Pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy
aa r X i v : . [ m a t h . DG ] J a n Pseudo-Riemannian almost quaternionichomogeneous spaces with irreducible isotropy
V. Cort´es and B. Meinke Department of Mathematicsand Center for Mathematical PhysicsUniversity of HamburgBundesstraße 55, D-20146 Hamburg, [email protected] Department of MathematicsUniversity of D¨usseldorfUniversit¨atsstraße 1, Raum 25.22.03.62D-40225 D¨usseldorf, [email protected]
Abstract
We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index4 and an H -irreducible isotropy group are locally isometric to a pseudo-Riemannianquaternionic K¨ahler symmetric space if the dimension is at least 16. In dimension 12we give a non-symmetric example. Keywords: Homogeneous spaces, symmetric spaces, pseudo-Riemannian manifolds,almost quaternionic structuresMSC classification: 53C26, 53C30, 53C35, 53C50
In [AZ] Ahmed and Zeghib studied pseudo-Riemannian almost complex homogeneousspaces of index 2 with a C -irreducible isotropy group. They showed that these spaces arealready pseudo-K¨ahler if the dimension is at least 8. If furthermore the Lie algebra of theisotropy group is C -irreducible then the space is locally isometric to one of five symmetricspaces.There are two different quaternionic analogues of K¨ahler manifolds, namely hyper-K¨ahlerand quaternionic K¨ahler manifolds. In the first case, the complex structure is replaced bythree complex structures assembling into a hyper-complex structure ( I, J, K ), in the sec-ond by the more general notion of a quaternionic structure Q ⊂ End
T M on the underlyingmanifold M . Riemannian as well as pseudo-Riemannian quaternionic K¨ahler manifoldsare Einstein and therefore of particular interest in pseudo-Riemannian geometry.In [CM] the authors investigated the hyper-complex analogue of the topic studied byAhmed and Zeghib, namely pseudo-Riemannian almost hyper-complex homogeneous spaces1f index 4 with an H -irreducible isotropy group. It turned out that these spaces of dimen-sion greater or equal than 8 are already locally isometric to the flat space H ,n except indimension 12, where non-symmetric examples exist.In this article we study the quaternionic analogue, that is we consider pseudo-Riemannianalmost quaternionic homogeneous spaces of index 4 with an H -irreducible isotropy group.The main result of our analysis is the following theorem. Theorem 1.1.
Let ( M, g, Q ) be a connected almost quaternionic pseudo-Hermitian man-ifold of index and dim M = 4 n + 4 ≥ , such that there exists a connected Lie subgroup G ⊂ Iso(
M, g, Q ) acting transitively on M . If the isotropy group H := G p , p ∈ M ,acts H -irreducibly, then ( M, g, Q ) is locally isometric to a quaternionic K¨ahler symmetricspace. Here Iso(
M, g, Q ) denotes the subgroup of the isometry group Iso(
M, g ) which preservesthe almost quaternionic structure Q of M . A consequence of the theorem is that thehomogeneous space M itself is quaternionic K¨ahler and locally symmetric. Notice thatpseudo-Riemannian quaternionic K¨ahler symmetric spaces have been classified in [AC]. InSection 3.2 we show, by construction of a non-symmetric example in dimension 12, thatthe hypothesis dim M ≥
16 in Theorem 1.1 cannot be omitted. Moreover, we classifyin Proposition 3.1 all examples with the same isotropy algebra h = so (1 , ⊕ so (3) ⊂ so (1 , ⊕ so (4) ⊂ gl ( R , ⊗ R ) ∼ = gl (12 , R ) in terms of the solutions of a system of fourquadratic equations for six real variables.The strategy of the proof of Theorem 1.1 is as follows. We consider the H -irreducibleisotropy group H as a subgroup of Sp(1 , n )Sp(1) and classify the possible Lie algebras.Then we consider the covering G/H of M = G/H and show by taking into account thepossible Lie algebras that it is a reductive homogeneous space. Finally, we show that theuniversal covering ˜ M is a symmetric space. The invariance of the fundamental 4-formunder G then implies that the symmetric space is quaternionic K¨ahler. Acknowledgments.
This work was partly supported by the German Science Foundation(DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the EarlyUniverse. n )Sp(1) Lemma 2.1 (Goursat’s theorem) . Let g , g be Lie algebras. There is a one-to-one cor-respondence between Lie subalgebras h ⊂ g ⊕ g and quintuples Q ( h ) = ( A, A , B, B , θ ) ,with A ⊂ g B ⊂ g Lie subalgebras, A ⊂ A , B ⊂ B ideals and θ : A/A → B/B is aLie algebra isomorphism.Proof: Let h ⊂ g ⊕ g be a Lie subalgebra and denote by π i : g ⊕ g → g i , i = 1 , A := π ( h ) ⊂ g , B := π ( h ) ⊂ g , A := ker( π | h ) and B := ker( π | h ). It is not hard to see that A and B can be identified with ideals in A B respectively. Now we can define a map ˜ θ : A → B/B as follows. For X ∈ A take any Y ∈ B such that X + Y ∈ h and define ˜ θ ( X ) := Y + B . It is easy to checkthat this map is well defined. Its kernel is A so ˜ θ induces a Lie algebra isomorphism θ : A/A → B/B . This defines a map h
7→ Q ( h ).Conversely, a quintuple Q = ( A, A , B, B , θ ) as above defines a Lie subalgebra h = G ( Q ) ⊂ g ⊕ g by setting h := { X + Y ∈ A ⊕ B | θ ( X + A ) = Y + B } . It is not hard to see that the maps G and Q are inverse to each other. (cid:3) We will use the following two classification results for H -irreducible subgroups of Sp(1 , n ). Theorem 2.1 ([CM, Corollary 2.1]) . Let H ⊂ Sp(1 , n ) be a connected and H -irreducibleLie subgroup. Then H is conjugate to one of the following groups: ( i ) SO (1 , n ) , SO (1 , n ) · U(1) , SO (1 , n ) · Sp(1) if n ≥ , ( ii ) SU(1 , n ) , U(1 , n ) , ( iii ) Sp(1 , n ) , ( iv ) U = { A ∈ Sp(1 , | A Φ = Φ A } ∼ = Spin (1 , with Φ = (cid:18) −
11 0 (cid:19) if n = 1 . Proposition 2.1 ([CM, Proposition 2.4]) . Let H ⊂ Sp(1 , n ) be an H -irreducible subgroup.Then one of the following is true. ( i ) H is discrete. ( ii ) H = U(1) · n +1 or H = Sp(1) · n +1 . ( iii ) H is H -irreducible. ( iv ) n = 1 and H is one of the groups SO (1 , , SO (1 , · U(1) , SO (1 , · Sp(1) or S = (cid:26) e ibt (cid:18) cosh( at ) sinh( at )sinh( at ) cosh( at ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t ∈ R (cid:27) , for some non-zero real numbers a, b . We denote by π : sp (1 , n ) ⊕ sp (1) → sp (1 , n ) and π : sp (1 , n ) ⊕ sp (1) → sp (1) thecanonical projections. Proposition 2.2.
Let n ≥ and H ⊂ Sp(1 , n )Sp(1) be an H -irreducible closed subgroup.Then the Lie algebra h is one of the following: ( i ) h = h ⊕ c with h ∈ {{ } , so (1 , n ) } , c ⊂ sp (1) · n +1 ⊕ sp (1) and π ( c ) = sp (1) · n +1 , π ( c ) = sp (1) , c ∩ sp (1 , n ) = { } , c ∩ sp (1) = { } , ( ii ) h = h ⊕ c with h ∈ {{ } , so (1 , n ) , su (1 , n ) } , c ⊂ u (1) · n +1 ⊕ u (1) and π ( c ) = u (1) · n +1 , π ( c ) = u (1) , c ∩ sp (1 , n ) = { } , c ∩ sp (1) = { } , iii ) h = h ⊕ c where h ⊂ sp (1 , n ) is one of the following Lie algebras sp (1 , n ) , u (1 , n ) , su (1 , n ) , so (1 , n ) ⊕ sp (1) · n +1 , so (1 , n ) ⊕ u (1) · n +1 , so (1 , n ) , sp (1) · n +1 , u (1) · n +1 , { } , and c ⊂ sp (1) is { } , u (1) or sp (1) .Proof: The idea is to apply Goursat’s theorem (Lemma 2.1) to h ⊂ sp (1 , n ) ⊕ sp (1). TheLie subalgebras A , A , B and B are given by π ( h ), h ∩ sp (1), π ( h ) and h ∩ sp (1). Let p : Sp(1 , n ) × Sp(1) → Sp(1 , n ) be the natural projection. Notice that H ⊂ Sp(1 , n )Sp(1)is H -irreducible if and only if p ( ˆ H ) ⊂ Sp(1 , n ) is H -irreducible, where ˆ H is the preimageof H under the two-fold covering Sp(1 , n ) × Sp(1) → Sp(1 , n )Sp(1). By Proposition 2.1and Theorem 2.1 we know that p ( ˆ H ) is either discrete or ( p ( ˆ H )) is one of the followingsubgroups of Sp(1 , n ):Sp(1 , n ) , U(1 , n ) , SU(1 , n ) , SO (1 , n ) (Sp(1) · n +1 ) , SO (1 , n ) (U(1) · n +1 ) , SO (1 , n ) , Sp(1) · n +1 , U(1) · n +1 . Since dp = π we immediately obtain all possibilities for π ( h ). Furthermore h ∩ sp (1 , n )is an ideal of the Lie algebra π ( h ). We can read off from the above list a decompositionof π ( h ) into ideals, which gives us all possibilities for h ∩ sp (1 , n ). The resulting list ofpairs ( A, A ) is displayed in a table below.On the other side there are only three Lie subalgebras of sp (1), namely sp (1) itself, u (1)and { } . It follows that π ( h ) is one of these three. Again, h ∩ sp (1) is an ideal of π ( h ).It follows that the only possibilites for h ∩ sp (1) are the same as for π ( h ).By Goursat’s theorem we have a Lie algebra isomorphism θ : A/A → B/B . Since weknow all possibilities for B and B , it follows that A/A is isomorphic to sp (1), u (1) or { } . Therefore we need to consider all possibilities for A and A , as listed in the followingtable, and keep only those for which A/A is isomorphic to sp (1), u (1) or { } . A A sp (1 , n ) sp (1 , n ) { } su (1 , n ) ⊕ u (1) su (1 , n ) ⊕ u (1) su (1 , n ) u (1) { } su (1 , n ) su (1 , n ) { } so (1 , n ) ⊕ sp (1) so (1 , n ) ⊕ sp (1) so (1 , n ) sp (1) { } so (1 , n ) ⊕ u (1) so (1 , n ) ⊕ u (1) so (1 , n ) u (1) { } o (1 , n ) so (1 , n ) { } sp (1) sp (1) { } u (1) u (1) { }{ } { } If B/B ∼ = sp (1) then B = sp (1) and B = { } . The possibilities for ( A, A ) are( so (1 , n ) ⊕ sp (1) · n +1 , so (1 , n )) and ( sp (1) · n +1 , { } ) . This gives us case ( i ). Analogously we get the remaining Lie algebras in ( ii ) and ( iii ). (cid:3) Lemma 3.1 ([CM, Lemma 3.1]) . Let n ≥ and α ∈ ⊗ V ∗ , where V = H ,n is consideredas real vector space. If α is SO (1 , n ) -invariant, then α = 0 . Remark 3.1.
The SO (1 , n ) -invariant elements of ⊗ V ∗ are in one-to-one correspondenceto the SO (1 , n ) -equivariant bilinear maps from V × V to V . It follows from Lemma . that the corresponding bilinear maps also vanish.Proof of Theorem 1.1: Let ρ : H → GL( T p M ) be the isotropy representation. We iden-tify H with its image ρ ( H ). Since H preserves the metric g and the almost quaternionicstructure Q , we can consider H as a subgroup of Sp(1 , n )Sp(1).In our first step we consider the covering G/H of M = G/H and show that it is a re-ductive homogeneous space, i.e. there exists an H -invariant subspace m ⊂ g such that g = h ⊕ m .We apply Proposition 2.2 to H . The existence of a subspace m is clear if h is one ofthe semi-simple Lie algebras in the list. If h is one of the abelian Lie algebras containedin u (1) · n +1 ⊕ u (1), then the closure of Ad( H ) ⊂ GL( g ) is compact and hence thereexists an Ad( H )-invariant subspace m . The remaining Lie algebras in the list have theform h = s ⊕ z where s is semi-simple containing so (1 , n ) and z is the non-trivial centre.Then g decomposes into g = s ⊕ z ⊕ m with respect to the action of s . If we consider theaction of s on m ∼ = H ,n as a complex representation, then m is either C -irreducible ordecomposes into two C -irreducible subrepresentations. Since the elements of z commutewith s , they preserve the sum of all non-trivial s -submodules, which is precisely m . Thuswe have shown that G/H is a reductive homogeneous space.Next we show that g = h ⊕ m is a symmetric Lie algebra. It is sufficient to show that[ m , m ] ⊂ h . We restrict the Lie bracket [ · , · ] to m × m and denote its projection to m by β . It is an antisymmetric bilinear map which is Ad( H )-equivariant. Since m ∼ = H ,n , we5an consider β as an element of ⊗ ( H ,n ) ∗ . It is also H Zar -invariant, where H Zar denotesthe Zariski closure. Since H Zar is an algebraic group, it has only finitely many connectedcomponents, see [Mi]. Now we show that ( H Zar ) is non-compact.Assume that ( H Zar ) is compact. Since H Zar has only finitely many connected com-ponents it follows that H Zar is compact and therefore contained in a maximal compactsubgroup of Sp(1 , n )Sp(1). Hence, H Zar is conjugate to a subgroup of (Sp(1) × Sp( n ))Sp(1)but this contradicts the H -irreducibility of H Zar . So we have shown that ( H Zar ) is non-compact.Now we apply Proposition 2.2 to H Zar . Since H Zar is non-compact we see from the listthere that ( H Zar ) contains SO (1 , n ). Hence, β is SO (1 , n )-equivariant. Since n ≥ β vanishes. This shows that g = h ⊕ m is a symmetricLie algebra and that the universal covering ˜ M = ˜ G/ ˜ G p of M is a symmetric space. Thefundamental 4-form Ω of ˜ M is ˜ G -invariant and since ˜ M is a symmetric space Ω is parallel.In particular Ω is closed. It is known that for dimension ≥
12 an almost quaternionicHermitian manifold is quaternionic K¨ahler if dΩ = 0, see [S]. This shows that ˜ M is fur-thermore a quaternionic K¨ahler manifold. Summarizing, we have shown that M is locallyisometric to a quaternionic K¨ahler symmetric space. (cid:3) In Theorem 1.1 we did not consider the dimension 12. This is because the argumentsused in the proof to show that M is a reductive homogeneous space do not apply in thisdimension, although still SO (1 , n ) ⊂ H Zar holds. In fact, the proof relies on Lemma 3.1which holds for dimension 4 n + 4 ≥
16. If dim M = 12 then n = 2 and then there existnon-trivial anti-symmetric bilinear forms H , × H , → H , which are invariant underSO (1 , g = h ⊕ m where h is a Lie algebra of the list in Proposition 2.2. The pair ( g , h ) defines a simplyconnected homogeneous space M = G/H where G is a connected and simply connectedLie group with Lie algebra g and H is the closed connected Lie subgroup of G with Liealgebra h .Let h = so (1 , ⊕ c with c = { ( X · , X ) ∈ sp (1) · ⊕ sp (1) | X ∈ sp (1) } , see Proposi-tion 2.2 ( i ). Then we consider the vector space direct sum g := h ⊕ m with m = H , anddefine a Lie bracket on g in the following way. For elements A, B ∈ h we take the standardLie bracket of h , i.e. [ A, B ] = AB − BA . Then we define [ A, x ] = − [ x, A ] = Ax for A ∈ h and x ∈ m . Note that, as an h -module, we can decompose m = H , = R , ⊗ H = R , ⊗ R ,where the action of so (1 ,
2) is by the defining representation on the first factor and trivialon the second and the action of c ∼ = so (3) ⊂ so (4) is trivial on the first factor and by thestandard four-dimensional representation H = R ⊕ Im H = R ⊕ R on the second. Finallywe have to define the Lie bracket for elements in m = R , ⊗ R .Let K : R , → so (1 ,
2) be an isomorphism of Lie algebras where R , is endowed with6he Lorentzian cross product, ι : sp (1) → c , X → X · + X , and let η be the standardLorentz metric on R , . Furthermore denote h· , ·i the standard inner product on R . Let x = u ⊗ p , y = v ⊗ q ∈ R , ⊗ R and write p = p + ~p , q = q + ~q , where p , q ∈ R and ~p, ~q ∈ Im H = R . We set[ x, y ] = h ~p, ~q i · K ( u × v ) − η ( u, v ) ι ( ~p × ~q ) | {z } ∈ h + u × v ( p q − h ~p, ~q i ) | {z } ∈ R , ⊂ H , = m , where ~p × ~q is the Euclidian cross product in Im H = sp (1) and u × v the Lorentzian crossproduct in R , . This extends the partially defined bracket to an anti-symmetric bilinearmap [ · , · ] : g × g → g , which satisfies the Jacobi-identity. Hence g becomes a Lie algebra.We claim that ( g , h ) is not a symmetric pair. In fact, every h -invariant complement m ′ of h in g contains R , ⊗ R (there is no other equivalent h -submodule in g ) and thus we seefrom the formula for the bracket that [ m ′ , m ′ ] * h .For a general classification of the homogeneous spaces with h = so (1 , ⊕ c we needto classify all the Lie algebra structures on the vector g = h ⊕ R , ⊗ R such that the Liebracket restricts to the Lie bracket of h and to the given representation of h on R , ⊗ R .For this one has to describe all the h -invariant tensors of Λ m ∗ ⊗ g ∼ = Λ m ∗ ⊗ h ⊕ Λ m ∗ ⊗ m which satisfy the Jacobi-identity. With the above notation, these bilinear maps have thefollowing form[ x, y ] = ( a · p q + b h ~p, ~q i ) · K ( u × v ) + η ( u, v ) ( c · ι ( ~p × ~q ) + d ( p ~q − q ~p ))+ u × v · (cid:16) a · p q + a · h ~p, ~q i + a p ~q + q ~p ) (cid:17) , where a, b, c, d, a , a , a ∈ R . The bracket satisfies the Jacobi-identity if and only if thefollowing equations hold 0 = d, a + a a − a , (1)0 = b + 2 c + a a , (2)0 = b + a a − a a , (3)0 = − ba aa . (4)Summarizing we obtain the following proposition. Proposition 3.1.
Every solution ( a, b, c, a , a , a ) of the quadratic system (1)-(4) definesa connected and simply connected homogeneous almost quaternionic pseudo-Hermitianmanifold G/H with isotropy algebra h = so (1 , ⊕ so (3) ⊂ so (1 , ⊕ so (4) ⊂ gl ( R , ⊗ R ) ∼ = gl (12 , R ) . Conversely, every such homogeneous space arises by this construction. The above example corresponds to a = 0, b = 1, c = − , d = 0, a = 1, a = − a = 0. 7 eferences [AC] D.V. Alekseevsky and V. Cort´es, Classification of pseudo-Riemannian symmetricspaces of quaternionic K¨ahler type , Amer. Math. Soc. Transl. (2) 213 (2005), 33-62.[AZ] A. ben Ahmed and A. Zeghib,
On homogeneous Hermite-Lorentz spaces , Preprint, arXiv:math.DG/1106.4145v1 (2011).[CM] V. Cort´es and B. Meinke,
Pseudo-Riemannian almost hypercomplex homogeneousspaces with irreducible isotropy , Preprint, arXiv:1606.06486 (2016).[Mi] J. Milnor,
On the Betti numbers of real varieties , Proc. Amer. Math. Soc. (1964)275-280. Press, 1974, 49-87.[S] A. F. Swann, HyperK¨ahler and Quaternionic K¨ahler Geometry , Math. Ann.289