Classification of indefinite hyper-Kähler symmetric spaces

Dmitri V. Alekseevsky^{1}, Vicente Cortés^{2}

daleksee@mpim-bonn.mpg.de , vicente@math.uni-bonn.de

1 ����� ������ ���� ���� �������� � � ��� ������ ������ and MPI für Mathematik, Bonn 2 Mathematisches Institut der Universität Bonn, Beringstr. 1, D-53115 Bonn

Abstract We classify indefinite simply connected hyper-Kähler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m, 4m). We establish a natural 1-1 correspondence between simply connected hyper-Kähler symmetric spaces of dimension 8m and orbits of the group GL(m, ℍ) on the space (S^{4}ℂ^{n})^{τ} of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τ is the real structure induced by the quaternionic structure of ℂ^{2m} = ℍ^{m}. We define and classify also complex hyper-Kähler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.

1 Introduction

We recall that a pseudo-Riemannian manifold (M, g) is called a symmetric space if any point x ∈ M is an isolated fixed point of an involutive isometry s_{x} (called central symmetry with centre x). Since the product of two central symmetries s_{x} and s_{y} with sufficiently close centres is a shift along the geodesic (xy), the group generated by central symmetries acts transitively on M and one can identify M with the quotient M = G/K, where G is the connected component of the isometry group Isom(M, g) and K is the stabilizer of a point o ∈ M.

A symmetric space (M = G/K, g) is called Kähler (respectively, hyper-Kähler) if its holonomy group Hol(M, g) is a subgroup of the pseudo-unitary group U(p, q) (respectively, of the pseudo-symplectic group Sp(p, q) ⊂ SU(2p, 2q)). Any hyper-Kähler symmetric space is in particular a homogeneous hypercomplex manifold. Homogeneous hypercomplex manifolds of compact Lie groups were constructed by Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen [SSTVP] and by D. Joyce [J] and homogeneous hypercomplex structures on solvable Lie groups by M.L. Barberis and I. Dotti-Miatello [BD].

The classification of simply connected symmetric spaces reduces to the classification of involutive automorphisms σ of a Lie algebra 𝔤, such that the adjoint representation ad𝔨|𝔪 preserves a pseudo-Euclidean scalar product g, where

𝔤 = 𝔨 + 𝔪 , σ|𝔨 = 1, σ|𝔪 = −1,

is the eigenspace decomposition of the involution σ. Note that the eigenspace decomposition of an involutive automorphism is characterized by the conditions

[𝔨, 𝔨] ⊂ 𝔨, [𝔨, 𝔪] ⊂ 𝔪, [𝔪, 𝔪] ⊂ 𝔨 .

Such a decomposition is called a symmetric decomposition.

In fact, for any pseudo-Riemannian symmetric space M = G/K the conjugation with respect to the central symmetry s_{o} with centre o = eK is an involutive automorphism of the Lie group G, which induces an involutive automorphism σ of its Lie algebra 𝔤. The pseudo-Riemannian metric of M induces a 𝔨-invariant scalar product on 𝔪 ≅ T_{o}M, where 𝔤 = 𝔨+𝔪 is the symmetric decomposition defined by σ. Conversely, a symmetric decomposition 𝔤 = 𝔨 + 𝔪 together with a 𝔨-invariant scalar product on 𝔪 determines a pseudo-Riemannian symmetric space M = G/K, where G is the simply connected Lie group with the Lie algebra 𝔤, K is the connected (and closed) subgroup of G generated by 𝔨, the pseudo-Riemannian metric on M is defined by g and the central symmetry is defined by the involutive automorphism σ associated to the symmetric decomposition.

Naturally identifying the space 𝔪 with the tangent space T_{o}M, the isotropy group is identified with Ad_{K}|𝔪 and the holonomy algebra is identified with ad𝔥, where 𝔥 = [𝔪, 𝔪]. If one assumes that the holonomy algebra is irreducible then one can prove that the Lie algebra 𝔤 is semisimple. Hence the classification of pseudo-Riemannian symmetric spaces with irreducible holonomy reduces to the classification of involutive authomorphisms of semisimple Lie algebras. Such a classification was obtained by M. Berger [B1, B2] and A. Fedenko [F]. It includes the classification of Riemannian symmetric spaces (obtained earlier by E. Cartan), since according to de Rham’s theorem any simply connected complete Riemannian manifold is a direct product of Riemannian manifolds with irreducible holonomy algebra and a Euclidean space.

A classification of pseudo-Riemannian symmetric spaces with non completely reducible holonomy is known only for signature (1, n) (Cahen-Wallach [CW]) and for signature (2, n) under the assumption that the holonomy group is solvable (Cahen-Parker [CP]). The classification problem for arbitrary signature looks very complicated and includes, for example, the classification of Lie algebras which admit a nondegenerate ad-invariant symmetric bilinear form. An inductive construction of solvable Lie algebras with such a form was given by V. Kac [K], see also [MR1], [Bo] and [MR2].

In this paper we give a classification of pseudo-Riemannian hyper-Kähler symmetric spaces. In particular, we prove that any simply connected hyper-Kähler symmetric space M has signature (4m,4m) and its holonomy group is commutative. The main result is the following, see Theorem 6.

Let (E, ω, j) be a complex symplectic vector space of dimension 4m with a quaternionic structure j such that ω(jx, jy) = \overline{ω(x, y)} for all x, y ∈ E and E = E_{+} ⊕ E_{−} a j-invariant Lagrangian decomposition. Such a decomposition exists if and only if the Hermitian form γ = ω(·, j·) has real signature (4m, 4m). We denote by τ the real structure in S^{2r}E defined by τ(e_{1}e_{2} … e_{2r}) := j(e_{1})j(e_{2}) … j(e_{2r}), e_{i} ∈ E. Then any element S ∈ (S^{4}E_{+})^{τ} defines a hyper-Kähler symmetric space M_{S} which is associated with the symmetric decomposition

𝔤 = 𝔥 + 𝔪 ,

where 𝔪 = (ℂ^{2} ⊗ E)^{ρ} is the fixed point set of the real structure ρ on ℂ^{2} ⊗ E given by ρ(h ⊗ e) = j_{H}h ⊗ je, where j_{H} is the standard quaternionic structure on ℂ^{2} = ℍ, 𝔥 = span{S_{e,e′}| e, e′ ∈ E}^{τ} ⊂ sp(E)^{τ} ≅ sp(m, m) with the natural action on 𝔪 ⊂ ℂ^{2}⊗E and the Lie bracket 𝔪∧𝔪 → 𝔥 is given by

[h ⊗ e, h′ ⊗ e′] = ω_{H}(h, h′)S_{e,e′},

where ω_{H} is the standard complex symplectic form on ℂ^{2}.

Moreover, we establish a natural 1-1 correspondence between simply connected hyper-Kähler symmetric spaces (up to isomorphism) and orbits of the group GL(m, ℍ) in (S^{4}E_{+})^{τ}.

We define also the notion of complex hyper-Kähler symmetric space as a complex manifold (M, g) of complex dimension 4n with holomorphic metric g such that for any point x ∈ M there is a holomorphic central symmetry s_{x} with centre x and which has holonomy group Hol(M, g) ⊂ Sp(n, ℂ) (Sp(n, ℂ) ↪ Sp(n, ℂ) × Sp(n, ℂ) ⊂ O(4n, ℂ) is diagonally embedded) and give a classification of such spaces. We establish a natural 1-1 correspondence between simply connected complex hyper-Kähler symmetric spaces and homogeneous polynomials of degree 4 in the vector space ℂ^{n} considered up to linear transformations from GL(n, ℂ).

2 Symmetric spaces

2.1 Basic facts about pseudo-Riemannian symmetric spaces

A pseudo-Riemannian symmetric space is a pseudo-Riemannian manifold (M, g) such that any point is an isolated fixed point of an isometric involution. Such a pseudo-Riemannian manifold admits a transitive Lie group of isometries L and can be identified with L/L_{o}, where L_{o} is the stabilizer of a point o. More precisely, any simply connected pseudo-Riemannian symmetric space M = G/K is associated with a symmetric decomposition

𝔤 = 𝔨 + 𝔪 , [𝔨, 𝔨] ⊂ 𝔨 , [𝔨, 𝔪] ⊂ 𝔪 , [𝔪, 𝔪] ⊂ 𝔨  (2.1)

of the Lie algebra 𝔤 = Lie G together with an Ad_{K}-invariant pseudo-Euclidean scalar product on 𝔪. We will assume that G acts almost effectively on M, i.e. 𝔨 does not contain any nontrivial ideal of 𝔤, that M and G are simply connected and that K is connected. Then, under the natural identification of the tangent space T_{o}M at the canonical base point o = eK with 𝔪, the holonomy group Hol ⊂ Ad_{K}|𝔪. We will denote by 𝔥 the holonomy Lie algebra. Since the isotropy representation is faithful it is identified with the subalgebra 𝔥 = [𝔪, 𝔪] := span{[x, y]| x, y ∈ 𝔪} ⊂ 𝔨. Recall that the curvature tensor R of a symmetric space M at o is 𝔥-invariant and determines the Lie bracket in the ideal 𝔥 + 𝔪 ⊂ 𝔤 as follows:

𝔥 = R(𝔪, 𝔪) := span{R(x, y)|x, y ∈ 𝔪} and [x, y] = −R(x, y) , x, y ∈ 𝔪 .

The following result is well known:

Proposition 1 The full Lie algebra of Killing fields of a symmetric space has the form

isom(M) = 𝔥̃ + 𝔪 ,

where the full isotropy subalgebra is given by

𝔥̃ = aut(R) = {A ∈ so(𝔪)|A · R = [A, R(·, ·)] − R(A·, ·) − R(·, A·) = 0} .  (2.2)

2.2 Symmetric spaces of semisimple Lie groups

We will prove that in the case when (M = G/K, g) is a pseudo-Riemannian symmetric space of a (connected) semisimple Lie group G then G is the maximal connected Lie group of isometries of M.

Proposition 2 Let (M = G/K, g) be a pseudo-Riemannian symmetric space associated with a symmetric decomposition 𝔤 = 𝔨 + 𝔪. If G is semisimple and almost effective then

(i) the restriction of the Cartan-Killing form B of 𝔤 to 𝔨 is nondegenerate and hence 𝔨 is a reductive subalgebra of 𝔤 and 𝔤 = 𝔨 + 𝔪 is a B-orthogonal decomposition,

(ii) 𝔨 = [𝔪, 𝔪] and

(iii) 𝔤 = isom(M, g) is the Lie algebra of the full isometry group of M.

Proof: For (i) see [O-V] Ch. 3 Proposition 3.6. (ii) It is clear that 𝔤̄ = [𝔪, 𝔪] + 𝔪 is an ideal of 𝔤. The B-orthogonal complement 𝔞 := 𝔤̄^{⊥} ⊂ 𝔨 is a complementary ideal of 𝔤. Since [𝔞, 𝔪] = 0 the Lie algebra 𝔞 acts trivially on M. From the effectivity of 𝔤 we conclude that 𝔞 = 0. (iii) By Proposition 1, 𝔤̃ = isom(M, g) = 𝔥̃ + 𝔪, where 𝔥̃ = aut(R) = {A ∈ so(𝔪)|A · R = 0}. Now 𝔥̃ preserves 𝔪 and by the identity A · R = [A, R(·, ·)] − R(A·, ·) − R(·, A·) it also normalizes 𝔨. This shows that 𝔥̃ normalizes 𝔤 and hence 𝔤 ⊂ 𝔤̃ is an ideal. Since 𝔤 is semisimple there exists a 𝔤-invariant complement 𝔟 in 𝔤̃. Note that [𝔤, 𝔟] ⊂ 𝔤 ∩ 𝔟 = 0. We can decompose any X ∈ 𝔟 as X = Y + Z, where Y ∈ 𝔥̃ and Z ∈ 𝔪. From [𝔤, 𝔟] = 0 it follows that [𝔤, Y] = [𝔤, Z] = 0 and in particular [𝔪, Y] = 0. This implies that Y = 0 and X = Z ∈ 𝔟 ∩ 𝔪 = 0. This shows that 𝔟 = 0 proving (iii).

We recall that a pseudo-Riemannian Hermitian symmetric space is pseudo-Riemannian symmetric space (M = G/K, g) together with an invariant (and hence parallel) g-orthogonal complex structure J.

Proposition 3 Let (M = G/K, g, J) be a pseudo-Riemannian Hermitian symmetric space of a semisimple and almost effective Lie group G. Then the Ricci curvature of M is not zero.

Proof: From Proposition 2 it follows that 𝔤 = isom(M, g) = 𝔥̃ + m, where 𝔥̃ = 𝔨 = [𝔪, 𝔪]. It is well known that the curvature tensor R of any pseudo-Kähler manifold (and in particular of any pseudo-Riemannian Hermitian symmetric space) is invariant under the operator J. This shows that J ∈ 𝔥̃ = aut(R) = [𝔪, 𝔪] = 𝔥 (holonomy Lie algebra), which implies that the holonomy Lie algebra is not a subalgebra of su(𝔪) ≅ su(p, q). Hence M is not Ricci-flat. In fact, we can write J = ∑ ad[X_{i}, Y_{i}], for X_{i}, Y_{i} ∈ 𝔪. Then using the formulas −2Ric(X, JY) = trJR(X, Y) for the Ricci curvature of a pseudo-Kähler manifold and R(X, Y) = −ad_{[X,Y]}|𝔪 for the curvature of a symmetric space we calculate:

−2 ∑ Ric(X_{i}, JY_{i}) = ∑ trJR(X_{i}, Y_{i}) = − ∑ trJad[X_{i}, Y_{i}] = −trJ^{2} ≠ 0 .

3 Structure of hyper-Kähler symmetric spaces

3.1 Definitions

A (possibly indefinite) hyper-Kähler manifold is a pseudo-Riemannian manifold (M^{4n}, g) of signature (4k, 4l) together with a compatible hypercomplex structure, i.e. three g-orthogonal parallel complex structures (J_{1}, J_{2}, J_{3} = J_{1}J_{2}). This means that the holonomy group Hol ⊂ Sp(k, l). Two hyper-Kähler manifolds (M, g, J_{α}) (α = 1, 2, 3) and (M′, g′, J′_{α}) are called isomorphic if there exists a triholomorphic isometry φ : M → M′, i.e. φ^{∗}J′_{α} = J_{α} and φ^{∗}g′ = g.

A hyper-Kähler symmetric space is a pseudo-Riemannian symmetric space (M = G/K, g) together with an invariant compatible hypercomplex structure. Consider now a simply connected hyper-Kähler symmetric space (M = G/K, g, J_{α}). Without restriction of generality we will assume that G acts almost effectively. M being hyper-Kähler is equivalent to Ad_{K}|𝔪 ⊂ Sp(k, l), or, since K is connected, to ad𝔨|𝔪 ⊂ sp(k, l). This condition means that 𝔨 commutes with the Lie algebra Q = sp(1) ⊂ so(𝔪) = so(4k, 4l) spanned by three anticommuting complex structures J_{1}, J_{2}, J_{3}.

3.2 Existence of a transitive solvable group of isometries and solvability of the holonomy

In this subsection we prove that any simply connected hyper-Kähler symmetric space (M, g, J_{α}) admits a transitive solvable Lie group G ⊂ Aut(g, J_{α}) of automorphisms and has solvable holonomy group.

Proposition 4 Let (M = G/K, g, J_{α}) be a simply connected hyper-Kähler symmetric space and A = Aut_{0}(g, Q) ⊃ Aut_{0}(g, J_{α}) ⊃ G the connected group of isometries which preserve the quaternionic structure Q = span{J_{α}}. Then

(i) the stabilizer A_{o} of a point o ∈ M contains a maximal semisimple subgroup of A,

(ii) the radical R of A acts transitively and triholomorphically on M and

(iii) the holonomy group of M is solvable.

Proof: We consider the quaternionic Kähler symmetric space (M = A/A_{o}, g, Q). The Lie algebra 𝔞_{o} of the stabilizer is given by

𝔞_{o} = aut(R, Q) = {A ∈ so(T_{o}M)|A · R = 0, [A, Q] ⊂ Q} .

Since the curvature tensor of a quaternionic Kähler manifold is invariant under the quaternionic structure Q we conclude that Q ⊂ 𝔞_{o} and 𝔞_{o} = Q⊕Z𝔞(Q), where Z𝔞(Q) denotes the centralizer of Q in 𝔞. Since Q ≅ sp(1) is simple, we may choose a Levi-Malcev decomposition 𝔞 = 𝔰+𝔯 such that the Levi subalgebra 𝔰 ⊃ Q. We put 𝔪_{r} := [Q, 𝔯] and denote by 𝔪_{s} a Q ⊕ Z𝔰(Q)-invariant complement of Q in [Q, 𝔰]. The stabilizer has the decomposition 𝔞_{o} = Q ⊕ (Z𝔰(Q) + Z𝔯(Q)).

Lemma 1 The complement 𝔪 = 𝔪_{s} + 𝔪_{r} to 𝔞_{o} in 𝔞 is 𝔞_{o}-invariant and the decomposition

𝔞 = 𝔞_{o} + 𝔪

is a symmetric decomposition.

Proof: It is clear that 𝔪_{r} is 𝔞_{o}-invariant and 𝔪_{s} is invariant under Q ⊕ Z𝔰(Q) by construction. It remains to check that [Z𝔯(Q), 𝔪_{s}] ⊂ 𝔪. Since 𝔪_{s} = [Q, 𝔪_{s}], we have

[Z𝔯(Q), 𝔪_{s}] = [Z𝔯(Q), [Q, 𝔪_{s}]] = [Q, [Z𝔯(Q), 𝔪_{s}]] ⊂ [Q, 𝔯] = 𝔪_{r} ⊂ 𝔪 .

This shows that 𝔞 = 𝔞_{o} + 𝔪 is an 𝔞_{o}-invariant decomposition. We denote by 𝔞 = 𝔞_{o} + 𝔭 a symmetric decomposition. Any other 𝔞_{o}-invariant decomposition is of the form 𝔞 = 𝔞_{o} + 𝔭_{φ}, where φ : 𝔭 → 𝔞_{o} is an 𝔞_{o}-equivariant map and 𝔭_{φ} = {X + φ(X)|X ∈ 𝔭}. If such non-zero equivariant map φ exists then 𝔭 and 𝔞_{o} contain non-trivial isomorphic Q-submodules. Since 𝔭 is a sum of 4-dimensional irreducible Q-modules and 𝔞_{o} is the sum of the 3-dimensional irreducible Q-module Q and the trivial complementary Q-module Z_{𝔞_{o}}(Q), we infer that there exists a unique 𝔞_{o}-invariant decomposition, which coincides with the symmetric decomposition 𝔞 = 𝔞_{o} + 𝔭.

To prove (i) we have to check that 𝔪_{s} = 0. We note that by the previous lemma 𝔰 = (Q⊕Z𝔰(Q))+𝔪_{s} is a symmetric decomposition of the semisimple Lie algebra 𝔰. Since [𝔪_{s}, 𝔪_{s}] ⊂ Z𝔰(Q) it defines a hyper-Kähler symmetric space N = L/L_{o}, where L is the simply connected semisimple Lie group with Lie algebra 𝔩 = Z𝔰(Q) + 𝔪_{s} and L_{o} is the Lie subgroup generated by the subalgebra Z𝔰(Q) ⊂ 𝔩. Since N is in particular a Ricci-flat pseudo-Riemannian Hermitian symmetric space, from Proposition 3 we obtain that N is reduced to point. Therefore 𝔪_{s} = 0. This proves (i) and (ii). Finally, since the holonomy Lie algebra 𝔥 is identified with 𝔥 = [𝔪, 𝔪] = [𝔪_{r}, 𝔪_{r}] ⊂ 𝔯 it is solvable as subalgebra of the solvable Lie algebra 𝔯.

3.3 Hyper-Kähler symmetric spaces and second prolongation of symplectic Lie algebras

Let (M = G/K, g, J_{α}) be a simply connected hyper-Kähler symmetric space associated with a symmetric decomposition (2.1). Without restriction of generality we will assume that G acts almost effectively and that 𝔨 = [𝔪, 𝔪] = 𝔥 (holonomy Lie algebra). The complexification 𝔪^{ℂ} as 𝔥^{ℂ}-module can be written as 𝔪^{ℂ} = H ⊗ E, such that 𝔥^{ℂ} ⊂ Id ⊗ sp(E) ≅ sp(E), where H = ℂ^{2} and E = ℂ^{2n} are complex symplectic vector spaces with symplectic form ω_{H} and ω_{E}, respectively, such that g^{ℂ} = ω_{H} ⊗ ω_{E} is the complex bilinear metric on 𝔪^{ℂ} induced by g. Note that the symplectic forms are unique up to the transformation ω_{H} ↦ λω_{H}, ω_{E} ↦ λ^{−1}ω_{E}, λ ∈ ℂ^{∗}. We have also quaternionic structures j_{H} and j_{E} on H and E, such that ω_{H}(j_{H}x, j_{H}y) = \overline{ω_{H}(x, y)} for all x, y ∈ H and ω_{E}(j_{E}x, j_{E}y) = \overline{ω_{E}(x, y)} for all x, y ∈ E, where the bar denotes complex conjugation. This implies that γ_{H} := ω_{H}(·, j_{H}·) and γ_{E} := ω_{E}(·, j_{E}·) are Hermitian forms on H and E. For fixed ω_{H} and ω_{E} the quaternionic structures j_{H} and j_{E} are uniquely determined if we require that γ_{H} is positive definite and that ρ = j_{H} ⊗ j_{E} is the real structure on 𝔪^{ℂ}, i.e. the complex conjugation with respect to 𝔪. The metric g^{ℂ} and the Hermitian form g^{ℂ}(·, ρ·) = γ_{H} ⊗ γ_{E} restrict to a real valued scalar product g of some signature (4k, 4l) on 𝔪 = (H ⊗ E)^{ρ}, where (2k, 2l) is the (real) signature of the Hermitian form γ_{E} = ω_{E}(·, j_{E}·). Note that for the holonomy algebra we have the inclusion

𝔥 = Id⊗(𝔥^{ℂ})^{j_{E}} ↪ sp(E)^{j_{E}} = {A ∈ sp(E)|[A, j_{E}] = 0} = aut(E, ω_{E}, j_{E}) ≅ aut(𝔪, g, J_{α}) ≅ sp(k, l) .

Using the symplectic forms we identify H = H^{∗} and E = E^{∗}. Then the symplectic Lie algebras are identified with symmetric tensors as follows:

sp(H) = S^{2}H , sp(E) = S^{2}E .

Since the curvature tensor R of any hyper-Kähler manifold M^{4n} at a point p ∈ M can be identified with an element R ∈ S^{2}sp(k, l) it is invariant under the Lie algebra sp(1) = span{J_{1}, J_{2}, J_{3}}. Let M = G/K be a hyper-Kähler symmetric space as above. By Proposition 1 we can extend the Lie algebra 𝔤 = 𝔨 + 𝔪 = 𝔥 + 𝔪 to a Lie algebra

𝔤̃ = sp(1) + 𝔥 + 𝔪

of Killing vector fields such that [sp(1), 𝔥] = 0. In the H ⊗ E-formalism the Lie algebra sp(1) is identified with sp(H)^{j_{H}} ⊗ Id ⊂ so(𝔪).

Lemma 2 Denote by 𝔤̃^{ℂ} = sp(1, ℂ) + 𝔥^{ℂ} + 𝔪^{ℂ} the complexification of the Lie algebra 𝔤̃. Then the Lie bracket [·, ·] : ∧^{2}𝔪^{ℂ} → 𝔥^{ℂ} can be written as

[h ⊗ e, h′ ⊗ e′] = ω_{H}(h, h′)S_{e,e′} ,  (3.1)

where S ∈ (𝔥^{ℂ})^{(2)} := 𝔥^{ℂ} ⊗ S^{2}E^{∗} ∩ E ⊗ S^{3}E^{∗} = 𝔥^{ℂ} ⊗ 𝔥^{ℂ} ∩ S^{4}E. Moreover S is sp(1, ℂ) ⊕ 𝔥^{ℂ}- invariant and satisfies the following reality condition: [S_{j_{E}e,e′} − S_{e,j_{E}e′}, j_{E}] = 0.

Proof: The Lie bracket [·, ·] : ∧^{2}𝔪^{ℂ} → 𝔥^{ℂ} is an sp(1, ℂ) ⊕𝔥^{ℂ}-equivariant map, due to the Jacobi identity. We decompose the sp(H) ⊕ sp(E)-module ∧^{2}𝔪^{ℂ}:

∧^{2}𝔪^{ℂ} = ∧^{2}(H ⊗ E) = ∧^{2}H ⊗ S^{2}E ⊕ S^{2}H ⊗ ∧^{2}E = ω_{H} ⊗ S^{2}E ⊕ S^{2}H ⊗ ∧^{2}E .

Since 𝔥^{ℂ} ⊂ S^{2}E the Lie bracket defines an sp(1, ℂ) ⊕ 𝔥^{ℂ}-invariant element of the space ω_{H} ⊗ S^{2}E ⊗ S^{2}E ⊕ S^{2}H ⊗ ∧^{2}E ⊗ S^{2}E. The second summand has no nontrivial sp(1, ℂ)-invariant elements. Hence the bracket is of the form (3.1), where S ∈ S^{2}E^{∗} ⊗ 𝔥^{ℂ} ⊂ S^{2}E ⊗ S^{2}E. The Jacobi identity reads:

0 = [h ⊗ e, [h′ ⊗ e′, h′′ ⊗ e′′]] − [[h ⊗ e, h′ ⊗ e′], h′′ ⊗ e′′] − [h′ ⊗ e′, [h ⊗ e, h′′ ⊗ e′′]]

= −ω_{H}(h′, h′′)h ⊗ S_{e′,e′′}e − ω_{H}(h, h′)h′′ ⊗ S_{e,e′}e′′ + ω_{H}(h, h′′)h′ ⊗ S_{e,e′′}e′ .

Since dim H = 2 we may assume that h, h′ = h′′ is a symplectic basis, i.e. ω_{H}(h, h′) = 1, and the equation implies: S_{e,e′′}e′ = S_{e,e′}e′′, i.e. S ∈ (𝔥^{ℂ})^{(2)}. The Lie bracket of two real elements h ⊗ e + j_{H}h ⊗ j_{E}e and h ⊗ e′ + j_{H}h ⊗ j_{E}e′ ∈ 𝔪 ⊂ 𝔪^{ℂ} is an element of 𝔥. This gives:

[h ⊗ e + j_{H}h ⊗ j_{E}e, h ⊗ e′ + j_{H}h ⊗ j_{E}e′] = ω_{H}(h, j_{H}h)(S_{e,j_{E}e′} − S_{j_{E}e,e′}) ∈ 𝔥 .

From the fact that the Hermitian form γ_{H} = ω_{H}(·, j_{H}·) is positive definite it follows that ω_{H}(h, j_{H}h) ≠ 0. This establishes the reality condition since 𝔥 = {A ∈ 𝔥^{ℂ}|[A, j_{E}] = 0}.

In fact any tensor S ∈ S^{4}E satisfying the conditions of the above lemma can be used to define a hyper-Kähler symmetric space as the following theorem shows. We can identify S^{4}E with the space ℂ[E]^{(4)} of homogeneous quartic polynomials on E ≅ E^{∗}.

Theorem 1 Let S ∈ S^{4}E, E = ℂ^{2n}, be a quartic polynomial invariant under all endomorphisms S_{e,e′} ∈ S^{2}E = sp(E) and satisfying the reality condition

[S_{j_{E}e,e′} − S_{e,j_{E}e′}, j_{E}] = 0 .  (3.2)

Then it defines a hyper-Kähler symmetric space, which is associated with the following complex symmetric decomposition

𝔤^{ℂ} = 𝔥^{ℂ} + H ⊗ E , 𝔥^{ℂ} = span{S_{e,e′}|e, e′ ∈ E} ⊂ sp(E) .  (3.3)

The bracket ∧^{2}(H ⊗ E) → 𝔥^{ℂ} is given by (3.1). The real symmetric decomposition is defined as ρ-real form 𝔤 = 𝔥 + 𝔪 of (3.3), where

𝔨 = 𝔥 = {A ∈ 𝔥^{ℂ}|[A, j_{E}] = 0} = span{S_{j_{E}e,e′} − S_{e,j_{E}e′}|e, e′ ∈ E} , 𝔪 = (H ⊗ E)^{ρ}.

The hyper-Kähler symmetric space M associated to this symmetric decomposition is the quotient M = M_{S} = G/K, where G is the simply connected Lie group with Lie algebra 𝔤 and K ⊂ G is the connected (and closed) subgroup with Lie algebra 𝔨 = 𝔥.

Moreover any simply connected hyper-Kähler symmetric space can be obtained by this construction. Two hyper-Kähler symmetric spaces M_{S} and M_{S′} defined by quartics S and S′ are isomorphic if and only if S and S′ are in the same orbit of the group Aut(E, ω_{E}, j_{E}) = {A ∈ Sp(E)|[A, j_{E}] = 0} ≅ Sp(k, l).

Proof: First of all we note that 𝔥^{ℂ} = S_{E,E} := span{S_{e,e′}|e, e′ ∈ E} is a subalgebra of sp(E) because

[S_{e,e′}, S_{f,f′}] = (S_{e,e′} · S)_{f,f′} − S_{S_{e,e′}f,f′} − S_{f,S_{e,e′}f′} = −S_{S_{e,e′}f,f′} − S_{f,S_{e,e′}f′} ∈ 𝔥^{ℂ} .

Since S is 𝔥^{ℂ}-invariant and completely symmetric we can check, as in Lemma 2, that the Jacobi identity is satisfied and that (3.3) defines a complex symmetric decomposition. We prove that 𝔥 := span{S_{j_{E}e,e′} − S_{e,j_{E}e′}|e, e′ ∈ E} ⊂ {A ∈ 𝔥^{ℂ}|[A, j_{E}] = 0} defines a real form of 𝔥^{ℂ}. Indeed for e, e′ ∈ E we have

S_{e,e′} = \frac{1}{2}(S_{e,e′} + S_{j_{E}e,j_{E}e′}) − \frac{\sqrt{−1}}{2}(\sqrt{−1}S_{e,e′} − \sqrt{−1}S_{j_{E}e,j_{E}e′})

= \frac{1}{2}(S_{j_{E}e′′,e′} − S_{e′′,j_{E}e′}) − \frac{\sqrt{−1}}{2}(S_{j_{E}e′′,\sqrt{−1}e′} − S_{e′′,j_{E} \sqrt{−1}e′}) ,

where e′′ = −j_{E}e. Due to the reality condition the restriction of the Lie bracket [·, ·] : ∧^{2}𝔪^{ℂ} → 𝔥^{ℂ} to ∧^{2}𝔪 has values in 𝔥 and 𝔤 = 𝔥 + 𝔪 is a symmetric decomposition with [𝔪, 𝔪] = 𝔥. The metric g^{ℂ} = ω_{H} ⊗ ω_{E} defines a real valued scalar product g of some signature (p, q) on 𝔪 = (H ⊗ E)^{ρ}, which is invariant under the Lie algebra 𝔥. Since [𝔥, j_{E}] = 0 the holonomy algebra 𝔥 ⊂ sp(k, l), p = 4k, q = 4l. Hence this symmetric decomposition defines a hyper-Kähler symmetric space.

By Lemma 2 any hyper-Kähler symmetric space can be obtained by this construction. It is well known that a simply connected symmetric space M of signature (p, q) is determined by its abstract curvature tensor R ∈ S^{2}(∧^{2}V), V = ℝ^{p,q}, and two tensors R and R′ define isometric symmetric spaces if and only if they belong to the same O(V) orbit. Similarly a simply connected hyper-Kähler symmetric space is determined up to isometry by its abstract curvature tensor R ∈ S^{2}(∧^{2}V), where V = ℝ^{4k,4l} is the pseudo-Euclidean vector space with fixed hypercomplex structure J_{α} ∈ O(V). For a hyper-Kähler symmetric space the complexified curvature tensor has the form

R(h ⊗ e, h′ ⊗ e′) = −ω_{H}(h, h′)S_{e,e′} ,

where S ∈ S^{4}E is the quartic form of Lemma 2. Two such curvature tensors define isomorphic hyper-Kähler symmetric spaces if and only if they belong to the same orbit of Aut(ℝ^{4k,4l}, J_{α}) = Sp(k, l). The group Sp(k, l) acts on V^{ℂ} = H ⊗ E as Id ⊗ Sp(E)^{j_{E}} = Id ⊗ Aut(E, ω_{E}, j_{E}). Hence two curvature tensors R = −ω_{H} ⊗ S and R′ = −ω_{H} ⊗ S′ are in the same Sp(k, l)-orbit if and only if S and S′ are in the same Sp(k, l)-orbit on S^{4}E.

4 Complex hyper-Kähler symmetric spaces

4.1 Complex hyper-Kähler manifolds

A complex Riemannian manifold is a complex manifold M equipped with a complex metric g, i.e. a holomorphic section g ∈ Γ(S^{2}T^{∗}M) which defines a nondegenerate complex quadratic form. As in the real case any such manifold has a unique holomorphic torsionfree and metric connection (Levi-Civita connection). A complex hyper-Kähler manifold is a complex Riemannian manifold (M^{4n}, g) of complex dimension 4n together with a compatible hypercomplex structure, i.e. three g-orthogonal parallel complex linear endomorphisms (J_{1}, J_{2}, J_{3} = J_{1}J_{2}) with J_{α}^{2} = −1. This means that the holonomy group Hol ⊂ Sp(n, ℂ) = Z_{O(4n,ℂ)}(Sp(1, ℂ)). The linear group Sp(n, ℂ) is diagonally embedded into Sp(n, ℂ) × Sp(n, ℂ) ⊂ GL(4n, ℂ). Two complex hyper-Kähler manifolds (M, g, J_{α}) (α = 1, 2, 3) and (M′, g′, J′_{α}) are called isomorphic if there exists a holomorphic isometry φ : M → M′ such that φ^{∗}J′_{α} = J_{α} and φ^{∗}g′ = g.

We will show that the complex hyper-Kähler structure can be described as a half-flat Grassmann structure of a certain type. A Grassmann structure on a complex Riemannian manifold (M, g) is a decomposition of the (holomorphic) tangent bundle T M ≅ H ⊗ E into the tensor product of two holomorphic vector bundles H and E of rank 2m and 2n with holomorphic nondegenerate 2-forms ω_{H} and ω_{E} such that g = ω_{H} ⊗ ω_{E}. The Grassmann structure will be called parallel if the Levi-Civita connection ∇ = ∇^{T M} can be decomposed as:

∇ = ∇^{H} ⊗ Id + Id ⊗ ∇^{E} ,

where ∇^{H} and ∇^{E} are (uniquely defined) symplectic connections in the bundles H and E. A parallel Grassmann structure will be called half-flat if ∇^{H} is flat. Note that a parallel Grassmann structure on a simply connected manifold is half-flat if and only if the holonomy group of the Levi-Civita connection is contained in Id ⊗ Sp(n, ℂ) ⊂ Sp(m, ℂ) ⊗ Sp(n, ℂ) ⊂ O(ℂ^{2m} ⊗ ℂ^{2n}).

Proposition 5 A complex hyper-Kähler structure (g, J_{α}) on a simply connected complex manifold M is equivalent to the following geometric data:

(i) a half-flat Grassmann structure (T M, g, ∇) ≅ (H, ω_{H}, ∇^{H}) ⊗ (E, ω_{E}, ∇^{E}) and

(ii) an isomorphism of flat symplectic vector bundles H ≅ M × ℂ^{2}. Under this isomorphism ω_{H} = h_{1}^{∗} ∧ h_{2}^{∗}, where (h_{1}, h_{2}) is the standard basis of ℂ^{2} considered as parallel frame of the trivial bundle H = M × ℂ^{2}.

More precisely,

J_{1} = R_{i} ⊗ Id , J_{2} = R_{j} ⊗ Id , and J_{3} = R_{k} ⊗ Id ,  (4.1)

where we have identified ℂ^{2} = ℂh_{1} ⊕ ℂh_{2} with ℍ = span_{ℝ}{1, i, j, k} = span_{ℂ}{1, j} = ℂ1 ⊕ ℂj with the complex structure defined by left-multiplication by i and R_{x} denotes the rightmultiplication by the quaternion x ∈ ℍ.

Proof: It is easy to check that the geometric data (i) and (ii) define a complex hyper-Kähler structure on M. Conversely let (g, J_{α}) be a complex hyper-Kähler structure on M. The endomorphism J_{1} has eigenvalues ±i and the tangent space can be decomposed into a sum of eigenspaces

T M = E_{+} ⊕ E_{−} .

From the J_{1}-invariance of the metric g it follows that g(E_{±}, E_{±}) = 0 and we can identify E_{−} = E^{∗} with the dual space of E = E_{+}. Since J_{2} anticommutes with J_{1} it interchanges E and E^{∗} and hence defines an isomorphism E →^{∼} E^{∗}. Now g(·, J_{2}·) defines a symplectic form ω_{E} on E. Let H = M × ℂ^{2} = M × (ℂh_{1} ⊕ ℂh_{2}) be the trivial bundle with 2-form ω_{H} = h_{1}^{∗} ∧ h_{2}^{∗}. Then we can identify

T M = E ⊕ E^{∗} = E ⊕ E = h_{1} ⊗ E ⊕ h_{2} ⊗ E = H ⊗ E .

We check that under this identification we have g = ω_{H} ⊗ ω_{E}. Note that both sides vanish on h_{1} ⊗ E and h_{2} ⊗ E and ω_{H}(h_{1}, h_{2}) = 1. We calculate for e, e′ ∈ E = E_{+} = h_{1} ⊗ E:

g(e, J_{2}e′) = ω_{E}(e, e′) = ω_{H}(h_{1}, h_{2})ω_{E}(e, e′) = (ω_{H} ⊗ ω_{E})(h_{1} ⊗ e, h_{2} ⊗ e′) .

Hence we have a Grassmann structure. The eigenspaces E_{±} of the parallel endomorphism J_{1} are invariant under parallel transport. Therefore the Levi-Civita connection ∇ induces a connection ∇^{E} in the bundle E. Since ∇g = 0 and ∇J_{2} = 0 we have ∇^{E}ω_{E} = 0. We define a flat connection ∇^{H} on the trivial bundle H = M × ℂ^{2} by the condition ∇^{H}h_{1} = ∇^{H}h_{2} = 0. Then ∇ = ∇^{H} ⊗ Id + Id ⊗ ∇^{E}. So the Grassmann structure is half-flat.

Finally, using the standard identification ℂ^{2} = ℍ, one can easily check that the J_{α} are given by (4.1).

4.2 Complexification of real hyper-Kähler manifolds

Let (M, g, J_{α}) be a (real) hyper-Kähler manifold. We will assume that it is real analytic. This is automatically true if the metric g is positive definite since it is Ricci-flat and a fortiori Einstein. Using analytic continuation we can extend (M, g, J_{α}) to a complex hyper-Kähler manifold (M^{ℂ}, g^{ℂ}, J_{α}^{ℂ}) equipped with an antiholomorphic involution T. In complex local coordinates z^{j} = x^{j} +iy^{j} which are extension of real analytic coordinates x^{j}, y^{j} the involution is given by the complex conjugation z^{j} → z̄^{j} = x^{j} − iy^{j}. We can reconstruct the (real) hyper-Kähler manifold as the fixed point set of T. We will call (M, g, J_{α}) a real form of (M^{ℂ}, g^{ℂ}, J_{α}^{ℂ}) and (M^{ℂ}, g^{ℂ}, J_{α}^{ℂ}) the complexification of (M, g, J_{α}).

In general a complex hyper-Kähler manifold has no real form. A necessary condition is that the holonomy group of ∇^{E} is contained in Sp(k, l), n = k+l, and hence preserves a quaternionic structure. Then we can define a parallel antilinear endomorphism field j_{E} : E → E such that j_{E}^{2} = −1 and ω_{E}(j_{E}x, j_{E}y) = \overline{ω_{E}(x, y)} for all x, y ∈ E, where the bar denotes complex conjugation. We define a parallel antilinear endomorphism field j_{H} : H → H as the leftmultiplication by the quaternion j on H = M × ℍ. Then ρ = j_{H} ⊗ j_{E} defines a field of real structures in T M = H ⊗ E. We denote by D ⊂ T M the real eigenspace distribution of ρ with eigenvalue 1. Here T M is considered as real tangent bundle of the real manifold M. If M^{ρ} ⊂ M is a leaf of D of real dimension 4n then the data (g, J_{α}) induce on M^{ρ} a (real) hyper-Kähler structure.

4.3 Complex hyper-Kähler symmetric spaces

A complex Riemannian symmetric space is a complex Riemannian manifold (M, g) such that any point is an isolated fixed point of an isometric holomorphic involution. Like in the real case one can prove that it admits a transitive complex Lie group of holomorphic isometries and that any simply connected complex Riemannian symmetric M is associated to a complex symmetric decomposition

𝔤 = 𝔨 + 𝔪 , [𝔨, 𝔨] ⊂ 𝔨 , [𝔨, 𝔪] ⊂ 𝔪 , [𝔪, 𝔪] = 𝔨  (4.2)

of a complex Lie algebra 𝔤 together with an ad𝔨-invariant complex scalar product on 𝔪. More precisely M = G/K, where G is the simply connected complex Lie group with the Lie algebra 𝔤 and K is the (closed) connected subgroup associated with 𝔨. The holonomy group of such manifold is H = Ad_{K}|𝔪. Any pseudo-Riemannian symmetric space M = G/K associated with a symmetric decomposition 𝔤 = 𝔨 + 𝔪 has a canonical complexification M^{ℂ} = G^{ℂ}/K^{ℂ} defined by the complexification 𝔤^{ℂ} = 𝔨^{ℂ} + 𝔪^{ℂ} of the symmetric decomposition. Proposition 1 remains true for complex Riemannian symmetric spaces. Ignoring the reality condition we obtain the following complex version of Theorem 1.

Theorem 2 Let S ∈ S^{4}E, E = ℂ^{2n}, be a quartic polynomial invariant under all endomorphisms S_{e,e′} ∈ S^{2}E = sp(E). Then it defines a complex hyper-Kähler symmetric space, which is associated with the following complex symmetric decomposition

𝔤 = 𝔥 + H ⊗ E , 𝔥 = S_{E,E} = span{S_{e,e′}|e, e′ ∈ E} ⊂ sp(E) .  (4.3)

The bracket ∧^{2}(H ⊗ E) → 𝔥 is given by (3.1). The complex hyper-Kähler symmetric space M associated to this symmetric decomposition is the quotient M = M_{S} = G/K, where G is the (complex) simply connected Lie group with Lie algebra 𝔤 and K ⊂ G is the connected (and closed) subgroup with Lie algebra 𝔨 = 𝔥.

Moreover any simply connected complex hyper-Kähler symmetric space can be obtained by this construction. Two complex hyper-Kähler symmetric spaces M_{S} and M_{S′} defined by quartics S and S′ are isomorphic if and only if S and S′ are in the same orbit of Aut(E, ω_{E}) = Sp(E) ≅ Sp(n, ℂ).

Corollary 1 There is a natural bijection between simply connected complex hyper-Kähler symmetric spaces of dimension 4n up to isomorphism and Sp(n, ℂ)-orbits on the space of quartic polynomials S ∈ S^{4}E in the symplectic vector space E = ℂ^{2n} such that

S_{e,e′} · S = 0 for all e, e′ ∈ E .  (4.4)

4.4 Classification of complex hyper-Kähler symmetric spaces

The following complex version of Proposition 4 (with similar proof) will be a crucial step in the classification of complex hyper-Kähler symmetric spaces.

Proposition 6 Let (M = G/K, g, J_{α}) be a simply connected complex hyper-Kähler symmetric space. Then the holonomy group of M is solvable and M admits a transitive solvable Lie group of automorphisms.

Due to Corollary 1 the classification of simply connected complex hyper-Kähler symmetric spaces reduces to the determination of quartic polynomials S satisfying (4.4). Below we will determine all such polynomials. We will prove that the following example gives all such polynomials.

Example 1: Let E = E_{+} ⊕ E_{−} be a Lagrangian decomposition, i.e. ω(E_{±}, E_{±}) = 0, of the symplectic vector space E = ℂ^{2n}. Then any polynomial S ∈ S^{4}E_{+} ⊂ S^{4}E satisfies the condition (4.4) and defines a simply connected complex hyper-Kähler symmetric space M_{S} with Abelian holonomy algebra 𝔥 = S_{E_{+},E_{+}} ⊂ S^{2}E_{+} ⊂ S^{2}E = sp(E).

In fact, since E_{+} is Lagrangian the endomorphisms from S^{2}E_{+} form an Abelian subalgebra of sp(E), which acts trivially on E_{+} and hence on S^{4}E_{+}.

Theorem 3 Let S ∈ S^{4}E be a quartic polynomial satisfying (4.4). Then there exists a Lagrangian decomposition E = E_{+} ⊕ E_{−} such that S ∈ S^{4}E_{+}.

Proof: According to Theorem 2 the quartic S defines a hyper-Kähler symmetric space with holonomy Lie algebra 𝔥 = S_{E,E}. Since, by Proposition 6, 𝔥 is solvable, Lie’s theorem implies the existence of a one-dimensional 𝔥-invariant subspace P = ℂp ⊂ E. There exists an ωnondegenerate subspace W ⊂ E such that the ω-orthogonal complement of P is P^{⊥} = P ⊕ W. We choose a vector q ∈ E such that ω(p, q) = 1 and ω(W, q) = 0 and put Q := ℂq. Then we have

E = P ⊕ W ⊕ Q .

Since 𝔥 preserves P we have the following inclusion

𝔥 ⊂ P E + W^{2} = P^{2} + P W + P Q + W^{2} ,

where we use the notation XY = X ∨ Y for the symmetric product of subspaces X, Y ⊂ E. Then the second prolongation 𝔥^{(2)} = {T ∈ S^{4}E|T_{e,e′} ∈ 𝔥 for all e, e′ ∈ E} has the following inclusion

𝔥^{(2)} ⊂ P^{3}E + P^{2}W^{2} + P W^{3} + W^{4} = P^{4} + P^{3}Q + P^{3}W + P^{2}W^{2} + P W^{3} + W^{4} .  (4.5)

Indeed 𝔥^{(2)} ⊂ 𝔥^{2} = P^{4} + P^{3}Q + P^{3}W + P^{2}Q^{2} + P^{2}W Q + P^{2}W^{2} + P QW^{2} + P W^{3} + W^{4}. The projection 𝔥^{(2)} → P^{2}Q^{2} + P^{2}W Q + P QW^{2} is zero because otherwise S_{q,q} ∈ 𝔥 ⊂ P E + W^{2} would have a nonzero projection to Q^{2} + W Q or S_{w,q} ∈ 𝔥 would have a nonzero projection to QW for appropriate choice of w ∈ W. By (4.5) we can write the quartic S as

S = p^{3}(λp + μq + w_{0}) + p^{2}B + pC + D ,

where λ, μ ∈ ℂ, w_{0} ∈ W, B ∈ S^{2}W, C ∈ S^{3}W and D ∈ S^{4}W. From now on we will identify S^{d}E with the space ℂ[E^{∗}]^{(d)} of homogeneous polynomials on E^{∗} of degree d. Then the ωcontraction T_{x} = ι_{ωx}T = T (ωx, …) of a tensor T ∈ S^{d}E with a vector x ∈ E is identified with the following homogeneous polynomial of degree d − 1:

T_{x} = \frac{1}{d}∂_{ωx}T ,

where ∂_{ωx}T is the derivative of the polynomial T ∈ ℂ[E^{∗}]^{(d)} in the direction of ωx = ω(x, ·) ∈ E^{∗}. For example p_{q} = ⟨p, ωq⟩ = ω(q, p) = ∂_{ωq}p = −∂_{p^{∗}}p = −1 = −q_{p}.

From S_{p,q} = −\frac{1}{4}μp^{2} and the condition S_{p,q} · S = 0 we obtain μ = 0, since p^{2} · S = μp^{4}. This implies S_{p,·} = 0. Next we compute:

S_{q,q} = \frac{1}{6}(6λp^{2} + 3pw_{0} + B)
S_{q,w} = −\frac{1}{12}(−3p^{2}ω(w_{0}, w) + 2p∂_{ωw}B + ∂_{ωw}C)
= −\frac{1}{12}(−3p^{2}ω(w_{0}, w) + 4pB_{w} + 3C_{w})
S_{w,w′} = \frac{1}{6}(p^{2}B_{w,w′} + 3pC_{w,w′} + 6D_{w,w′})

for any w, w′ ∈ W.

Now the condition (4.4) can be written as follows:

0 = 6S_{q,q} · S = (3pw_{0} + B) · S = (\frac{3}{2}(p ⊗ w_{0} + w_{0} ⊗ p) + B) · S
= \frac{3}{2}(2p^{3}B_{w_{0}} + 3p^{2}C_{w_{0}} + 4pD_{w_{0}}) + p^{3}Bw_{0} + p^{2}B · B + pB · C + B · D
= −2p^{3}Bw_{0} + \frac{9}{2}p^{2}C_{w_{0}} + p(6D_{w_{0}} + B · C) + B · D .

Note that Bw_{0} = −B_{w_{0}} and B · B = [B, B] = 0.

0 = −12S_{q,w} · S = (4pB_{w} + 3C_{w}) · S
= 2(p^{4}ω(B_{w}, w_{0}) + 2p^{3}B^{2}w − 3p^{2}C_{Bw} − 4pD_{Bw})

+3(p^{3}C_{w}w_{0} + p^{2}C_{w} · B + pC_{w} · C + C_{w} · D)

= 2p^{4}ω(B_{w}, w_{0}) + p^{3}(4B^{2}w + 3C_{w}w_{0}) + p^{2}(−6C_{Bw} + 3C_{w} · B)

+p(−8D_{Bw} + 3C_{w} · C) + 3C_{w} · D

0 = 2S_{w,w′} · S = \frac{1}{2}(p ⊗ C_{w,w′} + C_{w,w′} ⊗ p) · S + 2D_{w,w′} · S
= \frac{1}{2}(p^{4}ω(C_{w,w′}, w_{0}) − 2p^{3}BC_{w,w′} + 3p^{2}C_{C_{w,w′}} + 4pD_{C_{w,w′}}) +

2(p^{3}D_{w,w′}w_{0} + p^{2}D_{w,w′} · B + pD_{w,w′} · C + D_{w,w′} · D)

= \frac{1}{2}p^{4}ω(C_{w,w′}, w_{0}) + p^{3}(−BC_{w,w′} + 2D_{w,w′}w_{0}) +

p^{2}(\frac{3}{2}C_{C_{w,w′}} + 2D_{w,w′} · B) + p(2D_{C_{w,w′}} + 2D_{w,w′} · C) + 2D_{w,w′} · D .

This gives the following system of equations:

(1) Bw_{0} = 0
(2) C_{w_{0}} = 0
(3) 6D_{w_{0}} + B · C = 0
(4) B · D = 0
(5) ω(B_{w}, w_{0}) = 0
(6) 4B^{2}w + 3C_{w}w_{0} = 0
(7) −2C_{Bw} + C_{w} · B = 0
(8) −8D_{Bw} + 3C_{w} · C = 0
(9) C_{w} · D = 0
(10) ω(C_{w}w′, w_{0}) = 0
(11) −BC_{w}w′ + 2D_{w,w′}w_{0} = 0
(12) \frac{3}{2}C_{C_{w}w′} + 2D_{w,w′} · B = 0
(13) D_{C_{w}w′} + D_{w,w′} · C = 0
(14) D_{w,w′} · D = 0 .

Note that (5) and (10) follow from (1) and (2) and that using (2) equation (6) says that the endomorphism B has zero square:

(6′) B^{2} = 0 .

Eliminating D_{w_{0}} in equations (3) and (11) we obtain:

(15) 0 = (B·C)_{w}w′+3BC_{w}w′ = BC_{w}w′−C_{Bw}w′−C_{w}Bw′+3BC_{w}w′ = 4BC_{w}w′−C_{Bw}w′−C_{w}Bw′ .

We can rewrite (7) as:

(7′) − 2C_{Bw}w′ + C_{w}Bw′ − BC_{w}w′ = 0 .

Eliminating C_{Bw}w′ in (7′) and (15) we obtain:

(16) − 3BC_{w}w′ + C_{w}Bw′ = 0 .

Since the first summand is symmetric in w and w′ we get

(17) C_{w}Bw′ = C_{w′}Bw = C_{Bw}w′ .

Now using (17) we can rewrite (15) as:

(15′) 2BC_{w}w′ − C_{w}Bw′ = 0 .

The equations (15′) and (16) show that BC_{w}w′ = C_{w}Bw′ = C_{Bw}w′ = 0 and hence also B·C = 0. This implies D_{w_{0}} = 0, by (3). Now we can rewrite (1-14) as:

Bw_{0} = C_{w_{0}} = D_{w_{0}} = 0  (4.6)
B · C = B · D = 0  (4.7)
B^{2} = 0  (4.8)
C_{Bw} = C_{w}B = BC_{w} = 0  (4.9)
−8D_{Bw} + 3C_{w} · C = 0  (4.10)
C_{w} · D = 0  (4.11)
\frac{3}{2}C_{C_{w}w′} + 2D_{w,w′} · B = 0  (4.12)
D_{C_{w}w′} + D_{w,w′} · C = 0  (4.13)
D_{w,w′} · D = 0 .  (4.14)

Now to proceed further we decompose K := ker B = W_{0} ⊕ W′, where W_{0} = ker ω|K and W′ is a (nondegenerate) complement. Let us denote by W_{1} a complement to K in W such that ω(W′, W_{1}) = 0. Then W_{0} + W_{1} is the ω-orthogonal complement to the B-invariant nondegenerate subspace W′. This shows that BW_{1} ⊂ (W_{0} + W_{1}) ∩ K = W_{0}. Moreover since W_{1} ∩K = 0 the map B : W_{1} → W_{0} is injective and hence dim W_{1} ≤ dim W_{0}. On the other hand dim W_{1} ≥ dim W_{0}, since W_{0} is an isotropic subspace of the symplectic vector space W_{0} + W_{1}. This shows that B : W_{1} → W_{0} is an isomorphism.

Lemma 3 C ∈ S^{3}K and D ∈ S^{4}K. Proof: Since W_{0} = BW the equation (4.9) shows that C_{W_{0}} = 0, which proves the first statement. From (4.10) and the identity

(C_{x} · C)_{y} = [C_{x}, C_{y}] − C_{C_{x}y}  (4.15)

we obtain

D_{Bx,y} + D_{By,x} = \frac{3}{8}((C_{x} · C)_{y} + (C_{y} · C)_{x}) = −\frac{3}{4}C_{C_{x}y} .  (4.16)

The equation B · D = 0 (4.7) reads:

0 = (B · D)_{x,y} = [B, D_{x,y}] − D_{Bx,y} − D_{x,By} .

Using this (4.12) yields:

D_{Bx,y} + D_{By,x} = [B, D_{x,y}] = −D_{x,y} · B = \frac{3}{4}C_{C_{x}y} .  (4.17)

Now from (4.16) and (4.17) we obtain that

0 = C_{C_{x}y}z = C_{z}C_{x}y

for all x, y, z ∈ W. This implies [C_{x}, C_{y}] = 0 for all x, y ∈ W and hence

C_{x} · C = 0  (4.18)

for all x ∈ W, by (4.15). Finally this shows that D_{W_{0}} = 0 by (4.10). This proves the second statement. Lemma 4 D_{x,y}C_{z} = C_{z}D_{x,y} = 0 for all x, y, z ∈ W. Proof: Using (4.13) we compute:

D_{x,y}C_{z}w = D_{C_{z}w,x}y = −(D_{z,w} · C)_{x}y = −([D_{z,w}, C_{x}]y − C_{D_{z,w}x}y) .  (4.19)

From (4.11) we get: 0 = (C_{x}·D)_{z,w}y = [C_{x}, D_{z,w}]y−D_{C_{x}z,w}y−D_{z,C_{x}w}y = C_{x}D_{z,w}y−D_{z,w}C_{x}y−D_{y,w}C_{x}z−D_{z,y}C_{x}w , and hence:

[D_{z,w}, C_{x}]y = −D_{y,w}C_{x}z − D_{z,y}C_{x}w ,

and

C_{D_{z,w}x}y = C_{y}D_{z,w}x = D_{z,w}C_{x}y + D_{x,w}C_{y}z + D_{z,x}C_{y}w .

Now we eliminate the CD-terms from (4.19) arriving at:

D_{x,y}C_{z}w = (D_{y,w}C_{x}z + D_{z,y}C_{x}w + D_{z,w}C_{x}y + D_{x,w}C_{y}z + D_{z,x}C_{y}w) .  (4.20)

Considering all the permutations of (x, y, z, w) we get 6 homogeneous linear equations for the 6 terms of equation (4.20) with the matrix:

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

This is the matrix of the endomorphism −2 Id + e ⊗ e in the arithmetic space ℝ^{6}, where e = e_{1} + … + e_{6}; (e_{i}) the standard basis. It has eigenvalues (4, −2, −2, −2, −2, −2). This shows that the matrix is nondegenerate and proves the lemma.

For a symmetric tensor T ∈ S^{d}W we denote by

Σ_{T} := span{T_{x_{1},x_{2},...,x_{d−2}}x_{d−1}|x_{1}, x_{2}, … x_{d−1} ∈ W } ⊂ W

the support of T.

Lemma 5 The supports of the tensors B ∈ S^{2}W, C ∈ S^{3}W and D ∈ S^{4}W admit the following inclusions

Σ_{B} + Σ_{C} ⊂ kerB ∩ kerC ∩ kerD , Σ_{D} ⊂ kerB ∩ kerC .

Moreover Σ_{B} + Σ_{C} is isotropic and ω(Σ_{D}, Σ_{B} + Σ_{C}) = 0.

Proof: The first statement follows from B^{2} = BC_{x} = BD_{x,y} = C_{x}B = C_{x}C_{y} = C_{x}D_{y,z} = D_{x,y}B = D_{x,y}C_{z} = 0 for all x, y, z ∈ W. The second statement follows from the first and the definition of support, e.g. if z = C_{x}y ∈ Σ_{C} and w ∈ Σ_{B} + Σ_{C} + Σ_{D} ⊂ kerC we compute:

ω(z, w) = ω(C_{x}y, w) = −ω(y, C_{x}w) = 0 .

Lemma 6 The Lie algebra D_{W,W} ⊂ S^{2}W ≅ sp(W) is solvable.

Proof: This follows from Proposition 6, since D ∈ S^{4}W satisfies (4.4) and hence defines a complex hyper-Kähler symmetric space with holonomy Lie algebra D_{W,W}. It also follows from the solvability of S_{E,E} as we show now. In terms of the decomposition E = P + W + Q an endomorphism

S_{x,y} = (λp^{4} + p^{3}w_{0} + p^{2}B + pC + D)_{x,y} = (p^{2}B + pC + D)_{x,y} = B(x, y)p^{2} − pC_{x}y + D_{x,y} ,

where x, y ∈ W, is represented by

( \begin{matrix}0 & −(C_{x}y)^{t} & B(x, y) \\ 0 & D_{x,y} & −C_{x}y \\ 0 & 0 & 0 \end{matrix} ) .

Since the Lie algebra S_{E,E} is solvable this implies that the Lie algebra D_{W,W}, which corresponds to the induced representation of S_{E,E} on P^{⊥}/P ≅ W, is also solvable.

Lemma 7

ω(w_{0}, Σ_{B} + Σ_{C} + Σ_{D}) = 0 .

Proof: Note that w_{0} ∈ kerB ∩ kerC ∩ kerD, due to equations (1-3) and (4.7). This implies the lemma. In fact, if e.g. y = Bx ∈ Σ_{B} then

ω(w_{0}, y) = ω(w_{0}, Bx) = −ω(Bw_{0}, x) = 0 ,

which shows that ω(w_{0}, Σ_{B}) = 0

Now to finish the proof of Theorem 3 we will use induction on the dimension dim E = 2n. If n = 1 the (solvable) holonomy algebra 𝔥 is a proper subalgebra of S^{2}E ≅ sl(2, ℂ). Without loss of generality we may assume that either

a) 𝔥 = ℂp^{2} or

b) 𝔥 = ℂpq or

c) 𝔥 = ℂp^{2} + ℂpq, where (p, q) is a symplectic basis of E. In the all three cases the Lie algebra S_{E,E} = 𝔥 ⊂ ℂp^{2} + ℂpq and hence

S = λp^{4} + μp^{3}q .

In the cases b) and c) we have that pq ∈ 𝔥 and since

pq · S = \frac{1}{2}(p ⊗ q + q ⊗ p) · S = −2λp^{4} − μp^{3}q = 0

it follows that S = 0. In the case a) from S_{E,E} = 𝔥 = ℂp^{2} we have that S = λp^{4}. This tensor is invariant under 𝔥 = ℂp^{2} and belongs to the fourth symmetric power of the Lagrangian subspace ℂp ⊂ E. This establishes the first step of the induction. Now by induction using equation (4.14) and Lemma 6 we may assume that Σ_{D} is isotropic. Now Lemma 5 and Lemma 7 show that ℂw_{0} + Σ_{B} + Σ_{C} + Σ_{D} is isotropic and hence is contained in some Lagrangian subspace E_{+} ⊂ E. This implies that S ∈ S^{4}E_{+}.

Now we give a necessary and sufficient condition for a symmetric manifold M = M_{S}, S ∈ S^{4}E_{+}, to have no flat de Rham factor.

Proposition 7 The complex hyper-Kähler symmetric space M_{S}, S ∈ S^{4}E_{+}, has no flat de Rham factor if and only if the support Σ_{S} = E_{+}.

Proof: If M = M_{S} = G/K has a flat factor M_{0}, such that M = M_{1} × M_{0}, then this induces a decomposition E = E^{1} ⊕ E^{0} and S ∈ S^{4}E_{1}; hence Σ_{S} ⊂ E_{1} ∩ E_{+} ≠ E_{+}. Conversely let S ∈ S^{4}E_{+}, assume that E_{+}^{1} = Σ_{S} ⊂ E_{+} is a proper subspace and choose a complementary subspace E_{+}^{0}. We denote by E_{−}^{1} and E_{−}^{0} the annihilator of ωE_{+}^{0} and ωE_{+}^{1} respectively. Let us denote E^{1} = E_{+}^{1} ⊕ E_{−}^{1}, E^{0} = E_{+}^{0} ⊕ E_{−}^{0}, 𝔪^{1} = H ⊗ E^{1} and 𝔪^{0} = H ⊗ E^{0}. Then E^{0}, E^{1} ⊂ E are ω-nondegenerate complementary subspaces and 𝔪^{0}, 𝔪^{1} ⊂ 𝔪 = T_{o}M are g-nondegenerate complementary subspaces. Since S ∈ S^{4}E_{+}^{1} the Lie algebra 𝔤 = 𝔥 + 𝔪 = (𝔥 + 𝔪^{1}) ⊕ 𝔪^{0} has the Abelian direct summand 𝔪^{0}, see (3.1), which gives rise to a flat factor M^{0} ⊂ M_{S} = M^{1} × M^{0}.

Theorem 4 Any simply connected complex hyper-Kähler symmetric space without flat de Rham factor is isomorphic to a complex hyper-Kähler symmetric space of the form M_{S}, where S ∈ S^{4}E_{+} and E_{+} ⊂ E is a Lagrangian subspace of the complex symplectic vector space E = ℂ^{2n}. Moreover there is a natural 1-1 correspondence between simply connected complex hyper-Kähler symmetric spaces without flat factor up to isomorphism and orbits O of the group Aut(E, ω, E_{+})|E_{+} = {A ∈ Sp(E)|AE_{+} = E_{+}}|E_{+} ≅ GL(E_{+}) ≅ GL(n, ℂ) on the space S^{4}E_{+} such that Σ_{S} = E_{+} for all S ∈ O.

Proof: This is a corollary of Theorem 2, Theorem 3 and Proposition 7.

Let M = G/K be a simply connected complex hyper-Kähler symmetric space without flat factor. By Theorem 3 and Proposition 7 it is associated to quartic polynomial S ∈ S^{4}E_{+} with support Σ_{S} = E_{+}. Now we describe the Lie algebra aut(M_{S}) of the full group of automorphisms, i.e. isometries which preserve the hypercomplex structure, of M_{S}.

Theorem 5 Let M_{S} = G/K be as above. Then the full automorphism algebra is given by

aut(M_{S}) = aut(S) + 𝔤 ,

where A ∈ aut(S) = {B ∈ gl(E_{+})|B · S = 0} acts on 𝔤 = 𝔥 + 𝔪 as follows. It preserves the decomposition and acts on 𝔥 = S_{E,E} by

[A, S_{x,y}] = S_{Ax,y} + S_{x,Ay}

for all x, y ∈ E and on 𝔪 = H ⊗ E by

[A, h ⊗ e] = h ⊗ Ae ,

where gl(E_{+}) is canonically embedded into sp(E).

Proof: By (the complex version of) Proposition 1 it is sufficient to determine the centralizer 𝔠 of sp(1, ℂ) in the full isotropy algebra 𝔥̃ = aut(R) ⊃ sp(1, ℂ) ⊕ 𝔥. Equation (2.2) shows that

𝔠 = {Id ⊗ A| A ∈ sp(E) , A · S = [A, S(·, ·)] − S(A·, ·) − S(·, A·) = 0} .

From A · S = 0 we obtain that the commutator [A, S_{x,y}] = S_{Ax,y} + S_{x,Ay} for all x, y ∈ E and AΣ_{S} = AE_{+} ⊂ E_{+}. This implies 𝔠 = aut(S).

5 Classification of hyper-Kähler symmetric spaces

Using the description of complex hyper-Kähler symmetric spaces given in Theorem 4 we will now classify (real) hyper-Kähler symmetric spaces. Recall that a simply connected pseudo-Riemannian manifold is called indecomposable if it is not a Riemannian product of two pseudo-Riemannian manifolds. Any simply connected pseudo-Riemannian manifold can be decomposed into the Riemannian product of indecomposable pseudo-Riemannian manifolds. By Wu’s theorem [W] a simply connected pseudo-Riemannian manifold is indecomposable if and only if its holonomy group is weakly irreducible, i.e. has no invariant proper nondegenerate subspaces. Therefore it is sufficient to classify (real) hyper-Kähler symmetric spaces with indecomposable holonomy.

Let (M = G/K, g, J_{α}) be a hyper-Kähler symmetric space associated to a symmetric decomposition (2.1). The complexified tangent space of M is identified with 𝔪^{ℂ} = H ⊗ E, the tensor product of to complex symplectic vector spaces with quaternionic structure j_{H} and j_{E} such that ρ = j_{H} ⊗ j_{E} is the complex conjugation of 𝔪^{ℂ} with respect to 𝔪. By Theorem 1 it is defined by a quartic polynomial S ∈ S^{4}E satisfying the conditions of the theorem. Moreover the holonomy algebra 𝔥 acts trivially on H and is identified with the real form of the complex Lie algebra S_{E,E} ⊂ sp(E) given by 𝔥 = span{S_{je,e′} − S_{e,je′}|e, e′ ∈ E} = {A ∈ S_{E,E}|[A, j_{E}] = 0} ⊂ sp(E)^{j_{E}}.

The quartic polynomial S defines also a complex hyper-Kähler symmetric space M^{ℂ} = G^{ℂ}/K^{ℂ}, which is the complexification of M = G/K. By Theorem 3, S ∈ S^{4}L for some Lagrangian subspace L ⊂ E. Recall that the symplectic form ω = ω_{E} together with the quaternionic structure j = j_{E} define a Hermitian metric γ = γ_{E} = ω_{E}(·, j_{E}·) of (real) signature (4k, 4l), n = k + l, which coincides with the signature of the pseudo-Riemannian metric g (we normalize γ_{H} = ω_{H}(·, j_{H}·) to be positive definite). We may decompose γ-orthogonally L = L^{0} ⊕ L^{+} ⊕ L^{−}, such that γ vanishes on L^{0} is positive definite on L^{+} and negative definite on L^{−}.

Lemma 8 (i) jL^{0} = L^{0} and

(ii) L^{+} + L^{−} + jL^{+} + jL^{−} ⊂ E is an ω-nondegenerate and γ-nondegenerate 𝔥-invariant subspace (with trivial action of 𝔥).

Proof: We show first that L + jL^{0} is ω-isotropic and hence L + jL^{0} = L since L is Lagrangian. Indeed L ⊃ L^{0} is ω-isotropic and also jL^{0} because ω is j-invariant. So it suffices to remark that ω(L, jL^{0}) = 0:

ω(L, jL^{0}) = γ(L, L^{0}) = 0 .

This implies that jL^{0} ⊂ L. Since γ(L, jL^{0}) = −ω(L, L^{0}) = 0, we conclude that jL^{0} ⊂ kerγ|L = L^{0}. This proves (i).

To prove (ii) it is sufficient to check that the subspace L^{+} + L^{−} + jL^{+} + jL^{−} ⊂ E is nondegenerate with respect to γ, since it is j-invariant. First we remark that γ is positive definite on L^{+} and jL^{+} and negative definite on L^{−} and jL^{−}, due to the j-invariance of γ: γ(jx, jx) = γ(x, x), x ∈ E. So to prove (ii) it is sufficient to check that jL^{+}⊕jL^{−} is γ-orthogonal to the γ-nondegenerate vector space L^{+} + L^{−}:

γ(L^{+} + L^{−}, jL^{+} + jL^{−}) = ω(L^{+} + L^{−}, L^{+} + L^{−}) = 0 .

By Theorem 1 the quartic polynomial S must satisfy the reality condition [S_{je,e′} −S_{e,je′}, j] = 0. Now we describe all such polynomials.

The quaternionic structure j on E is compatible with ω, i.e. ω(jx, jy) = \overline{ω(x, y)} for all x, y ∈ E and it induces a real structure (i.e. an antilinear involution) τ := j ⊗ j ⊗ ⋯ ⊗ j on all even powers S^{2r}E ⊂ E ⊗ E ⊗ ⋯ ⊗ E. For S ∈ S^{2r}E and x_{1}, ⋯, x_{2r} ∈ E we have

(τS)(x_{1}, ⋯, x_{2r}) = \overline{S(jx_{1}, ⋯, jx_{2r})} .

Note that the fixed point set sp(E)^{τ} = {A ∈ sp(E)|[A, j] = 0} ≅ sp(k, l).

Proposition 8 Let (E, ω, j) be a complex symplectic vector space with a quaternionic structure j such that ω(jx, jy) = \overline{ω(x, y)} for all x, y ∈ E. Then a quartic polynomial S ∈ S^{4}E satisfies the reality condition [S_{je,e′} − S_{e,je′}, j] = 0 if and only if S ∈ (S^{4}E)^{τ} = span{T + τT |T ∈ S^{4}E}.

Proof: The reality condition for S ∈ S^{4}E can be written as

[S_{jx,jy} + S_{x,y}, j]z = 0

for all x, y, z ∈ E. Contracting this vector equation with jw ∈ E by means of ω and using the compatibility between j and ω we obtain the equivalent condition

0 = −ω(jw, [S_{jx,jy} + S_{x,y}, j]z)
= S(jx, jy, jz, jw) − \overline{S(jx, jy, z, w)} + S(x, y, jz, jw) − \overline{S(x, y, z, w)} .  (5.1)

Now putting x = y = z = w = u we obtain:

0 = S(ju, ju, ju, ju) − \overline{S(ju, ju, u, u)} + S(u, u, ju, ju) − \overline{S(u, u, u, u)}

and putting x = iu and y = z = w = u we obtain:

0 = −iS(ju, ju, ju, ju) − i\overline{S(ju, ju, u, u)} + iS(u, u, ju, ju) + i\overline{S(u, u, u, u)} .

Comparing these two equations we get S(ju, ju, juj, u) = \overline{S(u, u, u, u)}, i.e. S = τS. This shows that the reality condition implies that S ∈ (S^{4}E)^{τ}. Conversely the condition S = τS can be written as

S(jx, jy, jz, jw) = \overline{S(x, y, z, w)} for all x, y, z, w ∈ E .

Changing z → jz and w → jw in this equation we obtain

S(jx, jy, z, w) = \overline{S(x, y, jz, jw)} for all x, y, z, w ∈ E .

These two equations imply (5.1) and hence the reality condition.

Now we are ready to classify simply connected hyper-Kähler symmetric spaces. We will show that the following construction gives all such symmetric spaces.

Let (E, ω, j) be a complex symplectic vector space of dimension 2n with a quaternionic structure j such that ω(jx, jy) = \overline{ω(x, y)} for all x, y ∈ E and E = E_{+} ⊕ E_{−} a j-invariant Lagrangian decomposition. Such a decomposition exists if and only if the Hermitian form γ = ω(·, j·) has real signature (4m, 4m), where dim_{ℂ} E = 2n = 4m. Then any polynomial S ∈ (S^{4}E_{+})^{τ} = S^{4}E_{+} ∩(S^{4}E)^{τ} satisfies the condition (4.4) and the reality condition, by Proposition 8. Hence by Theorem 1 it defines a (real) simply connected hyper-Kähler symmetric space M_{S} with Abelian holonomy algebra 𝔥 = (S_{E_{+},E_{+}})^{τ} = S_{E_{+},E_{+}} ∩ (S^{2}E)^{τ} = span{S_{je,e′} − S_{e,je′}|e, e′ ∈ E} ⊂ sp(E)^{τ} ≅ sp(m, m).

Theorem 6 Any simply connected hyper-Kähler symmetric space without flat de Rham factor is isomorphic to a hyper-Kähler symmetric space of the form M_{S}, where S = T +τT, T ∈ S^{4}E_{+} and E_{+} ⊂ E is a j-invariant Lagrangian subspace of the complex symplectic vector space E with compatible quaternionic structure j. A hyper-Kähler symmetric space of the form M_{S} has no flat factor if and only if it complexification has no flat factor, which happens if and only if the support Σ_{S} = E_{+}. Moreover there is a natural 1-1 correspondence between simply connected hyper-Kähler symmetric spaces without flat factor up to isomorphism and orbits O of the group Aut(E, ω, j, E_{+})|E_{+} = {A ∈ Sp(E)|[A, j] = 0, AE_{+} = E_{+}}|E_{+} ≅ GL(m, ℍ) on the space (S^{4}E_{+})^{τ} such that Σ_{S} = E_{+} for all S ∈ O.

Proof: Let M be a simply connected hyper-Kähler symmetric space. We first assume that it is indecomposable. Then the holonomy algebra 𝔥 is weakly irreducible. By Theorem 1, M = M_{S} for some quartic polynomial S ∈ S^{4}E satisfying (4.4) and the reality condition (3.2). By Proposition 8 the reality condition means that S ∈ (S^{4}E)^{τ}. On the other hand, by Theorem 3 S ∈ S^{4}L for some Lagrangian subspace L of E. Now the weak irreducibility of 𝔥 and Lemma 8 imply that L = L^{0} is j-invariant. This proves that S ∈ (S^{4}E_{+})^{τ}, where E_{+} = L = L^{0} is a j-invariant Lagrangian subspace of E. This shows that M is obtained from the above construction. Any simply connected hyper-Kähler symmetric space M without flat factor is the Riemannian product of indecomposable ones, say M = M_{1} ×M_{2} ×⋯×M_{r}, and we may assume that M_{i} = M_{S_{i}}, S_{i} ∈ S^{4}E_{i}. Therefore M is associated to the quartic polynomial S = S_{1} ⊕ S_{2} ⊕ ⋯ ⊕ S_{r} ∈ S^{4}E, E = E_{1} ⊕ E_{2} ⊕ ⋯ ⊕ E_{r}. Moreover S satisfies (4.4) and (3.2) if the S_{i} satisfy (4.4) and (3.2). This shows that any simply connected hyper-Kähler symmetric space is obtained from the above construction.

It is clear that the complexification M_{S}^{ℂ} has a flat factor if M_{S} has a flat factor. Conversely let us assume that M_{S}^{ℂ} has a flat factor, hence Σ_{S} ⊂ E_{+} is a proper subspace. Since jΣ_{S} = jS_{E,E}E = S_{E,E}jE = S_{E,E}E = Σ_{S} there exists a j-invariant complementary subspace E′_{+} in E_{+}. Denote by E′_{−} the annihilator of Σ_{S} in E_{−} then E′ = E′_{+} ⊕ E′_{−} is an ω-nondegenerate and j-invariant subspace of E on which the holonomy 𝔥^{ℂ} = S_{E,E} ⊂ S^{2}Σ_{S} acts trivially. Then the corresponding real subspace (H ⊗ E′)^{ρ} ⊂ 𝔪 = (H ⊗ E)^{ρ} is a g-nondegenerate subspace on which the holonomy 𝔥 acts trivially. By Wu’s theorem [W] it defines a flat de Rham factor.

Now the last statement follows from the corresponding statement in Theorem 1.

Corollary 2 Any hyper-Kähler symmetric space without flat factor has signature (4m, 4m). In particular its dimension is divisible by 8.

Corollary 3 Let M = M_{S} be a complex hyper-Kähler symmetric space without flat factor associated with a quartic S ∈ S^{4}E_{+}, where E_{+} ⊂ E is a Lagrangian subspace. It admits a real form if and only if there exists a quaternionic structure j on E compatible with ω preserving E_{+} such that τS = S, where τ is the real structure on S^{4}E induced by j. In particular dim_{ℂ} M has to be divisible by 8.

Let M = G/K be a simply connected hyper-Kähler symmetric space without flat factor. By Theorem 6 it is associated to a quartic polynomial S ∈ (S^{4}E_{+})^{τ} with support Σ_{S} = E_{+}. Now we describe the Lie algebra aut(M_{S}) of the full group of automorphisms, i.e. isometries which preserve the hypercomplex structure of M_{S}.

Theorem 7 Let M_{S} = G/K be as above. Then the full automorphism algebra is given by

aut(M_{S}) = aut(S) + 𝔤 ,

where aut(S) = {A ∈ gl(E_{+})| [A, j] = 0 , A · S = 0} acts on

𝔤 = 𝔥 + 𝔪 , 𝔥 = {A ∈ S_{E,E}|[A, j] = 0} = span{S_{jx,y} − S_{x,jy}|x, y ∈ E} , 𝔪 = (H ⊗ E)^{ρ}

as in Theorem 5.

Proof: The proof is similar to that of Theorem 5.

6 Low dimensional hyper-Kähler symmetric spaces

6.1 Complex hyper-Kähler symmetric spaces of dimension ≤ 8

Dimension 4 Assume that M is a simply connected complex hyper-Kähler symmetric space of dimension 4. Applying Theorem 4 we conclude that M = M_{S} for some S ∈ S^{4}E_{+}, where E_{+} ⊂ E is a one-dimensional subspace E_{+} = ℂe. This proves:

Theorem 8 There exists up to isomorphism only one non-flat simply connected complex hyper-Kähler symmetric space of dimension 4: M = M_{S} associated with the quartic S = e^{4}.

Dimension 8 Any eight-dimensional simply connected complex hyper-Kähler symmetric space is associated with a quartic S ∈ S^{4}E_{+}, where E_{+} ⊂ E is a Lagrangian subspace of E = ℂ^{4}. We denote by (e, e′) a basis of E_{+}.

Theorem 9 Eight-dimensional simply connected complex hyper-Kähler symmetric spaces are in natural 1-1 correspondence with the orbits of the group CO(3, ℂ) = ℂ^{∗} ·SO(3, ℂ) on the space S_{0}^{2}ℂ^{3} of traceless symmetric matrices. The complex hyper-Kähler symmetric space associated with a traceless symmetric matrix A is the manifold M_{S(A)}, where S(A) ∈ S^{4}ℂ^{2} is the quartic polynomial which corresponds to A under the SO(3, ℂ)-equivariant isomorphism S_{0}^{2}ℂ^{3} ≅ S_{0}^{2} ∧^{2} ℂ^{3} ≅ S_{0}^{2}S^{2}ℂ^{2} = S^{4}ℂ^{2}.

The classification of SO(3, ℂ)-orbits on S_{0}^{2}ℂ^{3} was given by Petrov [P] in his classification of Weyl tensors of Lorentzian 4-manifolds.

Proof: By Theorem 4 the classification of eight-dimensional simply connected complex hyper-Kähler symmetric spaces reduces to the description of orbits of the group GL(E_{+}) = GL(2, ℂ) on S^{4}ℂ^{2} ⊂ S^{2}S^{2}ℂ^{2}. Fixing a volume form σ on ℂ^{2} we can identify S^{2}ℂ^{2} with sp(1, ℂ) ≅ so(3, ℂ). Then the Killing form B is an SL(2, ℂ)-invariant and we have the GL(2, ℂ)-invariant decomposition: S^{2}S^{2}ℂ^{2} = S_{0}^{2}S^{2}ℂ^{2} ⊕ ℂB. The action of SL(2, ℂ) on S^{2}ℂ^{2} is effectively equivalent to the adjoint action of SO(3, ℂ). The problem thus reduces essentially to the determination of the orbits of SO(3, ℂ) on S_{0}^{2}ℂ^{3}.

6.2 Hyper-Kähler symmetric spaces of dimension ≤ 8

By Corollary 3 the minimal dimension of non-flat hyper-Kähler symmetric spaces is 8.

Theorem 10 Eight-dimensional simply connected hyper-Kähler symmetric space are in natural 1-1 correspondence with the orbits of the group ℝ^{+} · SO(3) on the space S_{0}^{2}ℝ^{3} of traceless symmetric matrices. The hyper-Kähler symmetric space associated with a traceless symmetric matrix A is the manifold M_{S(A)}, where S(A) ∈ (S^{4}ℂ^{2})^{τ} is the quartic polynomial which corresponds to A under the SO(3)-equivariant isomorphism S_{0}^{2}ℝ^{3} ≅ (S^{4}ℂ^{2})^{τ}.

References

[BD] M.L. Barberis, I. Dotti-Miatello, Hypercomplex structures on a class of solvable Lie groups, Quart. J. Math. Oxford II. Ser. 47 (1996), no. 188, 389–404.

[B1] M. Berger, Classification des espaces homogènes symétriques irréductibles, C.R. Acad. Sci. Paris 240 (1955), 2370–2372.

[B2] M. Berger, Structure et classification des espaces homogènes symétriques à groupe d’isométries semi-simple, C. R. Acad. Sci. Paris 241 (1955), 1696–1698.

[Bo] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comen., New Ser. 66 (1997), no. 2, 151-201.

[CP] M. Cahen, M. Parker, Sur des classes d’espaces pseudo-riemanniens symétriques, Bull. Soc. Math. Bely. 22 (1970), 339–354.

[CW] M. Cahen, N. Wallach, Lorentzian symmetric spaces, Bull. Amer. Math. Soc. 76 (1970), 585–591.

[F] A.S. Fedenko, Spaces defined by Lie group endomorphisms (Φ-spaces), Collection of articles dedicated to the memory of German Fedorovič Laptev, Trudy Geometr. Sem. 4 (1973), 231–267.

[J] D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Diff. Geom. 35 (1992), 743–761.

[K] V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press, 1990.

[MR1] A. Medina, P. Revoy, Algèbres de Lie et produit scalaire invariant, Ann. Sci. Ec. Norm. Super. IV. Ser. 18 (1985), 553-561.

[MR2] A. Medina, P. Revoy, Algèbres de Lie orthogonales. Modules orthogonaux, Commun. Algebra 21 (1993), no.7, 2295-2315.

[O-V] A.L. Onishchik, E.B. Vinberg (Eds.), Lie groups and Lie algebras III, Encyclodaedia

of Mathematical Sciences 41, Springer, 1994.

[P] A.Z. Petrov, Einstein spaces, Pergamon, Oxford, 1969.

[SSTVP] Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen Extended super-symmetric σmodels on group manifolds, Nuclear Phys. B 308 (1988), 662–698.

[W] H. Wu, On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291–311.