[
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 structures with Hermitian-Norden metrics]
Hypercomplex structures with Hermitian-
Norden metrics
on four-dimensional Lie
algebras
M. Manev]Mancho Manev
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 :
(1) |
where denotes the identity and moreover, it is valid
(2) |
for the Nijenhuis tensors of given by
(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
(4) |
where
Moreover, the associated (Kähler) 2-form and the associated neutral metrics and are determined by
(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
(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
(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
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]:
(11) |
Let us introduced a pseudo-Euclidian metric with neutral signature as follows
(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
(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
(14) |
By virtue of (14), (11) and (6), we obtain components , , as follows:
(15) | ||||
The only non-zero components , , of the corresponding Lee forms are
(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
(17) |
Then we compute covariant derivatives and the nontrivial ones are
(18) |
By virtue of (17), (11) and (6), we obtain that and the other components , , are as follows
(19) | ||||
The only non-zero components of the corresponding Lee forms are
(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.
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
(21) |
Then we compute covariant derivatives and the nontrivial ones are
(22) |
By virtue of (21), (11) and (6), we obtain the following components of :
(23) | ||||
The only non-zero components of the corresponding Lee forms are
(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:
(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.
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
(26) |
Then we compute covariant derivatives and the nontrivial ones are
(27) |
By similar computation as in the previous cases, the components , , are as follows:
(28) | ||||
The only non-zero components of the corresponding Lee forms are
(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
(30) |
Therefore, the nontrivial covariant derivatives are
(31) |
In a similar way we obtain:
(32) | ||||
The only non-zero components of the corresponding Lee forms are
(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
(34) |
Then we compute covariant derivatives and the nontrivial ones are
(35) | ||||
Analogously of the previous cases we obtain the non-zero components , , as follows:
(36) | ||||
The only non-zero components of the corresponding Lee forms are
(37) |
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
(38) |
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