Аннотация (русский):
As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

Ключевые слова:
Nilmanifolds, six-dimensional nilpotent Lie algebras left-invariant para-complex structures, Einstein manifolds, half-flat structures

INTRODUCTION Left-invariant Kӓhlerian structure on Lie group G is a triple (g, , J) consisting of a left-invariant Riemannian metric g, left-invariant symplectic form  and orthogonal left-invariat complex structure J, where g(X,Y) = (X,JY) for any left-invariant vectors fields X and Y on G. Therefore, such a structure on G can be given by a pair (, J), where  is a symplectic form, and J is a complex structure compatible with , that is, such that (JX,JY) = (X,Y). If (X,JX) > 0,  X  0, it is Kӓhlerian metrics. If the positivity condition is not satisfied, then g(X,Y) = (X,JY) is a pseudo- Riemannian metric and then (g,,J) is called a pseudo- Kähler structure on the Lie group G. Classification of real six-dimensional nilpotent Lie algebras admitting invariant complex structures were obtained in [8]. Classification of symplectic structures on six- dimensional nilpotent Lie algebras was obtained in [5]. Out of 34 classes of isomorphic simply connected six- dimensional nilpotent Lie groups, only 26 admit left- invariant symplectic structures. Condition of existence of left-invariant positively definite metric on Lie group G applies restrictions to the structure of its Lie algebra g. For example, it was shown in [2] that such a Lie algebra can not be nilpotent except for the abelian case. Although nilpotent Lie groups and nilmanifolds (except for torus) do not admit Kӓhlerian left-invariant metrics, on such manifolds left-invariant pseudo- Riemannian Kӓhlerian metrics may exist. It was shown in [4] that 14 classes of symplectic six-dimensional nilpotent Lie groups admit compatible complex structures and, therefore, define pseudo-Kähler metrics. A more complete study of the properties of the curvature of such pseudo-Kähler and almost pseudo- Kähler structures was carried out in [9, 10]. As mentioned before, 26 out of 34 classes of six- dimensional nilpotent Lie groups admit left-invariant symplectic structures. Out of last 8 classes of non- symplectic Lie groups, 5 Lie groups Gi do not also admit complex structures [8]; their Lie algebras gi are shown below: g1: (0, 0, 12, 13, 14+23, 34 -25), g2: (0, 0, 12, 13, 14, 34 -25), g3: (0, 0, 0, 12, 13, 14+35), g4: (0, 0, 0, 12, 23, 14+35), g5: (0, 0, 0, 0, 12, 15+34). In this paper we study precisely these Lie groups. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the failure of symplectic and complex structures. It is shown that on all such Lie groups Gi any left-invariant closed 2-form ω is degenerated. There are natural ways to weaken the closedness requirement of ω to preserve non-degeneracy ω, in ways that 3-form dω is also non- Please cite this article in press as: Smolentsev N.K. Left-invariant almost para-complex einsteinian structures on six-dimensional nilpotent Lie groups. Science Evolution, 2017, vol. 2, no. 2, pp. 88-95. DOI: 10.21603/2500-1418-2017-2-2-88-95. Copyright © 2017, Smolentsev et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license. This article is published with open access at http:// science- evolution.ru/. degenerated and property ωdω = 0 is satisfied. defined as an m-tuple based on a sequence of 3 m Hitchin's operator Kdω corresponding to non- degenerated 3-form dω, can define either almost complex structure, or almost para-complex, depending differentials (0, 0, de , ..., de ) of basic 1-forms, in the notation ij is used instead of eij = eiej. For example, notation (0, 0, 0, 0, 12, 34) denotes Lie algebra with on the chosen ω. Associated metric g(X,Y) = ω(X, JdωY) structural equations: de1 = de2 = de3 = 0, de4 = 0, is pseudo-Riemannian of signatures (3,3) or (2,4). The structural group is reduced to SL(3,R) in case of signature (3,3) and to SU(1,2) in case of signature (2,4). On groups G2 - G5 this metric is Einsteinian of signature (3,3). An explicit form of these metrics is presented. As a result, we obtain a compatible couple (ω,Ω), where Ω = dω. We present an explicit form of pseudo almost Hermitian half-flat and para-complex half-flat structures. For any nilpotent Lie group G with rational structure constants there exists a discrete subgroup Γ such that M = Γ\G is a compact manifold called a nilmanifold. Therefore, all the results hold for the corresponding six- dimensional compact nilmanifolds. All calculations were made in the Maple system according to the usual formulas (see eg [10]) for the geometry of left-invariant structures. MATERIALS AND METHODS Let G be a real Lie group of dimension m and g be its Lie algebra. Lower central series of Lie algebra g is decreasing sequence of ideals C0g, C1g, ..., being defined inductively: C0g = g, Ck+1g = [g, Ckg]. Lie algebra g is called nilpotent, if Ckg = 0 for some k. In this case, the minimum length of lower central series is called class (or step) of nilpotency. In other words, the Lie algebra class is equal to s, if Csg = 0 and Cs-1g  0. In this case, Cs-1g lies in the center Z(g) of the Lie algebra g. The increasing central sequence {gl} was defined for nilpotent s-step Lie algebra, g0 = {0}  g1  g2  ···  gs-1  gs = g, where the ideals gl were defined inductively by the rule: gl = {X  g | [X, g]  gl-1}, l ≥ 1. Particularly, g1 is a center of Lie algebra. One can see from this sequence that nilpotency property is equivalent to existence of basis {e1, ..., em} of the Lie algebra g, for which k de5 = e1e2 and de6 = e3e4. Left-invariant symplectic structure on Lie group G is a left-invariant closed 2-form ω of the maximal rank. It is given by 2-form ω of the maximal rank on Lie algebra g. Closedness of the 2-form is equivalent to condition ω([X,Y],Z) - ω([X,Z],Y) + ω([Y,Z],X) = 0, X,Y,Z  g. In this case, Lie algebra g and group G will be called symplectic ones. Left-invariant almost complex structure on Lie group G is left-invariant field of endomorphisms J: TG → TG of tangent bundle TG, having the property J2 = -Id. Since J is defined by linear operator J on Lie algebra g = TeG, we will say that J is a left-invariant almost complex structure on Lie algebra g. In order for the almost complex structure J to define a complex structure on the Lie group G, it is necessary and sufficient (according to the Newlender- Nirenberg theorem [7]) that the Nijenhuis tensor vanishes: [JX,JY] - [X,Y] - J [JX,Y] - J[X,JY] = 0, for any X,Y  g. For the left-invariant complex structure on Lie group G left shifts Lg : G → G, g  G are holomorphic. Left-invariant Kӓhlerian structure on Lie group G is a triple (g, , J) consisting of a left-invariant Riemannian metric g, left-invariant symplectic form  and orthogonal left-invariant complex structure J, where g(X,Y) = (X,JY) , X,Y  g. Therefore, such a structure on Lie group G can be specified by a couple (ω,J), where ω is a symplectic form, and J is a complex structure being compatible with ω, i.e. such that ω(JX, JY) = ω(X,Y), X,Y  g. If ω(X,JY) > 0, X  0, then it is Kӓhlerian metric g(X,Y) = ω(X,JY ). But if the positivity condition is not fulfilled, then g(X,Y) is pseudo-Riemannian metric and then (g,J,ω) is called pseudo-Kӓhlerian structure on Lie group G. In further, (pseudo)Kӓhlerian structure will be specified by pair (J,ω) of compatible left-invariant complex and symplectic structures. It follows from left-invariance that (pseudo)Kӓhlerian structure (g,J,ω) can be given by the [ei , e j ]  Cij ek , k  i, j 1  i  j  m. values J, ω and g on Lie algebra g of the Lie group G. In this case (g,J,ω,g) is called pseudo-Kӓhlerian Lie Nilpotency is also equivalent to the existence of basis {e1,..., em} of left-invariant 1-forms on G such that algebra. Almost para-complex structure on 2n-dimensional manifold M is a field P of endomorphisms of the tangent dei  2{e1,, ei 1}, 1  i  m, bundle TM such that P2 = Id, where ranks of eigendistributions T±M:= ker(IdP) are equal. Almost parawhere the right side is considered to be zero for i = 1. As is known, the exterior differential of a left-invariant 1-form is expressed through the structural constants of a Lie algebra by the formula [7]: ij dek  Ck ei  e j , complex structure P is called integrable if distributions T±M are involutive. In this case, P is called para- complex structure. A manifold M supplied by (almost) para-complex structure P, is called (almost) para- complex manifold. The Nijenhuis tensor NP of almost para-complex structure P is defined by equation i j where {e1,..., em} is the dual to {e1,...,em} basis in g*. Therefore, the structure of a Lie algebra is given either by specifying nonzero Lie brackets or by differentials of basis left-invariant 1-forms. The Lie algebra g is often NP(X,Y) = [X,Y] + [PX,PY] - P[PX,Y] - P[X,PY], for all vector fields X, Y on M. As in the case with complex structure, para-complex one P is integrable if and only if NP = 0. Para-Kӓhlerian manifold can be defined as pseudo- Riemannian manifold (M,g) with skew-symmetric para- complex structure P, that is parallel with respect to the of complex vector space on real vector space V as follows: Levi-Civita connection. If (g,P) is a para-Kӓhlerian structure on M, then ω = g◦P is symplectic structure, and eigen-distributions T±M, corresponding to eigen-values J    1 K  () , ±1 of field P, represent two integrable ω-Lagrangian distributions. Therefore the para-Kӓhlerian structure can But if λ(Ω) > 0, 3-form Ω defines the para-complex structure JΩ, i.e. JΩ Ω be identified with bi-Lagrangian structure (ω,T±M), where ω is a symplectic structure, and T±M are the similar formula: 2 = 1, J  1 on real vector space V by integrable Lagrangian distributions. In [1] presents a review of the theory para-complex structures, and the invariant para-complex and para-Kӓhlerian structures on homogeneous spaces of semi-simple Lie groups are considered in detail. It is shown that every invariant para-Kähler structure P on M = G/H defines a unique para-Kähler Einstein structure (g,P) with given non-zero scalar curvature. Since the 2-form ω is not closed, it is possible to consider the 3-form dω. In [6] Hitchin had defined the   J  1 K () . Recall that the structure of almost a product is called para-complex, if eigen-subspaces have the same dimension. The elements of GL(V) orbits of 3-form Ω, corresponding to λ(Ω) > 0, have stabilizer SL(3,R)×SL(3,R) in GL+(V ) and JΩ is para-complex structure, i.e. JΩ Ω 2 = 1, J  1. The elements of orbit concept of non-degeneracy (stability) for 3-forms Ω and built a linear operator KΩ, whose square is proportional to identity operator Id. Recall the basic Hitchin's constructions. Let V be a six-dimensional real vector space, µ be a 2 corresponding to λ(Ω) < 0, have stabilizer SL(3,C) in GL+(V ) and JΩ is almost complex structure, i.e. JΩ = -1. In both cases, dual to Ω form is defined by formula    J *  . If λ(Ω) > 0 and Ω = α + β, then Ω^ = α - β. form of volume on V, and Λ3V* be a 20-dimensional But if λ(Ω) < 0 and      , then   i(   ) . linear space of skew-symmetric 3-forms on V. We shall J Note that Ω^^ = -Ω in both cases and    J  , where take interior product ιXΩ  Λ2V* for the form Ω  Λ3V* and vector X  V. Then ιXΩΩ Λ5V*. Natural pairing by the exterior product V* Λ5V* → Λ6V*  Rµ defines the isomorphism A: Λ5V∗ → V. Using Λ5V*  V we define linear map KΩ: V → V as KΩ(X) = A(ιXΩΩ). In other words, ιKΩ(X)µ = ιXΩΩ. Define λ(Ω)  R as ε is the sign of λ(Ω). The additional 3-form Ω^ has a defining property: if λ(Ω) > 0, then Ω + Ω^ is decomposable, and if λ(Ω) < 0, then complex form Ψ = Ω + i Ω^ is decomposable. The pair (ω, Ω)  Λ2(V*)Λ3(V*) non-degenerated forms is called compatible if ωΩ = 0 (or, equivalently, Ω^ω = 0), and it is called normalized, if Ω^Ω = 2ω3/3. Each compatible pair (ω,Ω) uniquely defines ε- 2 a trace of the square of KΩ: complex structure JΩ (i.e. JΩ = ε), scalar product ()  1 tr K 2 g(ω,Ω)(X,Y) = ω(X,JΩY) (signatures (3,3) for ε = 1 and signatures (2,4) or (4,2) for ε = -1), and (para-)complex  6 , volume form Ψ = Ω+iε Ω^ of type (3,0) with respect to JΩ (where iε is a complex or para-complex imaginary The form Ω is called non-degenerated (or stable) if λ(Ω)  0. It is shown in [6] that if λ(Ω)  0, then λ(Ω) > 0 if and only if Ω =  + , where ,  are real decomposable 3-forms and   0; unit). In addition, the stabilizer of (ω,Ω) pair is SU(p,q) for ε = -1 and SL(3,R)  SO(3,3) for ε =1. Therefore, (ω,Ω) pair for ε = -1 defines pseudo almost Hermitian structure. But if ε = 1, it defines almost para-Hermitian structure. Such structures are also called special almost λ(Ω) < 0 if and only if      , where   ε-Hermitian. Λ3(V*C) is complex decomposable 3-form and     0 . It follows that if Ω is real and λ(Ω) > 0, then it lies in GL(V)-orbit of form  = θ1θ2θ3 + θ4θ5θ6 for some basis θ1, ..., θ6 in V*, and if λ(Ω) < 0, then Also recall that SU(3) structure on real sixdimensional almost Hermitian manifold (M, g, J, ω) is specified by (3,0) form Ψ. Almost Hermitian 6-manifold is called half-flat [3] if it admits a reduction to SU(3), for which dRe Ψ = 0 and ωdω = 0. In the case of pseudo-Riemannian manifold, each it lies in orbit of form (θ1 + iθ2)(θ3 + iθ4)(θ5 + iθ6).      , where α = compatible pair (ω,Ω) uniquely defines the reduction to SU(1,2) for ε = -1 and SL(3,R)  SO(3,3) for ε = 1. Then the real 20-dimensional vector space Λ3(V*) contains invariant quadratic hypersurface λ(Ω) = 0 dividing the Λ3(V*) to 2 open sets: λ(Ω) > 0 and λ(Ω) < 0. The component of the unit of the stabilizer of the 3-form lying in the first set is conjugate to the group SL(3,R)×SL(3,R), and in the other case to the group  SL(3,C). The linear transformation of KΩ has [6] the following properties: tr KΩ = 0 and K 2  ()Id . In the case λ(Ω) < 0, the real 3-form Ω defines the structure JΩ Therefore, 6-manifold with a (ω,Ω) pair possessing the properties dΩ = 0 and ωdω = 0, will be called half-flat pseudo almost Hermitian if it admits the reduction to SU(1,2) or half-flat almost para-complex if it admits the reduction to SL(3,R)  SO(3,3). Remark. In this work, we assume that exterior product and exterior differential are defined without normalizing constant. In particular, then dxdy = dxdy - dydx and dη(X, Y) = Xη(Y) - Yη(X) - η([X, Y]). Let  be the Levi-Civita connectivity corresponding to left- invariant (pseudo)Riemannian metric g. It is defined by 46  a2    2a46 2 a56 56  2a2 2 0 0 0  2   form for left-invariant vector fields X,Y,Z on Lie group: 2g(XY, Z) = g([X,Y],Z) + g([Z,X],Y) + g(X,[Z,Y]). If 0 Kd    0 0  a46 0 0   2a46a56 0 0 a 0 2 0 46  tensor Ric(X,Y) for (pseudo)Riemannian metric g is defined as a construction of a curvature tensor over the first and fourth (upper) indices.   0 0 46 0 0 0 56  46  a2  RESULTS AND DISCUSSION Determine the operator P  Kd 46 / a 2 . It defines left- In this section Lie groups that do not admit neither symplectic, nor left-invariant complex structures will be considered. Such Lie groups will be called singular. It will be shown that they admit non-degenerated leftinvariant almost para-complex structure P2 = Id, having the property ω(PX,PY) = -ω(X,Y). Eigen- subspaces E ± related to the eigen-values ± 1 of operator P are generated by the following vectors: 3 1 2 3 3 4 1 2 5 invariant 2-forms, whose exterior differentials are non- degenerated. In addition, they admit almost para- E+ = {a56 e - a56a46 e + a46 e , -a56 e + a46 e , e }, complex structures and Einstein pseudo-Riemannian 56 E- = {e1, -a56 e2 + a46 e3, - a 2 e5 + a 46 2 e6}. metrics of signature (3,3). Lie group G1 Singular group G1 that does not admit neither symplectic, nor complex structures. Non-zero commutation relations: It is easy to see that they are not closed relative to Lie bracket, so P defines non-integrable almost a para- complex structure. Define pseudo-Riemannian metric g(X,Y) = ω(X,PY) of signature (3,3). It is given by: 1 2 3 4 [e1,e2] = e3, [e1,e3] = e4,, [e1,e4] = e5, [e2,e3] = g = 2e (a13a46/a56 e +a13e +a14e )+ 2 2 3 4 = e5, [e3,e4] = e6, [e2,e5] = -e6. + 2e (a13 e +(2a13a56 +a14a46)/a46 e + 2a14a56/a46 e ) + 2 3 2 2 4 Lie algebra g has ideals: C1g = D1g = R{e3,e4,e5,e6}, C2g = R{e4,e5,e6}, C3g = R{e5,e6}, C4g = Z = R{e6} is + 2e3(a56(a13a56 +a14a46)/a46 e + 2a14a56 )/a46 e ) - the center of Lie algebra. Filiform Lie algebra. Does not 56 - 2a46 e4e6 - 2a56 e5e6 - 2a 3/a 46 2 e6e6. admit half-flat structures [3]. Let ω = aijeiej be any left-invariant 2-form. For the general form ω Hitchin’s operator Kω has quite complicated form and the following function λ(dω): Direct calculations of curvature tensor in Maple system show that this metric is not Einsteinian and that it has scalar curvature 7 6 8 2 2 2 2 R  3(8a13a56  8a14a46a56  a46 ) λ = 4(a16a56 + 4a35a56 + 4a36 a56 - 4a36a46 - 46 - 4a45a46a56)a56+a 4. Thus, in general, 3-form dω is non-degenerated. It is easy to see that 2-form ω is closed if and only if a16 = a26 = a36 = a35 = a45 = a46 = a56 = 0 and a34 = -a25. a24 = a15. However, such 2-form ω is degenerated. There are several natural ways to weaken the closedness requirement of the 2-form ω, so as not to lose the nondegeneracy of ω and dω. Option 1. In that case we will not suppose that coefficients a46 and a56, which essentially occur in the expression for function λ(dω), are non-zero. Moreover, we will suppose that a56  0. Then the property ωdω = 0 is fulfilled under condition a15 = 0, a25 = 0 and . a a2 a6 56 14 46 Option 2. Take the 2-form ω in the view ω = ω0 + ωC, where ω0 is a closed 2-form and ωC is a non-degenerated 2-form on the ideality C2g = R{e4,e5,e6}. We require that the 2-form ω should have the property ωdω = 0: a25 = 0, a15 = 0, a12 a56 = a13a46, a56 a23 + a14a56 = 0. Then ω is non-degenerated under condition a14a56  0. The ω and dω take the view: ω = e1( a13a46/a56 e2 + a13 e3 + a14 e4) - a14 e2e3 + a45 e4e5 + a46 e4e6 + a56 e5e6, a12 a56 =a13 a46, a23 = -a14. 2-form ω is non-degenerated dω = a45 e234 - a45 e135 -a46 e136 + a46 e245 - a56 e146 - under condition that a14a56  0 and ω and dω become ω = e1(a13a46/a56 e2 + a13 e3 + a14 e4) - a56 e236 + a56 e345. In this case, λ(dω) function is expressed by the λ = - a14 e2e3 + a46 e4e6 + a56 e5e6, a 56 46 4 -4a46a45a 2 and it can take both positive and dω = -a46 e136 + a46 e245 - a56 e146 - a56 e236 + a56 e345. negative values. Case 1. The function λ(dω) takes the value -1 when 4 + 1)/(4a a 2). Then the operator J = K The function λ(dω) of the Hitchin's operator [6] for 4 a45 = (a46 46 56 dω form dω becomes λ = a46 . The Hitchin's operator Kdω has a matrix defines almost complex structure compatible with ω and has the form: 2 - 2a a )2Id. Therefore, 3-form dω is non-   a2 2a a  2a2 0 0 0  Kdω = (a46 36 56  46 a4 1 46 56 56 46  degenerated if (dω) = a 2 - 2a36a56  0. The 2-form ω  46 a2 2a a 2a2 0 0   2a a 46 46 56 56  is closed only in the case when it has the form: 46 56   0 0 a2 2a a 0 0  1 2 3 4 5 2 3 46 46 56 4  ω = e (a12 e + a13 e + a14 e + a15 e ) + e (a23 e - J   0 0 a46 1 a2 0 0   2a a 46  - a34 e5) + a34 e3e4.    0 0 46 56 a4 1  46 a4 1  46 a2  46 2a2  Such 2-form ω is non-degenerated. In order to 2a 2 56  4 2 2a46a56 4 56   preserve the non-degeneracy of the ω and dω at the  0 0 (a46 1) 0  a46 1  a2  minimal weakening of closedness property of ω, two 2 4 2 46  8a46a56 2a56  variants are possible: a46  0, or a36  0 and a56  0. Specify the associated pseudo-Riemannian metric by formula g(X,Y) = ω(X, JY) of signature (2,4). Direct calculations of curvature tensor in Maple system show that this metric is not Einsteinian and that it has scalar curvature However, if a56  0, then simple calculations show that the property ωdω = 0 is incompatible with the non- degeneracy ω. 46 Therefore, consider a case when a46  0. Then Kdω = a 4 Id. In addition, ωdω = 0 under condition that a13 = 0 and a34 = 0. Then the 2-form ω is non- 7 6 a 2 R  8a13a56  8a14 a46 a56  1 a56 14 . Case 2. The function λ(dω) takes the value +1 when degenerated under condition a23a15a46  0, and the ω and dω take the view: ω = e1(a12 e2 + a14 e4+ a15 e5) + a23 e2∧e3 + a46 e4e6, 46 a45 = (a 4 56 dω - 1)/(4a46a 2). Then the operator P = K dω = a46(-e136 + e245). defines almost para-complex structure compatible with ω and P has the same matrix, as the above almost The operator Kdω for the 3-form dω has the diagonal 2 2 2 2 2 2 complex structure J has, where it is necessary to form: Kdω = diag{-a46 , a46 , -a46 , a46 , a46 , -a46 }. 46 substitute a4 46  1 instead of a4  1 . The corresponding Define the operator P  Kd 46 / a 2 . It defines leftmetric g(X, Y) = ω(X, PY) is pseudo-Riemannian of signature (3,3); it is not the Einsteinian one and has the same scalar curvature, as in the first case. Conclusions. Any left-invariant closed 2-form ω on Lie group G1 is degenerated. There are several ways to weaken the closedness requirement of ω to preserve non-degeneracy ω, in ways that 3-form dω is non- degenerated and the property ωdω = 0 is fulfilled. Hitchin's operator Kdω corresponding to non- degenerated 3-form dω, can define either almost complex structure, or almost para-complex, depending on the chosen ω. Associated metric g(X,Y) = ω(X,JdωY) is pseudo-Riemannian of signature (2,4) or (3,3). As a result, we have obtained a compatible pair (ω,Ω), where Ω = dω. Therefore, the properties dΩ = 0 and ωdω = 0 are fulfilled in an obvious way. The (3,0)-form has a view of Ψ = dω +iεdω^, where iε is a invariant almost para-complex structure P2 = Id, having the property ω(PX, PY) = -ω(X, Y). Eigen-subspaces E ± related to the eigen-values ± 1 of operator P are generated by the following vectors: E+ = {e2, e4, e5 }, E- = {e1, e3, e6}. It is easy to see that they are not closed relative to Lie bracket, so P defines non-integrable almost a para- complex structure. Define the pseudo-Riemannian metric g(X, Y) = ω(X, PY). It has a signature (3,3) and it is given by: g = 2e1(a12e2+a14e4+a15e5) - 2a23 e2e3 - 2a46 e4e6. Direct calculations in a Maple system show that this metric is Einsteinian and that its Ricci tensor and scalar curvature are specified by formulas: complex or para-complex unit. Thus, half-flat pseudo almost Hermitian and half-flat para-complex structures Ric  a46 g R   3a46 were naturally defined on Lie group G1. 2a15 a23 , a15 a23 . Lie group G2 Singular group G2 that does not admit neither symplectic, nor complex structures. Commutation relations [e1,e2] = e3, [e1,e3] = e4, [e1,e4] = e5,[e3,e4] = e6, [e2,e5] = -e6. Lie algebra g has ideals: C1g = D1g = R{e3,e4,e5,e6}, C2g = R{e4,e5,e6}, C3g = R{e5,e6}, C4g = Z = R{e6} is the center of Lie algebra. Filiform Lie algebra. Does not admit half-flat structures [3]. Lie group G3 Singular group G3 that does not admit neither symplectic, nor complex structures. Commutation relations [e1,e2] = e4, [e1,e3] = e5, [e1,e4] = e6,[e3,e5] = e6. Lie algebra g has ideals: C1g = D g = R{e4,e5,e6}, C2g = R{e6} = Z is the center of Lie algebra Does admit half-flat structure [3]. Let ω = aijeiej be any 2-form. The Hitchin's operator Kdω for generic 2-form ω has a quite complicated view. 2 4 4 Let ω = aijeiej be any left-invariant non-degenerated Moreover, Kdω = a46 Id. For λ = a46  0 the 3-form dω is 2-form. For such a generic 2-form the square of the Hitchin's operator [6] for 3-form dω has a diagonality: non-degenerated. The operator P = Kdω/a462 defines the left-invariant almost para-complex structure on g. The ωdω = 0 property is fulfilled under the following - a14 a56 + a15 a46 - a16 a45 = 0, conditions: a34 a56 - a35 a46 + a36 a45 = 0. a12 a46 - a14 a26 - a23 a56 + a24 a16 + a25 a36 - a35 a26 = 0, It is easy to see that the 2-form ω is closed only if a25 a46 - a24 a56 - a26 a45 = 0, a35 a46 - a36 a45 - - a34 a56 = 0. ω = e1(a 12e2+a 13e3+a 14e4+a 15e5)+ It is easy to see that the 2-form ω is closed only if ω = e1(a12e2+a13e3+a14e4+a15e5)+ +e2(a23e3+a24e4+a25e5)+e3(a25e4+a35e5). In order to preserve the non-degeneracy of the ω and dω at the minimal weakening of closedness property of ω, consider the case when a46  0. The ωdω = 0 property is fulfilled under the condition a12 = 0, a25 = 0, a35 = 0. Thus, 2-form ω is non-degenerated if a15a23a46  0 and then we obtain: ω = e1( a13e3+a14e4+a15e5) + e2(a23e3+a24e4) + +e2(a23e3+a24e4+a25e5)+e3(-a15e4+a35e5). In order to preserve the non-degeneracy of the ω and dω at the minimal weakening of closedness property of ω, consider the case when a46  0 and a56  0. The ωdω = 0 property is fulfilled under the following conditions: - a12 a46+ a23 a56 = 0, - a14 a56 + a15 a46 = 0, - a15 a56 - a35 a46 = 0 and dω is given by the sum of two decomposable 3-forms: dω = (a56 e1 - a46 e3)e45 + (- a46e1 + a56 e3)e26. The function λ(dω) of the Hitchin's operator for + a46 e4e6, dω = -a46(e126 + e345). The operator Kdω for the 3-form dω has the diagonal 3-form dω has the same view λ = (a operator Kdω is given by: - a 56 46 2 2)2. And 2 2 2 2 2 2 view, Kdω = diag{-a46 , -a46 , a46 , a46 , a46 , -a46 }. Kdω = (-a46 56 1 46 56 2 46 Define the operator P  K / a2 . It defines left- 2-a 2)e e1 + (-a 2+a 2) e e2 + (a 2+ d 46 2) e e3 +(a 2-a 2) e e4 + (a 2-a 2)e e5+ invariant almost para-complex structure P2 = Id, having the property ω(PX,PY) = -ω(X,Y). Eigen-subspaces +a56 3 +(-a46 56 46 56 4 6 6 46 56 5 3 1 E ± related to the eigen-values ± 1 of operator P are 2+a 2) e e +2a46 a56 e1e - 2a46 a56 e3e . generated by the following vectors: Define the operator / | 2 2 | P  Kd a46  a56 . It defines E+ = {e3, e4, e5 }, E- = {e1, e2, e6}. 2 left-invariant almost para-complex structure P2 = Id, having the property ω(PX,PY) = -ω(X,Y). Eigen- It is easy to see that they are not closed relative to subspaces related to eigen-values (a46 - a 56 - a | 46 56 2)/|a 2 2 Lie bracket, so P sets non-integrable almost a paraand (a56 -a 2 46 2)/|a - a 56 46 2 2| of the operator P are generated complex structure. Define the pseudo-Riemannian metric g(X,Y) = ω(X, PY) of signature (3,3). It is given by: g = 2e1(a13e3+a14e4+a15e5) + 2a23 e2e3 + 2a24 e2e4 - - 2a46 e4e6. Direct calculations in a Maple system show that this metric is Einsteinian and that its Ricci tensor and scalar curvature are specified by formulas: by the following vectors: E1 = {a56 e1+a46e3, e4, e5 }, E2 = {a46 e1+a56e3, e2, e6 }. It is easy to see that they are not closed relative to Lie bracket, so P sets non-integrable almost a para- complex structure. Define the pseudo-Riemannian metric g(X, Y) = ω(X, PY) of signature (3,3). Direct calculations in a Maple system show that this metric is Einsteinian and that its Ricci tensor and scalar curvature are specified Ric  a46 g 2a15 a23 , R   3a46 a15 a23 . by formulas: 2 2 Lie group G4 Singular group that does not admit neither Ric  | a46  a45 | g 2a13 (a24 a56  a25 a46 ) , symplectic, nor complex structures. Commutation 3 | a2  a2 | 46 45 relations: [e1,e2] = e4, [e2,e3] = e5, [e1,e4] = e6,[e3,e5] = e6. R   a13 (a24 a56  a25 a46 ) . Lie algebra g has ideals: C1g = D1g = R{e4,e5,e6}, a56 In particular case, when one of the arguments a46 and is equal to zero, the situation becomes much simpler. C2g = R{e6} = Z is the center of Lie algebra. Does admit half-flat structure [3]. For example, let a56 = 0. The property ωdω = 0 is fulfilled under the following conditions: a12 = 0, a15 = 0, Let ω = aijeiej be any 2-form. The Hitchin's operator a35 = 0. Then 2-form is non-degenerated under the Kdω for generic 2-form ω has a quite complicated form. condition a13a25a46  0, and we obtain: 2 2 2 2 Moreover, Kdω = (a46 - a56 ) Id. The ωdω = 0 property is fulfilled under the following conditions: - a12 a46+ a14 a26 - a16 a24 + a23 a56- a25 a36 + a26 a35 = 0, ω = e1(a13e3+a14e4)+e2( a23e3+a24e4+a25e5) + a46 e4e6, dω = - a46e126 - a46 e345, 2 2 2 2 2 Kdω = diag{-a462, -a46 , a46 , a46 , a46 , -a46 }. d 46 Define the operator P  K / a2 . It specifies almost para-complex structure P2 = Id, possessing the property ω(PX,PY) = -ω(X,Y). Eigen-subspaces related to the eigen-values ± 1 of operator P are generated by the following vectors: E+ = {e3, e4, e5 }, E- = {e1, e2, e6}. a12 = 0. The 2-form ω is non-degenerated under the condition a56(a13a24 - a14a23)  0, and the following formulas occur: ω = e1( a13e3+a14e4+a15e5)+e2(a23e3+a24e4+a25e5) + 56 + a56 e5e6, dω = -a56 e126 + a56 e345, Kdω = a 2·diag{+1, +1, -1, -1, -1, +1}. Define the operator P  Kd 56 / a2 . It defines left- It is easy to see that they are not closed relative to Lie bracket, so P sets non-integrable almost a para- complex structure. The pseudo-Riemannian metric g(X, Y) = ω(X, PY) of signature (3,3) is given by g = -2e1(a13e3+a14e4+a15e5) - 2e2(a23e3+a24e4+a25e5) + + 2a46 e4e6. Direct calculations in a Maple system show that this metric is Einsteinian and that its Ricci tensor and scalar curvature are specified by formulas: invariant almost para-complex structure P2 = Id, having the property ω(PX,PY) = -ω(X,Y). Eigen-subspaces related to the eigen-values ± 1 of operator P are generated by the following vectors: E+ = {e1, e2, e6 }, E- = {e3, e4, e5}. It is easy to see that they are not closed relative to Lie bracket, so P sets non-integrable almost a para- complex structure. The pseudo-Riemannian metric g(X, Y) = ω(X, PY) of signature (3,3) is given by g = -2e1(a13e3+a14e4+a15e5) - 2e2(a23e3+a24e4+a25e5) + Ric   a46 g R  3a46 4 6 2a13a25 , a13a25 . + 2a46 e e . Direct calculations in a Maple system show that this Lie group G5 Singular group that does not admit neither symplectic, nor complex structures. Commutation metric is Einsteinian and that its Ricci tensor and scalar curvature are specified by formulas: relations: [e1,e2] = e5, [e1,e5] = e6, [e3,e4] = e6. Ric  a56 g 2(a13 a24  a14 a23 ) , Lie algebra g has ideals: C1g = D1g = R{e5,e6}, 3a56 C2g = R{e6} = Z is the center of Lie algebra. Does admit half-flat structure [3]. R   a13 a24  a14 a23 . Let ω = aijeiej be any 2-form. The Hitchin's operator Kdω for generic 2-form ω is given by a quite complicated Conclusions. Any left-invariant closed 2-form ω on Lie groups G2 - G5 is degenerated. When the closedness ω form. Moreover, Kd 2 56 = a 4 Id. The ωdω = 0 property is requirement of ω is weakened in order to preserve the nonfulfilled under the following conditions: a34 a56 + a35 a46 - a36 a45 = 0, a12 a56 - a15a26 + a16 a25 - a23a46 + a24 a36 - a26 a34 = 0. It is easy to see that the 2-form ω is closed only if ω = e1(a12e2+a13e3+a14e4+a15e5) + e2(a23e3+a24e4+a25e5) + a34 e3e4. Such 2-form ω is non-degenerated. In order to preserve the non-degeneracy of the ω and dω at the minimal weakening of closedness property of ω, consider the case when a56  0. Then the property ωdω = 0 is fulfilled under the following conditions: a34 = 0 and degeneracy of ω and dω and of the property ωdω = 0, the Hitchin's operator Kdω corresponding to dω, defines almost para-complex structure P. Pseudo-Riemannian metric g(X,Y) = ω(X, PY) depending on 5 to 7 arguments is of signature (3,3) and is Einsteinian one. As a result, we have obtained a compatible pair (ω,Ω), where Ω = dω. Therefore, the properties dΩ = 0 and ωdω = 0 were fulfilled in an obvious way. The para-complex (3,0)-form is given by Ψ = dω +iεdω^, where iε is a para-complex unit. Thus, multiparametric families of Einsteinian almost para-complex half-flat structures were naturally defined on the Lie groups G2 - G5 and corresponding nilmanifolds. The structural group is reduced to SL(3,R)  SO(3,3).

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