aa r X i v : . [ m a t h . DG ] J a n SPACES ADMITTING HOMOGENEOUS G -STRUCTURES FRANK REIDEGELD
Abstract.
We classify all seven-dimensional spaces which admit a ho-mogeneous cosymplectic G -structure. The motivation for this classi-fication is that each of these spaces is a possible principal orbit of aparallel Spin(7)-manifold of cohomogeneity one. Introduction
The aim of this article is to classify all spaces which admit a homogeneouscosymplectic G -structure. Moreover, we not only classify the spaces them-selves, but also the transitive group actions which preserve at least onecosymplectic G -structure.In the literature, many of such spaces are known. Friedrich, Kath, Mo-roianu, and Semmelmann [11] classify all simply connected, compact spaceswhich admit a homogeneous nearly parallel G -structure. The product ofa space with a homogeneous SU (3)-structure and a circle carries a canoni-cal homogeneous G -structure. The spaces from the article of Cleyton andSwann [7] which admit a homogeneous SU (3)-structure should therefore bementioned in this context, too.One reason for our interest in this kind of spaces is that any principal orbitof a parallel Spin(7)-manifold of cohomogeneity one carries a homogeneouscosymplectic G -structure. Conversely, any homogeneous cosymplectic G -structure can be extended to a parallel Spin(7)-manifold of cohomogeneityone. A discussion of these facts can be found in Hitchin [12]. The aim ofthis article is to prove the following theorem: Theorem 1. (1)
Let
G/H be a seven-dimensional, compact, connected, G -homogeneous space which admits a G -invariant G -structure. Weassume that G/H is a product of a circle and another homogeneousspace and that G acts almost effectively on G/H . Furthermore, weassume that G and H are both connected. In this situation, G , H ,and G/H are up to a covering one of the spaces from the table below:
G H G/HU (1) { e } T SU (2) × U (1) { e } S × T SU (2) × U (1) { e } S × S × S SU (2) × U (1) U (1) S × S × S SU (2) × U (1) U (1) SU (2) /U (1) × T SU (2) × U (1) SU (2) S × S × S SU (3) × U (1) SU (2) S × T SU (3) × U (1) U (1) SU (3) /U (1) × S Sp (2) × U (1) Sp (1) × U (1) CP × S G × U (1) SU (3) S × S Conversely, any of the above spaces admits a G -invariant G -structure. (2) Let G , H , and G/H satisfy the same conditions as in (1) with thesingle exception that
G/H is not a product of a circle and anotherhomogeneous space. In this situation, G , H , and G/H are up to acovering one of the spaces from the table below:
G H G/HSU (3) U (1) N k,l with k, l ∈ Z SO (5) SO (3) V , Sp (2) Sp (1) S SO (5) SO (3) B SU (2) U (1) Q , , SU (3) × U (1) U (1) N k,l with k, l ∈ Z SU (3) × SU (2) SU (2) × U (1) M , , SU (3) × SU (2) SU (2) × U (1) N , Sp (2) × U (1) Sp (1) × U (1) S Sp (2) × Sp (1) Sp (1) × Sp (1) S SU (4) SU (3) S Spin (7) G S Conversely, any of the above spaces admits a G -invariant G -struc-ture. (3) Any of the spaces
G/H from (1) or (2) even admits a G -invariantcosymplectic G -structure. In table (2), N k,l denotes an Aloff-Wallach space, V , denotes the Stiefelmanifold of all orthonormal pairs in R , and B is the seven-dimensionalBerger space. In the fourth, fifth, and sixth row of table (1) and in the fifthand seventh row of table (2), the embedding of H into G has to be specialin order to make G/H a space which admits a G -invariant G -structure. PACES ADMITTING HOMOGENEOUS G -STRUCTURES 3 The details of those embeddings are described in Section 5 and 6. In theother cases, the information in the above tables is sufficient to determinethe embedding of H into G .From the theorem it follows that either G/H is a product of a circle anda space which admits a homogeneous SU (3)-structure or that it cannot bedecomposed into factors of lower dimension. We remark that we not onlyprove the existence of a homogeneous cosymplectic G -structure on each ofthe spaces but also the existence of cosymplectic G -structures which areinvariant under any of the transitive group actions. The space ( SU (2) × SU (2)) /U (1) × T admits a homogeneous G -structure but seems not to bementioned in the literature before.The proof of Theorem 1 consists of three steps: After two introductory sec-tions, we classify all connected Lie subgroups of G . This is necessary, sincein the situation of the theorem H can be embedded into G . In Section 5and 6, we determine all G/H which admit a G -invariant, but not necessar-ily cosymplectic G -structure. Finally, we have to prove the existence of a G -invariant cosymplectic G -structure on all of the spaces which we havefound. This will be done in Section 7.2. The group G Before we classify the connected subgroups of G , we collect some facts onthis group. For a more comprehensive introduction into this issue, see Baez[2] or Bryant [3].The group G can be defined with help of the octonions: We recall thata normed division algebra is a pair ( A, h· , ·i ) of a real, not necessarily as-sociative algebra with a unit element and a scalar product which satisfies h x · y, x · y i = h x, x ih y, y i for all x, y ∈ A . There exists up to isomorphisms ex-actly one eight-dimensional normed division algebra, namely the octonions O .The quaternions H are a subalgebra of O . We fix an octonion ǫ in theorthogonal complement of H such that k ǫ k = 1. We call ( x , . . . , x ) :=(1 , i, j, k, ǫ, iǫ, jǫ, kǫ ) the standard basis of O . Let Im( O ) := span(1) ⊥ be the imaginary space of O . The map ω : Im( O ) × Im( O ) × Im( O ) → R ω ( x, y, z ) := h x · y, z i (1)is a three-form. From now on, we denote dx i ∧ . . . ∧ dx i k shortly by dx i ...i k .With this notation, we have:(2) ω = dx + dx − dx + dx + dx + dx − dx . FRANK REIDEGELD
Remark . The multiplication table of O is uniquely determined by thecoefficients of ω . Let ǫ ′ be an octonion with the same properties as ǫ . Sincethere exists an automorphism of O which is the identity on H and maps ǫ to ǫ ′ , ω is independent of the choice of ǫ .We are now able to define the Lie group G : Definition and Lemma 2.2. (1) Any automorphism ϕ of O satisfies ϕ (Im( O )) ⊆ Im( O ) and thus can be identified with a map fromIm( O ) onto itself. G is defined as the stabilizer group of ω orequivalently as the automorphism group of O .(2) The Lie algebra of G we denote by g .(3) The seven-dimensional representation which is induced by the actionof G on Im( O ) by automorphisms we call the standard representa-tion of G .A proof of the fact that the stabilizer of ω is the same as the automorphismgroup of O can be found in Bryant [3]. Later on, we work with the Hodgedual ∗ ω ∈ V Im( O ) ∗ of ω which is taken with respect to h· , ·i and theorientation which makes ( x , . . . , x ) positive:(3) ∗ ω = − dx + dx + dx + dx − dx + dx + dx . Finally, we fix a Cartan subalgebra t of g , which we will need for our explicitcalculations. With respect to the standard basis of Im( O ), t is the followingset of matrices:(4) t := λ − λ λ − λ λ + λ − λ − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ , λ ∈ R Some remarks on G -structures In this section, we introduce the different types of G -structures which weconsider in this article. We refer the reader to Bryant [3] or the books ofJoyce [13] and Salamon [15] for further facts on these structures. A G -structure can be defined as a three-form which is at each point stabilized by G : PACES ADMITTING HOMOGENEOUS G -STRUCTURES 5 Definition 3.1.
Let M be a seven-dimensional manifold and ω be a three-form on M with the following property: For any p ∈ M there exists aneighborhood U of p and vector fields X , . . . , X on U such that(5) ω q ( X i , X j , X k ) = ω ( x i , x j , x k ) ∀ q ∈ U, i, j, k ∈ { , . . . , } . The ω on the right hand side of the above formula is the three-form (2) and x i , x j , and x k are elements of the standard basis of O . In this situation, ω is called a G -structure on M and the pair ( M, ω ) is called a G -manifold .On any G -manifold ( M, ω ) there exist a metric g and a volume form volwhich are defined by:(6) g ( X, Y ) vol := −
16 ( X ⌊ ω ) ∧ ( Y ⌊ ω ) ∧ ω . We call g the associated metric and vol the associated volume form . g andvol induce a Hodge star operator ∗ : V ∗ T ∗ M → V ∗ T ∗ M and we thereforehave a four-form ∗ ω on M , which is invariant under the stabilizer G of ω .On the flat G -manifold ( R , ω ) this four-form coincides with (3). For ourconsiderations, we need the following types of G -structures: Definition 3.2. A G -manifold ( M, ω ) is called(1) parallel if dω = 0 and d ∗ ω = 0,(2) nearly parallel if there exists a λ ∈ R \ { } such that dω = λ ∗ ω andthus d ∗ ω = 0,(3) cosymplectic if d ∗ ω = 0.Further information on the different types of G -structures can be foundin the article by Fern´andez and Gray [10]. We will deal first of all withhomogeneous G -manifolds: Definition 3.3. A G -manifold ( M, ω ) is called ( G -)homogeneous if thereexists a transitive smooth action by a Lie group G which leaves ω invariant.In the above situation, M is G -equivariantly diffeomorphic to a quotient G/H . G can be chosen in such a way that it acts effectively on G/H andpreserves ω . H acts on the tangent space of G/H by its isotropy represen-tation. Since G acts on the tangent space as the stabilizer of ω and ω is G -invariant, we have proven the following lemma: Lemma 3.4.
Let
G/H be a seven-dimensional G -homogeneous space whichadmits a G -invariant G -structure. We assume that G acts effectively on G/H . In this situation, there exists a vector space isomorphism ϕ : T p G/H → R such that ϕHϕ − ⊆ G , where H is identified with its isotropy represen-tation and G with its seven-dimensional irreducible representation. FRANK REIDEGELD
The converse of the above lemma is also true:
Lemma 3.5.
Let
G/H be a seven-dimensional G -homogeneous space suchthat G acts effectively and there exists a vector space isomorphism ϕ : T p G/H → R with ϕHϕ − ⊆ G . In this situation, there exists a G -invariant G -structure on G/H . Proof : The action of G on the tangent bundle determines a G -invariant H -structure on G/H . Its extension to a principal bundle with structuregroup G is a G -invariant G -structure. (cid:3) Subgroups of G In this section, we classify all connected subgroups of G . First, we describeall of these subgroups explicitly. After that, we prove that the list which wehave found is complete.On page 4, we have described a Cartan subalgebra t of g . t is generatedby the two elements which satisfy ( λ , λ ) = (1 ,
0) and ( λ , λ ) = (0 , O ) splits with respect to the action of t into V C , ⊕ V C , ⊕ V C , ⊕ V R , .The subscripts denote the weights with which the two generators of t actand the superscript indicates if the submodule is complex or real. Since anyabelian subalgebra of g is conjugate to a subalgebra of t , we have finishedthe abelian case.Next, we describe the subgroups of G whose Lie algebra has an ideal oftype su (2). In an article by Cacciatori et al. [5], the authors introduce thefollowing Lie group homomorphism: ϕ : Sp (1) × Sp (1) → G ϕ ( h, k )( x + yǫ ) := hxh − + ( kyh − ) ǫ , (7)where x, y ∈ H and Sp (1), which is isomorphic to SU (2), is identified withthe unit quaternions. The kernel of ϕ is { (1 , , ( − , − } and its image thusis isomorphic to SO (4). The first factor of Sp (1) × Sp (1) acts irreducibly onIm( H ) and H ǫ and the second factor acts irreducibly on H ǫ and trivially onits orthogonal complement. The splitting of Im( O ) into irreducible 2 su (2)-modules therefore is V R , ⊕ V C , . Analogously to above, the subscripts ofthe modules denote the weights of the 2 su (2)-action with respect to thefirst and second summand. By a straightforward calculation, we can provethat the group Sp (1) which is diagonally embedded into Sp (1) × Sp (1) actsirreducibly on Im( H ) and Im( H ) ǫ and trivially on span( ǫ ). Im( O ) thus splitsinto 2 V R ⊕ V R with respect to that subgroup. PACES ADMITTING HOMOGENEOUS G -STRUCTURES 7 In his article ”Semisimple subalgebras of semisimple Lie algebras” [9], Dynkinproves the existence of another subalgebra of g which is isomorphic to su (2)and acts irreducibly on Im( O ) with weight 6. Since we do not need an ex-plicit description of that subalgebra, we simply state its existence.According to the non zero weights of their action on Im( O ), we denote thefour subalgebras of g which are isomorphic to su (2) by su (2) , , su (2) , su (2) , , and su (2) .By a short calculation, we see that the element of t with λ = 2 and λ = − su (2) and the element with λ = 0 and λ = 1 commuteswith su (2) , . g therefore contains a subalgebra of type su (2) ⊕ u (1) and asubalgebra of type su (2) , ⊕ u (1). Both of them are a direct sum of an idealof su (2) ⊕ su (2) ⊆ g and a one-dimensional subalgebra of the other ideal.The group of all automorphisms of O which fix i is a compact, connected,eight-dimensional Lie group. Its action on C is trivial and it acts irreduciblyon the orthogonal complement of C ⊆ O . These conditions force the groupto be isomorphic to SU (3).Our next step is to prove that there are up to conjugation by an element of G no further connected subgroups. g is a Lie algebra of rank 2 and twomaximal tori of a Lie group are always conjugate to each other. Therefore,further connected, abelian subgroups of G cannot exist.According to Dynkin [9], all semisimple subalgebras of g are isomorphic to su (2), 2 su (2), su (3), or g . Any of these algebras acts by the restriction ofthe adjoint representation on g . The weights of this action are computedin [9], too. The list Dynkin obtains is the same as our list of semisimpleLie subalgebras. Moreover, Dynkin [9] proves that his list is complete up toconjugation by elements of G .It remains to prove that there are no further subalgebras of type su (2) ⊕ u (1). Let x be a generator of the center. Since su (2) commutes with u (1),the action of x on Im( O ) has to be su (2)-equivariant. With help of thereal version of Schur’s Lemma, we are able to classify all su (2)-equivariantendomorphisms of Im( O ) for any embedding of su (2) into g . Since theaction of x on Im( O ) has to be a restricted automorphism of O , we canreduce the number of those endomorphisms even further. After that, we seethat all x ∈ g which commute with su (2) ( su (2) , ) are conjugate to thetwo matrices which we have already found. The conjugation is with respectto an element of the Lie subgroup of G which is associated to su (2) , ( su (2) ). By the same method, we are able to prove that no subalgebras oftype su (2) , ⊕ u (1) or su (2) ⊕ u (1) do exist. We finally have proven thefollowing theorem: Theorem 4.1.
Let H be a connected Lie subgroup of G . We denote theLie algebra of H by h . The irreducible action of G on Im ( O ) induces anaction of H on Im ( O ) . In this situation, h , H , and the action of H on FRANK REIDEGELD Im ( O ) are contained in the table below. Moreover, any two connected Liesubgroups of G whose action on Im ( O ) is equivalent are conjugate not onlyby an element of GL (7) but even by an element of G . h H Splitting of Im ( O ) into irreducible summands { } { e } u (1) U (1) V C a ⊕ V C b ⊕ V C − a − b ⊕ V R u (1) U (1) V C , ⊕ V C , ⊕ V C , ⊕ V R , su (2) SU (2) V C ⊕ V R su (2) SU (2) V R ⊕ V C su (2) SO (3) 2 V R ⊕ V R su (2) SO (3) V R su (2) ⊕ u (1) U (2) V C ⊕ V R w.r.t. su (2) su (2) ⊕ u (1) U (2) V R ⊕ V C w.r.t su (2)2 su (2) SO (4) V R , ⊕ V C , su (3) SU (3) V C , ⊕ V R , g G V R , The subscripts of the modules in the above table denote the weights of the H -action and the superscript indicates if the module is complex or real. Furtherdetails of the embeddings, in particular of those of U (2) and SO (4) into G ,we have described on the preceding pages.Remark . Most statements of Theorem 4.1 can also be proven by ele-mentary calculations which make use of the octonions. In order to keep ourpresentation of this issue short, we often made use of the results of Dynkin[9]. 5.
The reducible case
We divide the spaces which admit a homogeneous G -structure into twoclasses: Definition 5.1.
Let
G/H be a G -homogeneous space. We call G/H S -reducible if it is G -equivariantly covered by a product of a circle and anotherhomogeneous space. Otherwise, G/H is called S -irreducible .In this section, we classify all S -reducible spaces which admit a homoge-neous G -structure, and in the next section, we classify the S -irreducibleones. We will see that none of the S -irreducible spaces is covered by a prod-uct of lower-dimensional homogeneous spaces. The S -irreducible spaceswhich we will find are thus irreducible in the classical sense, too. PACES ADMITTING HOMOGENEOUS G -STRUCTURES 9 Throughout this article we denote the Lie algebra of G by g and the Liealgebra of H by h . In order to simplify our considerations, we assume that G/H is compact and that G is connected and acts almost effectively on G/H ,i.e. the subgroup of G which acts as the identity map is finite. Moreover,we classify the possible G/H and G only up to coverings. Before we startour classification, we collect some helpful facts:(1) We have dim g = dim h + 7. This fact reduces the number of possible g which we have to consider.(2) Since G/H is compact and G is a subgroup of the isometry group ofthe metric on G/H , G has to be compact, too. We thus can assumethat g is the direct sum of a semisimple and an abelian Lie algebra.(3) Since the roots of a semisimple Lie algebra are paired, we havedim k ≡ rank k ( mod
2) for any Lie algebra k of a compact Lie group.It follows from dim g = dim h + 7 that rank g rank h ( mod h can be considered as a subalgebra of g and thus is trivial or of rank1 or 2.(a) If h is of rank 1, rank g has to be even. The Cartan subalgebraof h has to act on the tangent space in the same way as a one-dimensional subalgebra of t on Im( O ). The maximal trivial h -submodule of the tangent space therefore is at most three-dimensional. It follows that the center z ( g ) of g is at mostthree-dimensional, too.(b) If rank h = 2, its Cartan subalgebra has to act as t on Im( O ).The maximal trivial h -submodule therefore is at most one-di-mensional and we have dim z ( g ) ≤
1. Moreover, rank g has tobe odd.(4) Let G = G ′ × U (1) and H = H ′ × U (1). If the second factors of bothgroups coincide, U (1) acts trivially on G/H . If the second factorof H is transversely embedded into G ′ × U (1), G/H is covered by G ′ /H ′ . Since the group which acts on G ′ /H ′ is G ′ × U (1) instead of G ′ , we consider this case as a new one. The only other case whichwe have to consider is where H ′ × U (1) ⊆ G ′ .(5) Let m be the orthogonal complement of h in g with respect to abiinvariant metric on g . The restriction of the adjoint action to amap h → gl ( m ) is equivalent to the action of h on the tangent spaceof G/H . This identification helps us to compute the action of h explicitly. In general, we omit that computation and give the readera description of the isotropy action instead.In this section, we assume that G = G ′ × U (1) and G/H = G ′ /H × S . G ′ /H admits a G ′ -invariant SU (3)-structure. We can prove by similar ar-guments as in Lemma 3.4 and 3.5 that our task reduces to classifying allsix-dimensional G ′ -homogeneous spaces G ′ /H with H ⊆ SU (3). The pos-sibilities for h are thus fewer than in the general situation. We prove ourclassification result, by considering each possible h ⊆ su (3) separately. For reasons of brevity, we mostly mention only those g which cannot be excludedby the above techniques. h = { } : In this case, G/H simply is a seven-dimensional, compact, con-nected Lie group. Up to coverings, the only groups of this kind are U (1) , SU (2) × U (1) , and SU (2) × U (1). h = u (1): Since dim g = 8 and spaces of type SU (3) /U (1) are irreducible,the only remaining possibilities for G are SU (2) × U (1) and SU (2) × U (1) .The first case can be excluded, since the center of G is too large. If G = SU (2) × U (1) , H is embedded into G by a map of type:(8) e iϕ (cid:18) (cid:18) e ik ϕ e − ik ϕ (cid:19) , (cid:18) e ik ϕ e − ik ϕ (cid:19) , e ik ϕ , e ik ϕ (cid:19) , where k , . . . , k ∈ Z . We repeat the argument from page 9 twice and seethat G/H is covered by S × S × S or that H ⊆ SU (2) . The action of H on the tangent space has at most two non zero weights. We comparethe weights of that action with the weights with which the one-dimensionalsubgroups of G act on Im( O ). After that, we see that we can assume | k | = | k | = 1. Since we obtain the same space for different choices of thesigns of k and k , we can even assume that k = k = 1. If ( k , k ) = (1 , G/H is diffeomorphic to S × S × S , and if ( k , k ) = (0 , G/H which is not covered by S × S × S . h = su (2): In this situation, G has to be a ten-dimensional compact Liegroup. On the one hand, dim z ( g ) has to be positive, since G/H is S -reducible. On the other hand, we have dim z ( g ) ≤
3. The only remainingpossibilities for G therefore are SU (2) × U (1) and SU (3) × U (1) .In the first case, we can embed H diagonally, i.e. by the map g ( g, g, g, H on the tangent space is the same as of su (2) , on Im( O )and G/H is diffeomorphic to S × S × S . If we had embedded H differ-ently, it would act as the identity on a four-dimensional subspace, which isimpossible.In the second case, there are two possible embeddings of H into SU (3):The first embedding is induced by he standard representation of SO (3) on R ⊆ C . The only elements of SU (3) which commute with all of SO (3)are the multiples of the identity. Since those elements are a discrete set,the action of H splits the tangent space into a trivial and a five-dimensionalirreducible submodule. There is no connected subgroup of G which acts inthis way on Im( O ) and we thus can exclude this case. The second embeddingis given by the following map from SU (2) to SU (3):(9) A (cid:18) A (cid:19) . PACES ADMITTING HOMOGENEOUS G -STRUCTURES 11 In this situation, H acts as V C ⊕ V R on the tangent space. Since su (2) actsin the same way, we have to put the space SU (3) /SU (2) × U (1) = S × T on our list. There are no further embeddings of H into SU (3). This canbe seen by considering the splitting of C into su (2)-submodules which isinduced by the embedding of su (2) into su (3). h = 2 u (1): Since rank h = 2 and G/H is S -reducible, we have dim z ( g ) = 1.The group G has to be nine-dimensional. Therefore, we can assume that G = SU (3) × U (1). Since su (3) ⊆ g and rank su (3) = rank g , any Cartansubalgebra of su (3) acts on C in the same way as t on span( j, k, . . . , kǫ ).We thus have to put the space G/H = SU (3) /U (1) × U (1) on our list. h = su (2) ⊕ u (1): For similar reasons as in the previous case, g has to bethe direct sum of a semisimple Lie algebra and u (1). With help of theclassification of the semisimple Lie algebras, we see that g = sp (2) ⊕ u (1). Wedescribe a possible G/H in detail. Sp (2) has a subgroup of type Sp (1) × U (1)which is given by:(10) H = (cid:26)(cid:18) h h (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) h ∈ H , h ∈ C , | h | = | h | = 1 (cid:27) . The Lie algebra of H acts in the same way on the tangent space of G/H ,which is diffeomorphic to CP × S , as su (2) ⊕ u (1) on Im( O ). The kernelof the isotropy representation of H is isomorphic to Z . Therefore, we havean effective action by ( Sp (1) × U (1)) / Z on the tangent space. Since thatgroup is isomorphic to U (2), our example does not contradict the fact that G contains no subgroup of type Sp (1) × U (1).We exclude the existence of further spaces of the above kind. If g = sp (2) ⊕ u (1) and h = sp (1) ⊕ u (1), either G/H is covered by the sphere Sp (2) /Sp (1),which is not reducible, or h ⊆ sp (2). There are three embeddings of sp (1)into sp (2), which is isomorphic to so (5). In the first case, sp (1) acts as so (3)on R ⊆ R , in the second case, it acts as su (2) on C ∼ = R ⊆ R , andin the last case, it acts irreducibly on R . The second embedding yieldsthe homogeneous space CP × S , which we have described above. If thesemisimple part of h was embedded by the first map, it would act as su (2) , on the tangent space. Since g has no subalgebra of type su (2) , ⊕ u (1),this is not possible. It follows from Schur’s Lemma that there is no non zeroelement of so (5) which commutes with the third possible embedding of thesemisimple part. This case can therefore be excluded, too. h = su (3): As in the previous cases, G has to be a product of a 14-dimensionalsemisimple Lie group G ′ and U (1). With help of the classification of thesemisimple Lie algebras, we conclude that G ′ is SU (3) × SU (2) or G . Inthe first case, SU (3) acts trivially on G/H and in the second case we obtain G /SU (3) × U (1), which is diffeomorphic to S × S . We can verify that H acts in the same way as the subgroup SU (3) of G on Im( O ). Therefore, we have to put this space on our list and have finally proven the first partof Theorem 1. Remark . There is a one-to-one correspondence between the spaces fromTheorem 1.1 and the six-dimensional spaces which admit a homogeneous SU (3)-structure. These spaces are considered by Cleyton and Swann [7],too. They obtain a list of homogeneous spaces which coincides with our listwith the single exception of SU (2) /U (1) × T , which seems to be missingin [7]. 6. The irreducible case
In this section, we classify the S -irreducible spaces which admit a homoge-neous G -structure. As in the previous section, we consider each possible h separately. h = { } : Since any seven-dimensional compact Lie group is covered by aproduct of a semisimple Lie group and a torus of positive dimension, we canexclude this case. h = u (1): In the previous section, we have already proven that if h = u (1)and G/H is S -irreducible, we necessarily have G = SU (3). G/H thereforeis an Aloff-Wallach space, i.e. a quotient N k,l := SU (3) /U (1) k,l with k, l ∈ Z and(11) U (1) k,l := e ikt e ilt
00 0 e − i ( k + l ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ∈ R . By an explicit calculation, we see that there exists a one-dimensional Liesubalgebra of t which acts in the same way on Im( O ) as the Lie algebra of U (1) k,l on the tangent space of N k,l . h = su (2): Since h has to be embedded into the semisimple part of g , z ( g )has to be trivial. Otherwise, G/H would not be S -irreducible. The onlyremaining possibility for g therefore is so (5). As we have mentioned before,there are three embeddings of su (2) into so (5), which are distinguished bythe splitting of R with respect to su (2):(1) R = V R ⊕ V R : In this situation, G/H is the Stiefel manifold V , = SO (5) /SO (3) of all orthonormal pairs in R . The action of su (2)splits the tangent space into 2 V R ⊕ V R . Since su (2) , splits Im( O )in the same way, V , admits an SO (5)-invariant G -structure.(2) R = V C ⊕ V R : If this is the case, G/H is covered by the seven-sphere Sp (2) /Sp (1). The action of Sp (1) splits the tangent spaceinto V C ⊕ VR . su (2) acts in the same way on Im( O ) and S thusadmits an Sp (2)-invariant G -structure. PACES ADMITTING HOMOGENEOUS G -STRUCTURES 13 (3) R = V R : If su (2) acts irreducibly on R , it also acts irreducibly onthe tangent space of G/H . Since the action of su (2) on Im( O ) isirreducible, too, we have found another space which admits a homo-geneous G -structure, namely the seven-dimensional Berger space B . h = 2 u (1): Since h is of rank 2, dim z ( g ) is either 0 or 1. If the center is one-dimensional, we have g = su (3) ⊕ u (1) and h is transversely embedded intothat direct sum. In this situation, G/H is covered by an Aloff-Wallach space N k,l , on which SU (3) × U (1) acts transitively. The group SU (3) acts as usualby left multiplication on N k,l . Moreover, a certain one-dimensional subgroupof the normalizer Norm SU (3) U (1) k,l acts on N k,l by right multiplication.This subgroup can be identified with the second factor of SU (3) × U (1).If g is semisimple, we can assume that g = 3 su (2). We describe the possibleembeddings of 2 u (1) into 3 su (2). A Cartan subalgebra of 3 su (2) is given by:(12) ix − ix iy − iy iz − iz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, y, z ∈ R We fix the biinvariant metric q ( X, Y ) := − tr( XY ) on 3 su (2). Let k k,l,m ,where k, l, m ∈ Z , be the one-dimensional subalgebra of 3 su (2) which isgenerated by the matrix with x = k , y = l , and z = m . Furthermore,let 2 u (1) k,l,m be the q -orthogonal complement of k k,l,m in the above Cartansubalgebra. Any connected two-dimensional Lie subgroup of SU (2) is con-jugate to a connected subgroup with a Lie algebra of type 2 u (1) k,l,m . Wedenote the quotient of SU (2) by that subgroup by Q k,l,m .By the action of the group ( Z ) ⋊ S of outer automorphisms of 3 su (2),we can change the signs and the order of ( k, l, m ) arbitrarily. We maytherefore assume without loss of generality that k ≥ l ≥ m ≥
0. Theisotropy representation of 2 u (1) k,l,m on the tangent space of Q k,l,m is withrespect to a suitable basis given by:(13) x − x y − y z − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xk + yl + zm = 0 By comparing (13) with the Cartan subalgebra (4) of g , we see that only Q , , admits an SU (2) -invariant G -structure. h = su (2) ⊕ u (1): Since rank su (2) ⊕ u (1) = 2, the center of g is at most one-dimensional. g has to be an eleven-dimensional Lie algebra and therefore iseither su (3) ⊕ su (2) or so (5) ⊕ u (1).We start with the first of the two cases. The semisimple part of h we denoteby su (2) ′ . In order to classify the homogeneous spaces which we can obtainin this situation, we have to describe the possible embeddings of su (2) ′ into su (3) ⊕ su (2). su (2) ′ ∩ su (3) has to be nontrivial. Otherwise, G would notact almost effectively on G/H . The projection of su (2) ′ onto su (3) thereforehas to be one of the two maps which we have described on page 10. If su (2) ′ acted irreducibly on C , the tangent space of G/H would contain afive-dimensional su (2) ′ -submodule. This follows by the same arguments ason page 10. Since no subalgebra of g acts in this way on Im( O ), su (2) ′ hasto split C into V C ⊕ V C . Next, we consider the projection of su (2) ′ ontothe second summand of su (3) ⊕ su (2). We first assume that su (2) ′ ⊆ su (3).In this situation, the center of h is without loss of generality generated by amatrix of type(14) ki ki
00 0 − ki li − li where k and l are integers. su (2) ′ acts as su (2) on the tangent space of G/H . There is up to conjugation only one one-dimensional subalgebra of g which commutes with su (2) . Therefore, the weights with which the centerof h acts on the tangent space are uniquely determined. By computing theaction of the above matrix on the tangent space, we see that we necessarilyhave l = ± k . The quotient G/H is in both cases up to an SU (3) × SU (2)-equivariant diffeomorphism the same and admits an SU (3) × SU (2)-invariant G -structure. We use the same notation as Castellani [6] and call our space M , , .If the projection of su (2) ′ onto the second summand of su (3) ⊕ su (2) isbijective, there is up to conjugation only one one-dimensional subalgebraof su (3) ⊕ su (2) which commutes with su (2) ′ . In this situation, G/H isdiffeomorphic to the exceptional Aloff-Wallach space N , . SU (3) acts ona gU (1) , by matrix multiplication from the left. Since U (1) , commuteswith S ( U (2) × U (1)) which is isomorphic to SU (2), gU (1) , gh − U (1) , defines a left action by SU (2) on N , which commutes with the action of SU (3). The isotropy group of the SU (3) × SU (2)-action which we havedefined is SU (2) × U (1). The embedding of its Lie algebra into su (3) ⊕ su (2) PACES ADMITTING HOMOGENEOUS G -STRUCTURES 15 is the same as we have described above. We thus have found another groupaction on N , which we have to include in our list.In both of the above two cases, there exists an element of G which is oforder two and acts trivially on G/H . For the same reasons as on page 11,the fact that G contains no subgroup of type SU (2) × U (1) therefore doesnot contradict the statement of our theorem.Next, we assume that g = so (5) ⊕ u (1). The embedding of su (2) ′ into so (5) has to be one of the three subalgebras which we have described onpages 11 and 12. Furthermore, the projection of z ( h ) onto so (5) should notbe trivial. If su (2) ′ was embedded by its five-dimensional representationinto so (5), there would be no element of so (5) left which commutes with su (2) ′ . Since this contradicts our statement on z ( h ), we can exclude this case.If su (2) ′ was embedded by its three-dimensional representation, it woulddecompose its complement in so (5) into 2 V R ⊕ VR . g has no subalgebraof type su (2) , ⊕ u (1) and we thus can exclude this case, too. The onlyremaining case is where su (2) ′ is embedded by its two-dimensional complexrepresentation. Since z ( h ) has to commute with su (2) ′ , its projection onto so (5) has to be an element of the second summand of su (2) ′ ⊕ su (2), which weidentify with the Lie subalgebra so (4) of so (5). If h ⊆ so (5), we obtain thespace CP × S , which we already have considered in the previous section.If this is not the case, G/H is covered by S , which is equipped with anaction of Sp (2) × U (1). h = 2 su (2): Since dim h = 6, the dimension of g has to be 13. There isno non zero element of Im( O ) on which the subalgebra 2 su (2) of g actstrivially. Therefore, z ( g ) has to be trivial. The only remaining possibilityfor g is so (5) ⊕ su (2).It follows from Lemma 3.4 and Theorem 4.1 that h has to decompose thetangent space into V R , ⊕ V C , . Let ı : 2 su (2) → so (5) ⊕ su (2) be theembedding of h into g , π : so (5) ⊕ su (2) → so (5) be the projection onthe first summand, and π : so (5) ⊕ su (2) → su (2) be the projection onthe second one. The tangent space contains a submodule of type V C , onlyif ( π ◦ ı )(2 su (2)) is the standard embedding of so (4) into so (5). The firstsummand of 2 su (2) has to act irreducibly on a three-dimensional submoduleof the tangent space and we therefore can assume that(15) ( π ◦ ı )( x, y ) = x ∀ x, y ∈ su (2) . We are now able to describe
G/H explicitly. Let S ⊆ H be the seven-sphere. Sp (2) acts on S from the left by matrix multiplication. We identify Sp (1) with the group of all unit quaternions. Since the scalar multiplicationon a quaternionic vector space acts from the right, scalar multiplicationwith h − where h ∈ Sp (1) defines a left action of Sp (1) on S . We thus haveconstructed a transitive Sp (2) × Sp (1)-action on S . The isotropy group of this action is Sp (1) × Sp (1) and the isotropy action has the properties whichwe have demanded above. Analogously to the case where H = SU (2) × U (1),the kernel of the isotropy representation of Sp (1) × Sp (1) is Z and the groupwhich acts effectively on the tangent space is in fact ( Sp (1) × Sp (1)) / Z ,which is isomorphic to SO (4). h = su (3): G has to be a Lie group of dimension 15 which contains SU (3).With help of the classification of the compact Lie groups, we see that G iscovered either by a product of SU (3) and a seven-dimensional Lie group orby SU (4). In the first case, G would not act almost effectively on G/H . Inthe second case,
G/H is covered by S . h = g : For similar reasons as above, we have g = so (7). Therefore, G/H iscovered by the seven-dimensional sphere Spin(7) /G and we have completedthe proof of Theorem 1.2. Remark . Friedrich, Kath, Moroianu, and Semmelmann [11] have classi-fied all spaces which admit a homogeneous nearly parallel G -structure. Inparticular, the authors prove that all spaces from Theorem 1.2 admit such a G -structure. In the table of our theorem, we also have listed all transitivegroup actions on those spaces which preserve a G -structure. On the sphere S , for example, there are G -structures which are invariant under Spin(7), SU (4), Sp (2) × Sp (1), Sp (2) × U (1), or Sp (2). We remark that some ofthe Aloff-Wallach spaces are diffeomorphic or homeomorphic to each other,although they are not SU (3)-equivariantly diffeomorphic. This phenome-non is discussed by Kreck and Stolz [14]. Their results prove that on thesame space there can exist G -structures which are preserved by differenttransitive Lie group actions.7. Existence of the cosymplectic G -structures In the previous two sections, we have classified all spaces which admit ahomogeneous G -structure. The aim of this section is to prove that a tran-sitive group action which leaves at least one G -structure invariant alsoleaves a cosymplectic G -structure invariant. We prove this fact by a case-by-case analysis. Although most of this work has already been done by otherauthors, there are still some cases left open.Since any nearly parallel G -structure is also cosymplectic, the article ofFriedrich et al. [11] answers our question for many subcases of the irreduciblecase. More precisely, we only have to consider those irreducible spaces onwhich we have more than one transitive group action.Let S ⊆ O be the unit sphere. We equip the tangent space Im( O ) of 1 ∈ O with the canonical G -structure ω from page 3. By the action of Spin(7), wecan extend ω to a nearly parallel G -structure on all of S . Since we have Sp (2) ⊆ SU (4) ⊆ Spin(7), ω is invariant with respect to the action of thethree groups. In [11], the authors describe a homogeneous nearly parallel PACES ADMITTING HOMOGENEOUS G -STRUCTURES 17 G -structure on S . The associated metric on S is the squashed one andits isometry group is Sp (2) × Sp (1). Since the G -structure is homogeneous,it has to be at least Sp (2)-invariant. We assume that the second factor of Sp (2) × Sp (1) does not preserve the G -structure. In that situation, thereexists a one-dimensional subgroup of Sp (1) which generates a continuousfamily of nearly parallel G -structures but preserves the associated metric.Any nearly parallel G -structure induces a Killing spinor and the dimensionof the space of all Killing spinors thus is at least two. Since it is known (cf.[11]) that this dimension is in fact one, the G -structure is Sp (2) × Sp (1)-and in particular Sp (2) × U (1)-invariant. All in all, we have found for eachtransitive action on S an invariant cosymplectic G -structure.Next, we consider the Aloff-Wallach spaces. Cvetiˇc et al. [8] have proventhat any Aloff-Wallach space admits two SU (3)-invariant nearly parallel G -structures, which coincide for k = − l . It is known (cf. [11]) that theisometry group of the associated metric is SU (3) × U (1). Since the space ofall Killing spinors is one-dimensional (cf. [8], [11]), we can conclude by thesame arguments as above that both G -structures are not only SU (3)- butalso SU (3) × U (1)-invariant.The nearly parallel G -structure on N , which is considered in [11] is pre-served by SU (3) × SU (2). Since SU (3) ⊆ SU (3) × U (1) ⊆ SU (3) × SU (2),that G -structure is invariant with respect to all of the three group actionsfrom Theorem 1.2.We proceed to the reducible case. Butruille [4] has proven that the only six-dimensional manifolds which admit a homogeneous nearly K¨ahler structureare S , CP , SU (3) /U (1) , and S × S . These four manifolds have alsobeen considered by B¨ar [1], since they carry a real Killing spinor. The groupswhich preserve the nearly K¨ahler structure on the first three spaces are G , Sp (2), and SU (3). In [1] it is also proven that S × S admits a nearlyK¨ahler structure which is invariant under an SU (2) -action. The isotropygroup of this action is SU (2), which is embedded as the diagonal subgroupby(16) g ( g, g, g ) . We denote the metric, the real two-form, and the complex (3 , SU (3)-structure by g , α , and θ . Furthermore, we denotethe real (imaginary) part of θ by θ Re ( θ Im ). We have dα = 3 λ θ Re and dθ Im = − λ α ∧ α for a λ ∈ R \ { } , since the four spaces are nearly K¨ahler.These equations are discussed in more detail by Hitchin [12]. On a productof a circle and a nearly K¨ahler manifold of real dimension six, we can definea G -structure by ω := α ∧ dt + θ Im . Here, ” t ” denotes the coordinate ofthe circle. By a straightforward calculation, it follows that d ∗ ω = 0. All in all, we have proven our statement for the last three spaces from Theorem1.1 and for all three actions on S × S × S .On the torus T , we have the flat G -structure, which is of course cosym-plectic. On C × T ( C × T ), there exists a flat Spin(7)-structure Ω. It ispreserved by the action of SU (2) × U (1) ( SU (3) × U (1) ), where the firstfactor acts on C ( C ) and the second one by translations on the torus. Theprincipal orbits of this action, which is of cohomogeneity one, are S × T ( S × T ). Ω induces an SU (2) × U (1) ( SU (3) × U (1) )-invariant G -structure on any principal orbit. This G -structure is cosymplectic, since d Ω = 0.The only remaining space is SU (2) /U (1) × T . The issue of homogeneous G -structures on this space is not yet discussed in the literature. In thefollowing, we construct an explicit SU (2) × U (1) -invariant cosymplectic G -structure on SU (2) /U (1) × T . First, we choose the following basis of su (2):(17) σ := (cid:18) i − i (cid:19) , σ := (cid:18) −
11 0 (cid:19) , σ := (cid:18) ii (cid:19) . The above basis obeys the following commutator relations:(18) [ σ , σ ] := − σ , [ σ , σ ] := − σ , [ σ , σ ] := − σ . As usual, the tangent space of SU (2) /U (1) × T can be identified with thecomplement m of the isotropy algebra in 2 su (2) ⊕ u (1). We construct abasis ( e , . . . , e ) of m and supplement it with a generator e of h to a basisof 2 su (2) ⊕ u (1). Let e and e be generators of the center of 2 su (2) ⊕ u (1).Furthermore, let(19) e := (cid:18) σ − σ (cid:19) , e := (cid:18) σ (cid:19) , e := (cid:18) σ (cid:19) ,e := (cid:18) σ (cid:19) , e := (cid:18) σ (cid:19) , e := (cid:18) σ σ (cid:19) . We define by e i ( e j ) := δ ij a basis of left invariant one-forms on SU (2) × U (1) . With help of the formula de i ( e j , e k ) = − e i ([ e j , e k ]) and the commu-tator relations on 2 su (2) ⊕ u (1) it follows that: PACES ADMITTING HOMOGENEOUS G -STRUCTURES 19 (20) de = 0 de = 0 de = e ∧ e − e ∧ e de = − e ∧ e + 2 e ∧ e de = 2 e ∧ e − e ∧ e de = 2 e ∧ e + 2 e ∧ e de = − e ∧ e − e ∧ e de = e ∧ e + e ∧ e In order to construct a homogeneous cosymplectic G -structure, we intro-duce a further basis ( f , . . . , f ) of m :(21) f := e , f := e , f := e ,f := e + e , f := − e − e ,f := e − e , f := − e + e . With respect to this basis, the action of e on m is represented by a matrixwhich is contained in the Cartan subalgebra (4). Therefore, we can identify( f , . . . , f ) with the standard basis of Im( O ). This identification yieldsan SU (2) × U (1) -invariant G -structure ω on SU (2) /U (1) × T , whichsatisfies:(22) ∗ ω = − e + 2 e − e − e − e + 2 e + 4 e . As in Section 2, e ijkl is an abbreviation of e i ∧ e j ∧ e k ∧ e l . With help of theequations (20) and the fact that the projection of e onto SU (2) /U (1) × T vanishes, we are able to compute d ∗ ω and see that our G -structure isindeed cosymplectic. This calculation finishes the proof of Theorem 1. Remark . Our proof that any G -homogeneous space G/H which admitsan arbitrary G -invariant G -structure also admits a cosymplectic one is doneby a case-by-case analysis. If G/H is irreducible, there even exists a G -invariant nearly parallel G -structure on G/H . The author suspects that itis possible to prove these facts more directly.
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