[

# [

[ Paisii Hilendarski University of Plovdiv \brFaculty of Mathematics, Informatics and IT\brDepartment of Algebra and Geometry \br236 Bulgaria blvd \brPlovdiv 4027 \brBulgaria
###### Abstract

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by M.L. Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.

hypercomplex structure, 4-dimensional Lie algebra, Hermitian metric, Norden metric, indefinite metric

Hypercomplex structures with Hermitian-Norden metrics] Hypercomplex structures with Hermitian-
Norden metrics on four-dimensional Lie
algebras

M. Manev]Mancho Manev

thanks: This paper is partially supported by project NI13-FMI-002 of the Scientific Research Fund, Plovdiv University, Bulgaria and the German Academic Exchange Service (DAAD)\subjclass

53C15, 53C50, 22E30, 53C55, 53C56

## Introduction

The present work is inspired by the work of Barberis [2] where invariant hypercomplex structures on 4-dimensional real Lie groups are classified. In that case the corresponding metric is positive definite and Hermitian with respect to the triplet of complex structures of . Our main goal is to classify 4-dimensional real Lie algebras which admit hypercomplex structures with Hermitian and Norden metrics.

We equip a hypercomplex structure with a metric structure, generated by a pseudo-Riemannian metric of neutral signature (see [9, 10]). In our case the one (resp., the other two) of the almost complex structures of acts as an isometry (resp., act as anti-isometries) with respect to in each tangent fibre. Thus, there exist three (0,2)-tensors associated by except the metric — a Kähler form and two metrics of the same type. The metric is Hermitian with respect to the one almost complex structure of and is a Norden (i.e. an anti-Hermitian) metric regarding the other two almost complex structures of . For this reason we call the derived almost hypercomplex structure an almost hypercomplex structure with Hermitian-Norden-metrics or briefly almost hypercomplex HN-metric structure.

Let us remark that in [13] and [12] are classified the invariant complex structures on 4-dimensional solvable simply-connected real Lie groups where the dimension of commutators is less than three and equal three, respectively.

A hypercomplex structure is called Abelian ([4]) if , for all (). Abelian hypercomplex structures are considered in [3], [5] and they can only occur on solvable Lie algebras ([6]). It is clear that the condition can be rewritten as for all . Thus, Abelian complex structures and therefore Abelian hypercomplex structure are integrable.

If the three almost complex structures of are parallel with respect to the Levi-Civita connection of then such hypercomplex HN-manifolds of Kähler type we call hyper-Kähler HN-manifolds, which are flat according to [10].

The paper is organized as follows. In Sect. 1 we recall some facts about the almost hypercomplex HN-manifolds known from [1, 9, 10, 11]. In Sect. 2 we construct different types of hypercomplex structures on Lie algebras following the Barberis classification.

The basic problem of this work is the existence and the geometric characteristics of hypercomplex HN-structures on 4-dimensional Lie algebras according to the Barberis classification. The main results of this paper is construction of the different types of the considered structures and their characterization.

## 1 Preliminaries

Let be a hypercomplex manifold, i.e. is a -dimensional differentiable manifold and is a triple of complex structures on with the following properties for all cyclic permutations of :

 Jα=Jβ∘Jγ=−Jγ∘Jβ,J2α=−I, (1)

where denotes the identity and moreover, it is valid

 Nα=0,α∈{1,2,3} (2)

for the Nijenhuis tensors of given by

 Nα(⋅,⋅)\allowbreak=[Jα⋅,Jα⋅]−Jα[Jα⋅,⋅]−Jα[⋅,Jα⋅]−[⋅,⋅] (3)

on .

A hypercomplex structure on a Lie group is said to be invariant if left translations by elements of are holomorphic with respect to for all . Obviously, if is the corresponding Lie algebra of the Lie group , a hypercomplex structure on induces an invariant hypercomplex structure on by left translations.

Let be a simply connected 4-dimensional real Lie group admitting an invariant hypercomplex structure. A left invariant metric on is called invariant hyper-Hermitian if it is hyper-Hermitian with respect to some invariant hypercomplex structure on . It is known that all such metrics on given are equivalent up to homotheties.

If denotes the Lie algebra of then it is known the following

###### Theorem 1.1 ([2]).

The only 4-dimensional Lie algebras admitting an integrable hypercomplex structure are the following types:

(hc1) is Abelian; (hc2) ; (hc3) ;

(hc4) is the solvable Lie algebra corresponding to ;

(hc5) is the solvable Lie algebra corresponding to ,
where is the Lie algebra of the Lie groups and ; is the Lie algebra of the affine motion group of – the unique 4-dimensional Lie algebra carrying an Abelian hypercomplex structure; is the real hyperbolic space; is the complex hyperbolic space.

Let be a neutral metric on with the properties

 g(⋅,⋅)=εαg(Jα⋅,Jα⋅), (4)

where

 εα={1,α=1;−1,α=2;3.

Moreover, the associated (Kähler) 2-form and the associated neutral metrics and are determined by

 gα(⋅,⋅)=g(Jα⋅,⋅)=−εαg(⋅,Jα⋅). (5)

The structure tensors of a such manifold are the following three -tensors

 (6)

where is the Levi-Civita connection generated by . The corresponding Lee 1-forms are defined by

 θα(⋅)=gijFα(ei,ej,⋅) (7)

for an arbitrary basis of , .

In [10] we study the so-called hyper-Kähler manifolds with HN-metric structure (or pseudo-hyper-Kähler manifolds), i.e. the almost hypercomplex HN-manifold in the class , where for all . A sufficient condition be in is this manifold be of Kähler-type with respect to two of the three complex structures of [9].

As is an indefinite metric, there exist isotropic vectors on , i.e. , . In [9] we define the invariant square norm

 ∥∇Jα∥2=gijgklg((∇iJα)ek,(∇jJα)el), (8)

where is an arbitrary basis of the tangent space at an arbitrary point of . We say that an almost hypercomplex HN-manifold is an isotropic hyper-Kähler HN-manifold if for each of . Clearly, if the manifold is a hyper-Kähler HN-manifold, then it is an isotropic hyper-Kähler HN-manifold. The inverse statement does not hold.

Let us note that according to (4) the manifold is almost Hermitian and the manifolds , , are almost complex manifolds with Norden metric [7]. The basic classes of the mentioned two types of manifolds are given in [8] and [7], respectively. The special class of the Kähler-type manifolds belongs to any other class within the corresponding classification. In the 4-dimensional case the four basic classes of the almost Hermitian manifolds are restricted to two: , the class of the almost Kähler manifolds and , the class of the Hermitian manifolds with respect to . They are determined for by:

 (9)

where is the cyclic sum by three arguments , , . The basic classes of the almost Norden manifolds (i.e., for or ) are determined for dimension as follows:

 (10)

It is known that the class of the complex manifolds with Norden metric is for ().

Then the class of hypercomplex manifolds with Hermitian-Norden metrics is

 H=W4(J1)∩(W1⊕W2)(J2)∩(W1⊕W2)(J3).

## 2 Four-dimensional Lie algebras with such structures

Let be a basis of a 4-dimensional real Lie algebra with center and derived Lie algebra . A standard hypercomplex structure on is defined as in [14]:

 J1e1=e2,J1e2=−e1,J1e3=−e4,J1e4=e3;J2e1=e3,J2e2=e4,J2e3=−e1,J2e4=−e2;J3e1=−e4,J3e2=e3,J3e3=−e2,J3e4=e1. (11)

Let us introduced a pseudo-Euclidian metric with neutral signature as follows

 g(x,y)=x1y1+x2y2−x3y3−x4y4, (12)

where , . This metric satisfies (4) and (5). Then the metric generates an almost hypercomplex HN-metric structure on .

Let us consider the different cases of Theorem 1.1.

### 2.1 Hypercomplex HN-metric structure of type (hc1)

Obviously, in this case the considered manifold belongs to the class .

### 2.2 Hypercomplex HN-metric structure of type (hc2)

Let be not solvable and let us determine it by

 [e2,e4]=e3,[e4,e3]=e2,[e3,e2]=e4. (13)

In this consideration the -unit , i.e. , is orthogonal to with respect to .

Then we compute covariant derivatives in the basis and the nontrivial ones are

 ∇e2e3=−32e4,∇e3e2=−12e4,∇e4e2=12e3,∇e2e4=32e3,∇e3e4=−12e2,∇e4e3=12e2. (14)

By virtue of (14), (11) and (6), we obtain components , , as follows:

 (F1)314=−(F1)323=(F1)332=−(F1)341==−(F1)413=−(F1)424=(F1)431=(F1)442=12; (15) (F2)214=−(F2)223=−(F2)232=(F2)241=32,(F2)322=−1,(F2)412=(F2)421=(F2)434=(F2)443=12,(F2)344=−1;(F3)213=(F3)224=(F3)231=(F3)242=32,(F3)422=1,(F3)312=(F3)321=−(F3)334=−(F3)343=12,(F3)433=1.

The only non-zero components , , of the corresponding Lee forms are

 (θ1)2=−1,(θ2)3=−2,(θ3)4=2. (16)

Using the results in (15), (16) and the classification conditions (9), (10), we obtain

###### Proposition 2.1.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (13), belongs to the largest class of the considered manifolds, i.e. , as well as this manifold does not belong to neither nor for and .

The other possibility is the signature of on to be , e.g. , where . By similar computations we establish the same class in the statement of Proposition 2.1.

### 2.3 Hypercomplex HN-metric structure of type (hc3)

We analyze separately the cases of signature (1,1), (0,2) and (2,0) of on .

#### 2.3.1

Firstly, we consider of signature (1,1) on .

Let us determine by

 [e2,e3]=[e1,e4]=e2,[e2,e1]=[e4,e3]=e4. (17)

Then we compute covariant derivatives and the nontrivial ones are

 ∇e2e1=∇e4e3=e4,∇e2e2=−∇e4e4=e3,∇e2e3=−∇e4e1=e2,∇e2e4=∇e4e2=e1. (18)

By virtue of (17), (11) and (6), we obtain that and the other components , , are as follows

 (F2)212 =(F2)221=(F2)234=(F2)243= (19) =−(F2)414=(F2)423=(F2)432=−(F2)441=2; (F3)211 =−(F3)222=−(F3)233=(F3)244= =(F3)413=(F3)424=(F3)431=(F3)442=−2.

The only non-zero components of the corresponding Lee forms are

 (θ2)1=(θ3)2=4. (20)

Using that , the results in (19), (20) and the classification conditions (9), (10), we obtain

###### Proposition 2.2.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (17), belongs to the subclass of the Kähler manifold with respect to of the largest class of the considered manifolds, i.e.

 W0(J1)∩(W1⊕W2)(J2)∩(W1⊕W2)(J3),

as well as this manifold does not belong to neither nor for and .

#### 2.3.2

Secondly, we consider of signature (2,0) on . The case for signature (0,2) is similar.

Let us determine by

 [e1,e3]=[e4,e2]=e1,[e1,e4]=[e2,e3]=e2. (21)

Then we compute covariant derivatives and the nontrivial ones are

 ∇e1e1=∇e2e2=e3,∇e2e3=−∇e4e1=e2,∇e1e3=∇e4e2=e1. (22)

By virtue of (21), (11) and (6), we obtain the following components of :

 (F1)114 =−(F1)123=(F1)132=−(F1)141= (23) =(F1)213=(F1)224=−(F1)231=−(F1)242=−1; (F2)111 =(F2)133=2, (F2)212 =(F2)221=(F2)234=(F2)243 =−(F2)414=(F2)423=(F2)432=−(F2)441=1; (F3)222 =(F3)233=2, −(F3)112 =−(F3)121=(F3)134=(F3)143 =(F3)413=(F3)424=(F3)431=(F3)442=−1.

The only non-zero components of the corresponding Lee forms are

 (θ1)4=−2,(θ2)1=(θ3)2=4. (24)

Using the results in (23), (24) and the classification conditions (9), (10), we obtain that the considered manifold belongs to the class . Remark that, according to [10], necessary and sufficient conditions a 4-dimensional almost hypercomplex HN-manifold to be in the class are:

 θ2∘J2=θ3∘J3=−2(θ1∘J1). (25)

These conditions are satisfied bearing in mind (24).

Let us consider the class , which is the class of the (locally) conformally equivalent -manifolds, where a conformal transformation of the metric is given by for a differential function on the manifold.

Using (24) and (25), we establish that the considered manifold belongs to the subclass .

###### Proposition 2.3.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (21), belongs to the class of the (locally) conformally equivalent -manifolds.

### 2.4 Hypercomplex HN-metric structure of type (hc4)

In this case, is solvable and the derived Lie algebra is 3-dimensional and Abelian.

#### 2.4.1

Firstly, we fix , for which , as an element orthogonal to with respect to . Therefore is determined by

 [e1,e2]=e2,[e1,e3]=e3,[e1,e4]=e4. (26)

Then we compute covariant derivatives and the nontrivial ones are

 ∇e2e1=−e2,∇e3e1=−e3,∇e4e1=−e4,∇e2e2=−∇e3e3=−∇e4e4=e1. (27)

By similar computation as in the previous cases, the components , , are as follows:

 (F1)314 =−(F1)323=(F1)332=−(F1)341= (28) =−(F1)413=−(F1)424=(F1)431=(F1)442=1; (F2)311 =(F2)333=−2, (F2)214 =−(F2)223=−(F2)232=(F2)241 =(F2)412=(F2)421=(F2)434=(F2)443=−1; (F3)411 =(F3)444=2, (F3)213 =(F3)224=(F3)231=(F3)242 =(F3)312=(F3)321=−(F3)334=−(F3)343=−1.

The only non-zero components of the corresponding Lee forms are

 (θ1)2=−(θ2)3=(θ3)4=−2. (29)

The results in (28), (29) and the classification conditions (9), (10) imply

###### Proposition 2.4.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (26), belongs to the largest class of the considered manifolds, i.e. , as well as this manifold does not belong to neither nor for and .

#### 2.4.2

Secondly, we choose , for which , as an element orthogonal to with respect to . Therefore, in this case is determined by

 [e4,e1]=e1,[e4,e2]=e2,[e4,e3]=e3. (30)

Therefore, the nontrivial covariant derivatives are

 ∇e1e1=∇e2e2=−∇e3e3=−e4,∇e1e4=−e1,∇e2e4=−e2,∇e3e4=−e3. (31)

In a similar way we obtain:

 (F1)113 =(F1)124=−(F1)131=−(F1)142= (32) =−(F1)214=(F1)223=−(F1)232=(F1)241=−1; (F2)222 =(F2)244=−2, (F2)112 =(F2)121=(F2)134=(F2)143 =(F2)314=−(F2)323=−(F2)332=(F2)341=−1; (F3)111 =(F3)144=2, −(F3)212 =−(F3)221=(F3)234=(F3)243 =(F3)313=(F3)324=(F3)331=(F3)342=−1.

The only non-zero components of the corresponding Lee forms are

 (θ1)3=−2,(θ2)2=−(θ3)1=−4. (33)

Then, analogously of Case 2.3.2, we obtain the following

###### Proposition 2.5.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (30), belongs to the class of the (locally) conformally equivalent -manifolds.

### 2.5 Hypercomplex HN-metric structure of type (hc5)

In this case, is solvable and is a 3-dimensional Heisenberg algebra.

#### 2.5.1

Firstly, we fix , for which , as an element orthogonal to with respect to . Then is determined by

 [e1,e2]=e2,[e1,e3]=12e3,[e1,e4]=12e4,[e3,e4]=12e2. (34)

Then we compute covariant derivatives and the nontrivial ones are

 ∇e2e2=−2∇e3e3=−2∇e4e4=e1, (35) −∇e2e1=4∇e3e4=−4∇e4e3=e2, −4∇e2e4=−2∇e3e1=−4∇e4e2=e3, 4∇e2e3=4∇e3e2=−2∇e4e1=e4.

Analogously of the previous cases we obtain the non-zero components , , as follows:

 (F1)314 =−(F1)323=(F1)332=−(F1)341= (36) =−(F1)413=−(F1)424=(F1)431=(F1)442=14; (F2)214 =−(F2)223=−(F2)232=(F2)241=−54, (F2)311 =−2(F2)322=(F2)333=−2(F2)344=−1, (F2)412 =(F2)421=(F2)434=(F2)443=−34; (F3)213 =(F3)224=(F3)231=(F3)242=−54, (F3)312 =(F3)321=−(F3)334=−(F3)343=−34, (F3)411 =−2(F3)422=−2(F3)433=(F3)444=1.

The only non-zero components of the corresponding Lee forms are

 (θ1)2=−12,(θ2)3=−(θ3)4=3. (37)

The results in (36), (37) and the classification conditions (9), (10) imply

###### Proposition 2.6.

The hypercomplex manifold with Hermitian-Norden metrics on a 4-dimensional Lie algebra, determined by (34), belongs to the largest class of the considered manifolds, i.e. , as well as this manifold does not belong to neither nor for and .

#### 2.5.2

The other possibility is to choose , for which , as an element orthogonal to with respect to . We rearrange the basis in (34) and then is determined by

 [e1,e2]=−12e3,[e1,e4]=−12e1,[e2,e4]=−12e2,[e3,e4]=−e3. (38)

By similar computations we establish the same statement as of Proposition 2.6 for the Heisenberg algebra introduced by (38).

## References

• [1] Alekseevsky, D.V., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. CLXXI (IV), 205–273 (1996)
• [2] Barberis, M.L.: Hypercomplex structures on four-dimensional Lie groups. Proc. AMS 128 (4), 1043–1054 (1997)
• [3] Barberis, M.L., Dotti, I.: Abelian complex structures on solvable Lie algebras. J. Lie Theory 14 (1), 25–34 (2004)
• [4] Barberis, M.L., Dotti, I., Miatello, R.: On certain locally homogeneous Clifford manifolds, Ann. Glob. Anal. Geom. 13, 289–301 (1995)
• [5] Dotti, I., Fino, A.: Hyper-Kähler with torsion structures invariant by nilpotent Lie groups, Class. Quantum Grav. 19, 1–12 (2002)
• [6] Fino, A., Grantcharov, G.: Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189, 439–450 (2004)
• [7] Ganchev, G., Borisov, A.: Note on the almost complex manifolds with a Norden metric. C. R. Acad. Bulgare Sci. 39, 31–34 (1986)
• [8] Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. CXXIII (IV), 35–58 (1980)
• [9] Gribachev, K., Manev, M.: Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure. J. Geom. 88 (1-2), 41–52 (2008)
• [10] Gribachev, K., Manev, M., Dimiev, S.: On the almost hypercomplex pseudo-Hermitian manifolds. In: Dimiev, S., Sekigawa, K. (eds.) Trends of Complex Analysis, Differential Geometry and Mathematical Physics, pp. 51–62. World Sci. Publ., Singapore (2003)
• [11] Manev, M.: A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics. J. Geom. Phys. 61 (1), 248–259 (2011)
• [12] Ovando, G.: Invariant complex structures on solvable real Lie groups. Manuscripta Math. 103, 19–30 (2000)
• [13] Snow, J. E.: Invariant complex structures on four-dimensional solvable real Lie groups. Manuscripta Math. 66, 397–412 (1990)
• [14] Sommese, A.: Quaternionic manifolds, Math. Ann. 212, 191–214 (1975)
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