable group L of isometries.

We can present M as metric Lie group (L, g) and consider its metric Lie algebra (𝔩, ⟨·, ·⟩). The quaternionic Kähler structure of M induces a quaternionic structure 𝔮 = span{J_{1}, J_{2}, J_{3}} on the Euclidean vector space (𝔩, ⟨·, ·⟩). The triple (𝔩, ⟨·, ·⟩, 𝔮) associated to the Alekseevsky space M is called its Alekseevskian Lie algebra. According to [A], [dW-VP] and [C] there are (up to symmetric spaces) 3 series of Alekseevsky spaces: T , W and V-spaces. Their Alekseevskian Lie algebras (𝔩, ⟨·, ·⟩, 𝔮) are constructed as quaternionic Kähler extensions of normal J-algebras 𝔲.

More precisely, given a normal J-algebra of the form 𝔲 = 𝔣_{0} ⊕^{⊥} 𝔲_{0} which admits a so called Q-representation T : 𝔲 → End(𝔲̃) we can canonically define the structure (𝔩, ⟨·, ·⟩, 𝔮) of Alekseevskian Lie algebra on the vector space 𝔩 = 𝔲 + 𝔲̃ such that (𝔲, ⟨·, ·⟩|𝔲, J = J_{1}|𝔲) is a Kählerian subalgebra, 𝔲̃ = J_{2}𝔲, [𝔲̃, 𝔲̃] ⊂ 𝔲 and [𝔲, 𝔲̃] ⊂ 𝔲̃ is given by T. We remark that the definition of Q-representation and this construction can be naturally generalized to pseudo Kählerian Lie algebras; the homogeneous spaces associated to (𝔩, ⟨·, ·⟩, 𝔮) being pseudo quaternionic Kähler manifolds.

The 3 series of T , W and V-spaces are defined by 3 series of normal J-algebras (𝔲_{0}, ⟨·, ·⟩, J) and corresponding Q-representations. By direct comparison of these series of normal J-algebras 𝔲_{0} with the cubic normal J-algebras (s. Theorem 2.8) one can easily check the following facts.

The Alekseevsky spaces V(ψ) are defined by the Q-representation of the normal J-algebras 𝔲 = 𝔣_{0}⊕𝔲_{0}(ψ) for which 𝔲_{0}(ψ) is a cubic normal J-algebra of rank 3 defined by a non zero special isometric map ψ.

For the spaces T (p), p = 0, 1, 2, …,

𝔲_{0}(p) = 𝔢(p + 1, 1)∈ +𝔢(1, \frac{1}{\sqrt{2}})

are the cubic normal J-algebras of rank 2, cf. Proposition 2.10.

Finally, for the spaces W(p, q) ≅ W(q, p), p, q = 0, 1, 2, …,

𝔲_{0}(p, q) = 𝔢(p + q + 1, 1)∈ +𝔢(1, 1)∈ +𝔢(1, 1)

are the cubic normal J-algebras of rank 3 associated to isometric maps of order zero. The preceding facts motivate the following definition.

Definition 2.7. Let M = (L, g) be an Alekseevsky space with Alekseevskian Lie algebra (𝔩, ⟨·, ·⟩, 𝔮), 𝔩 = 𝔲+𝔲̃, 𝔲 = 𝔣_{0}⊕𝔲_{0} the decompositions introduced above and (U_{0}, g, J̃) the Kähler Lie group associated to the normal J-algebra (𝔲_{0}, ⟨·, ·⟩, J = J_{1}|𝔲_{0}). It is naturally identified with a (totally geodesic) Kähler submanifold of M, which is called the special Kähler submanifold of M. The normal J-algebra (𝔲_{0}, ⟨·, ·⟩, J = J_{1}|𝔲_{0}) is called the special Kähler subalgebra of the Alekseevskian Lie algebra (𝔩, ⟨·, ·⟩, 𝔮).

We remark that to the cubic pseudo Kählerian Lie algebras of Proposition 2.20 one can associate pseudo Riemannian analogues of the Alekseevsky spaces.

Proposition 2.21. The cubic pseudo Kählerian Lie algebras (𝔲_{0}(p)′, ⟨·, ·⟩′, J) and 𝔲_{0}(ψ) associated to a an isometric map ψ of pseudo Euclidean vector spaces, which is special or of zero order, admit a Q-representation. In particular, to any such map ψ we can associate a homogeneous pseudo quaternionic Kähler manifold.

Proof: The proof does not depend on the signature of the scalar products, s. [A], [C]. □

Remark 6: Being a totally geodesic Kählerian submanifold of a symmetric space, the special Kähler submanifold of a symmetric Alekseevsky spaces is Hermitian symmetric. It can be described as follows. Let M = G/K be a symmetric Alekseevsky space, G its maximal connected isometry group and K = Sp(1) · H its isotropy subgroup; G and H are semisimple. We denote by (𝔩, ⟨·, ·⟩, 𝔮) the Alekseevskian Lie algebra corresponding to M and by (𝔲_{0}, ⟨·, ·⟩, J) its special Kähler subalgebra. Then 𝔩 is isomorphic to the Iwasawa Lie algebra of G and 𝔲_{0} is isomorphic to the Iwasawa Lie algebra of a non compact semisimple Lie group H^{0} ⊂ G. The special Kähler submanifold U ⊂ M is the orbit of the point K = eK ∈ M = G/K under the group H^{0} and is a Hermitian symmetric space of non compact type.

Consider the twistor space Z = G/(U(1)·H) of M. It carries a natural structure of homogeneous complex contact manifold, s. [W]. The contact hyperplane D_{S} ⊂ T_{S}Z, S ∈ Z, carries an H-invariant complex symplectic structure ω. The representation of H on D_{S} is irreducible, preserves this complex symplectic structure and the orbit of the highest root vector is the base of a Lagrangean cone Ĉ. Its projectivization P (Ĉ) is isomorphic to the compact dual U^{∗} of the Hermitian symmetric space U.

We can consider the action of the complex semisimple Lie group Hℂ on the complex symplectic vector space D_{S} and on the compact Hermitian symmetric space U^{∗} ≅ P (Ĉ). Now we remark that the Lie group H^{0} = Isom(U) is a (non compact) real form of Hℂ, hence we can also consider the action of H^{0} on D_{S} and on U^{∗} ≅ P (Ĉ). In this way we can realize U as open orbit U = P (C) ⊂ P (Ĉ) of H^{0} on the compact Hermitian symmetric space U^{∗} ≅ P (Ĉ).

One can check that D_{S} carries a H^{0}-invariant real structure τ, such that the canonical special Kähler metric of the projectivized cone P (C) defined by the data (D_{S}, ω, τ) coincides with the (nonpositively curved) Hermitian symmetric metric given by the inclusion U ⊂ M.

It is known^{1} that the compact projectivized cone P (Ĉ) ≅ U^{∗} is precisely the projectivization of the cone Ĉ ⊂ T_{S}Z^{∗} of tangent directions to minimal rational curves through a point S ∈ Z^{∗} in the twistor space Z^{∗} of the Wolf space M^{∗} = G^{∗}/K. M^{∗} = G^{∗}/K is the compact symmetric quaternionic Kähler manifold which is dual to the symmetric Alekseevsky space M = G/K.

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Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

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