Killing-Yano 2-forms on 2-step nilpotent Lie groups
Adri\'an Andrada, Isabel G. Dotti

TL;DR
This paper characterizes 2-step nilpotent Lie groups that admit non-degenerate left invariant Killing-Yano 2-forms, showing they are precisely the complex Lie groups, with a focus on those from connected graphs.
Contribution
It identifies the specific class of 2-step nilpotent Lie groups supporting non-degenerate Killing-Yano 2-forms and describes the structure of these forms in graph-derived cases.
Findings
Only complex Lie groups among 2-step nilpotent groups admit such forms.
The space of invariant Killing-Yano 2-forms is one-dimensional for graph-derived groups.
Non-degenerate forms are exclusive to complex Lie groups in this setting.
Abstract
In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensional.
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Killing-Yano -forms on 2-step nilpotent Lie groups
Adrián Andrada
and
Isabel G. Dotti
FaMAF-CIEM, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina
Abstract.
In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensional.
Key words and phrases:
Killing-Yano forms; parallel tensors; nilpotent Lie groups
2010 Mathematics Subject Classification:
53C15, 22E25, 53C30
Both authors were partially supported by CONICET, ANPCYT and SECYT-UNC (Argentina).
1. Introduction
A -form on a Riemannian manifold is called Killing-Yano if it satisfies the Killing-Yano equation,
[TABLE]
where is the Levi-Civita connection and are arbitrary vector fields on . These forms were introduced by K. Yano in [16] and they can be considered as generalizations of Killing vector fields. Indeed, a -form is Killing-Yano if and only if its dual vector field is Killing.
Clearly, any parallel -form is Killing-Yano. However, it is not easy to find examples of non-parallel Killing-Yano -forms, as the following results show:
- •
If is a compact manifold with holonomy then any Killing-Yano form is parallel [15].
- •
Every Killing-Yano -form on a compact quaternion-Kähler manifold is parallel for any [14].
- •
A compact simply connected symmetric space carries a non-parallel Killing-Yano -form () if and only if it is isometric to a Riemannian product , where is a round sphere and [6].
A natural source of examples in Riemannian geometry is provided by Lie groups equipped with left invariant Riemannian metrics. The study of left invariant Killing-Yano 2-forms on Lie groups with left invariant metrics started in [5] and was continued in [3, 4]. In particular, it was proved in [4] that there are no left invariant parallel non-degenerate 2-forms on nilpotent Lie groups. Therefore, if we are able to find non-degenerate Killing-Yano 2-forms on a nilpotent Lie group, then they will be automatically non-parallel. In this direction, the only known examples of left invariant Killing-Yano 2-forms on nilpotent Lie groups appear on 2-step nilpotent Lie groups, that is, when the commutator ideal is contained in the center of the group (see [5, Theorem 5.1]). In this article we refine this result by showing that non-degenerate Killing-Yano 2-forms appear only on complex 2-step nilpotent Lie groups endowed with a left invariant metric (Theorem 3.7) and, furthermore, any such Lie group admits a left invariant metric with non-degenerate Killing-Yano 2-forms (Proposition 3.5). In Section 4 we exhibit a family of examples where the dimension of the space of left invariant Killing-Yano 2-forms is 1.
2. Preliminaries
A -form on a Riemannian manifold is called Killing-Yano if it satisfies the Killing-Yano equation,
[TABLE]
where is the Levi-Civita connection and are arbitrary vector fields on (see [16]). The following result is clear.
Lemma 2.1**.**
Let be a Riemannian manifold, the Levi-Civita connection and a -form on . The following conditions are equivalent:
; 2.
**
Let be a Riemannian manifold, the Levi-Civita connection and a -form on satisfying any of the conditions of Lemma 2.1. Then the skew-symmetric endomorphism of defined by and , that is satisfies
[TABLE]
for all vector fields on . Conversely, if is a skew-symmetric endomorphism of satisfying (3) then the 2-form satisfies any of the conditions of Lemma 2.1. After this observation we will refer equivalently to a 2-form satisfying (2) or a skew-symmetric -tensor satisfying or (3) as Killing-Yano (KY). Note that if is an almost Hermitian structure, then the fundamental -form given by is Killing-Yano if and only if is nearly Kähler.
Let be an -dimensional Lie group and let be the associated Lie algebra of all left invariant vector fields on . If is the tangent space of at , the identity of , the correspondence from is a linear isomorphism. This isomorphism allows to define a Lie algebra structure on the tangent space setting, for , where are the left invariant vector fields defined by , respectively.
A left invariant metric on is a Riemannian metric such that , the left multiplication by , is an isometry for every . Conversely, every inner product on gives rise, by left translations, to a left invariant metric. A Lie group equipped with a left invariant metric is therefore a homogeneous Riemannian manifold where many geometric invariants can be computed at the Lie algebra level. In particular, the Levi-Civita connection associated to a left invariant metric , when applied to left invariant vector fields, is given by:
[TABLE]
where is the inner product induced by on . Note that is equivalent to for any .
A left invariant -form on is a -form such that for all . We will consider left invariant -forms on satisfying (2). Since and are left invariant as well, we will study \omega\in\raise 1.0pt\hbox{\bigwedge}^{2}\mathfrak{g}^{*} satisfying (2) for . We will say then that is a Killing-Yano (KY) -form on . Clearly, the corresponding skew-symmetric -tensor defined by and is left invariant, so it is determined by its value in , which will be denoted by . The endomorphism will be called a Killing-Yano (KY) tensor on .
Lemma 2.2**.**
Let be a KY tensor on a Lie algebra with inner product . If is a -invariant Lie subalgebra of then is a KY tensor on .
Proof.
Let denote the Levi-Civita connection on associated to the restriction of to , and let denote the Levi-Civita connection on associated to . Then it follows easily from (4) that
[TABLE]
Therefore if is KY on then is KY on . ∎
We recall also the following result proved in [4], which holds in particular for non abelian nilpotent Lie groups.
Theorem 2.3**.**
[4, Theorem 5.1]** If is a Lie algebra with an inner product such that , then there is no skew-symmetric invertible parallel tensor on . In particular, this holds for any inner product on a non-abelian nilpotent Lie algebra.
3. Main results
We will consider next the case of a -step nilpotent Lie algebra equipped with an inner product . We recall that a Lie algebra is called -step nilpotent if it is not abelian and , or equivalently, the commutator ideal is contained in the center of . Such a Lie algebra is unimodular and its associated simply connected Lie group is diffeomorphic to a Euclidean space via the exponential map.
If is a -step nilpotent Lie algebra equipped with an inner product , then there is an orthogonal decomposition , where is the center of and is its orthogonal complement.
In [5] a characterization of -step nilpotent Lie groups with left invariant metrics admitting a left invariant Killing-Yano tensor was obtained. Namely, the following result was proved:
Theorem 3.1**.**
[5, Theorem 3.1]** Let be a skew-symmetric endomorphism of . Then is a Killing-Yano tensor on if and only if
[TABLE]
and
[TABLE]
Note that (6) implies for any .
Corollary 3.2**.**
There are no nearly Kähler structures on 2-step nilpotent Lie algebras.
Proof.
If is a nearly Kähler structure on a 2-step nilpotent Lie algebra , then is a KY tensor on . Therefore, for ,
[TABLE]
and therefore , so that would be abelian. ∎
We show next that the existence of a KY tensor on a 2-step nilpotent Lie algebra imposes strong restrictions on . Indeed,
Theorem 3.3**.**
Let be a -step nilpotent Lie algebra with an inner product . If is a KY tensor on then:
* is isometrically isomorphic to a direct product of ideals , where , is -invariant and is an invertible KY tensor on ;* 2.
* is parallel if and only if is abelian. Moreover, if then .*
Proof.
Since is skew-symmetric we have an orthogonal decomposition into -invariant subspaces. Recall that we also have another orthogonal decomposition , where is the center of . According to Theorem 3.1 both and are -invariant, therefore, if is decomposed as with and , then with and . Hence, and , so that . In the same way it is shown that .
We will show next that both and are ideals of . Indeed:
- •
if , then , with , and , . Hence, , using (6). Thus is an ideal of .
- •
if and , then and with and . Then we have , thus is an ideal of .
Therefore, setting and we obtain . Clearly, , is -invariant and is invertible. Moreover, it follows from Lemma 2.2 that is a KY tensor on . This proves .
It is clear that is either abelian or -step nilpotent. If is -step nilpotent, it follows from Theorem 2.3 that is not parallel, since it is invertible. Now, if is abelian, then this means that . Hence and it follows from (6) that . It is easy to verify that these conditions imply that is parallel, and if . Therefore follows. ∎
Remark 3.4*.*
Note that when the KY tensor is invertible the KY 2-form associated to , , is non-degenerate.
As a consequence of Theorem 3.3, we may restrict ourselves to the study of invertible KY tensors on 2-step nilpotent Lie groups. A large number of such examples appear in complex 2-step nilpotent Lie groups, as the following result shows:
Proposition 3.5**.**
Let be a complex 2-step nilpotent Lie group equipped with a left invariant Hermitian metric . Then admits a non-parallel invertible KY tensor.
Proof.
Let denote the associated bi-invariant complex structure on . Therefore the restriction of to , the Lie algebra of , satisfies: , and for any , i.e., for any .
Let denote the induced inner product on . If denotes the center of , let , so that , orthogonal sum. Note that both and are -invariant, as well as the commutator ideal .
Let us define a skew-symmetric isomorphism of in the following way: if we decompose as an orthogonal sum, we set
[TABLE]
where is any skew-symmetric isomorphism of . It follows from Theorem 3.1 that is a left invariant Killing-Yano tensor on . Moreover, according to Theorem 2.3, is not parallel. ∎
Remark 3.6*.*
We point out that the classification of complex 2-step nilpotent Lie algebras, up to isomorphism, is not known; furthermore, this is considered to be a wild problem in the literature (see for instance [12], where several partial results are collected).
The next theorem will show that the complex 2-step nilpotent Lie groups exhaust the family of 2-step nilpotent Lie groups with left invariant metric admitting invertible KY tensors.
Theorem 3.7**.**
If a 2-step nilpotent Lie group equipped with a left invariant metric admits an invertible left invariant KY tensor then is a complex Lie group. Moreover, is Hermitian with respect to this complex structure.
Proof.
Let be a -step nilpotent Lie group equipped with a left invariant metric and a left invariant Killing-Yano tensor . We denote by the induced inner product on , and we also denote by the restriction of to . Therefore, according to Theorem 3.1, preserves the center of and (6) holds.
Let us consider the orthogonal decomposition , where and . Note that due to (6), preserves each of these subspaces, since is skew-symmetric. Therefore, there exists an orthonormal basis of such that
[TABLE]
for some non-zero . Moreover, interchanging with if necessary, we may assume that for all and, after a further reordering of the basis, we may assume also that . Let us denote , so that .
It follows from (6) that . Indeed, , hence . In particular, for each there exists such that , since otherwise . Let be a pair of indices such that . Then at least one of , , and is non-zero. Then, according to (6), we have that
[TABLE]
Since is invertible, it follows that all four Lie brackets are non-zero. Furthermore, we have that
[TABLE]
As the constants are positive, we obtain . Hence, , and , with , and is -invariant.
Let us define a series of integers in the following way:
[TABLE]
We define now the following subspaces of :
[TABLE]
Clearly and, moreover, if . It then follows that the Lie subalgebras of defined by are in fact ideals of .
Next note that , while , according to (6). As a consequence, since , we have that and furthermore,
[TABLE]
a direct product of ideals. In order to show that carries a bi-invariant complex structure, set as follows:
- (i)
is any almost complex structure on compatible with ; 2. (ii)
; 3. (iii)
.
Clearly, . In order to verify that it is bi-invariant, we need only check that for any for some . Indeed,
[TABLE]
It is clear from the construction that is skew-symmetric with respect to . ∎
Remark 3.8*.*
It follows from the proof of Theorem 3.7 that can be decomposed as , an orthogonal product of -invariant ideals, with abelian and 2-step nilpotent. As a consequence, if is irreducible (i.e., it cannot be decomposed as an orthogonal direct product of ideals) then , , and the endomorphisms and of can be written, according to the orthogonal decomposition , as
[TABLE]
in some orthonormal basis of and , for some . Here denotes the identity matrix, and .
Remark 3.9*.*
It was proved in [1] that any left invariant Hermitian metric on a unimodular complex Lie group is balanced, i.e. its fundamental 2-form is co-closed. It thus follows that the Hermitian metric in Theorem 3.7 is balanced with respect to the bi-invariant complex structure .
Remark 3.10*.*
It is known that if is a KY tensor on a Riemannian manifold then is a Killing tensor on (see for instance [15, 9]). This means that is symmetric and for any vector field on . In contrast with the KY case, it was shown in [9] that many 2-step nilpotent Lie groups, not only the complex ones, admit Killing tensors for certain left invariant metrics.
Remark 3.11*.*
A symmetric -tensor on a Riemannian manifold is called a Codazzi tensor if it satisfies for any vector fields on . In the case of a left invariant metric on a 2-step nilpotent Lie group, an analogous result to Theorem 3.1 holds. Indeed, a symmetric endomorphism of the Lie algebra is a Codazzi tensor if and only if and for all . However, in this case, it is easy to see that these conditions also imply that is parallel (compare [8, Proposition 2]).
4. Space of solutions
In this section we will show that for certain 2-step nilpotent complex Lie groups equipped with left invariant metrics, the space of Killing-Yano tensors is one-dimensional.
We will be concerned with 2-step nilpotent complex Lie groups arising from graphs. We recall briefly this construction, introduced in [7].
Let be a simple, undirected graph with set of vertices and set of edges , with . Let and for a field with characteristic different from 2, consider the Lie algebra whose underlying -vector space has as a basis, and the Lie bracket is given by:
[TABLE]
Clearly, is a 2-step nilpotent Lie algebra, and it satisfies the following very strong condition:
[TABLE]
Many properties of can be deduced directly from . For instance, the center of is spanned by , and is indecomposable if and only if is connected. Moreover, it was proved in [13] that given two graphs and , the Lie algebras and are isomorphic if and only if and are isomorphic.
Example 1**.**
The 3-dimensional Heisenberg Lie algebra over arises from the graph with two vertices joined by one edge. Moreover, it is clear from (8) that the -dimensional Heisenberg Lie algebra arises from a graph if and only if .
We will study next KY tensors on 2-step nilpotent complex Lie algebras arising from graphs. So, let be a graph as before, and let be the complex 2-step nilpotent Lie algebra arising from . When considered as a real Lie algebra, it will be denoted simply by . It has a real basis such that its natural bi-invariant complex structure is given by . The Lie bracket on satisfies
[TABLE]
Note that is not a real 2-step nilpotent Lie algebra arising from a graph, since it does not satisfy (8).
Let us define an inner product on by declaring the basis to be orthonormal. If has no isolated vertices then the center of is , and its orthogonal complement is .
According to Proposition 3.3, admits KY tensors. In the next result we determine the dimension of the space of KY tensors on when is connected.
Theorem 4.1**.**
When the graph is connected, the space of KY tensors on has dimension 1.
Proof.
Let be a KY tensor on . According to Theorem 3.1, preserves both and and, moreover, for any . Since is an orthonormal basis of , we have that
[TABLE]
Let us fix two indices with . We will show that
[TABLE]
First, let us assume that . It follows from that
[TABLE]
The first sum in the left-hand side lies in , whereas the second sum lies in . Therefore, each sum vanishes and, taking (8) into account, we have that and , which imply . Beginning with , the same computation gives .
Let us now assume that . There exists such that , since otherwise would be an isolated vertex of . Since , and using (9), we obtain
[TABLE]
The first sum in each side lies in , whereas the second sum in each side lies in . Therefore we obtain that
[TABLE]
The non-zero element appears in the left-hand side of (11), but if it appeared also in the right-hand side, according to (8) we would have that for some , and this is impossible since both and are different from . It follows that . The same reasoning can be applied in (12) (since also ), hence we obtain .
Analogously, beginning with , we prove that .
Thus, (10) holds and since is skew-symmetric we have that and for some , . We will show next that for all . Let us assume first that . It follows from (6) that and therefore we obtain that
[TABLE]
Therefore, . If, on the other hand, , we can choose a sequence of indices such that , since is connected. As a consequence, we have that . Setting for any , the KY tensor is given by
[TABLE]
Hence the space of KY tensors on has dimension 1. ∎
Corollary 4.2**.**
When is connected, any non-zero Killing-Yano tensor on is invertible, thus non-parallel.
Note that in the proof of Theorem 4.1, the main properties used were the skew-symmetry of and equation (6). The constant appearing in this formula is not relevant for the proof, and as a consequence we obtain the following result (compare with [11, Section 5.1]).
Proposition 4.3**.**
If is a bi-invariant complex structure on which is Hermitian with respect to , then .
We learnt recently that V. del Barco and A. Moroianu obtained in [10] the uniqueness, up to sign, of the orthogonal bi-invariant complex structure on a 2-step nilpotent metric Lie group which is de Rham irreducible.
Remark 4.4*.*
As mentioned in Example 1, the only complex Heisenberg Lie algebra which can be obtained from a graph is . However, it is easy to see that for , with its canonical -type Hermitian metric has a space of KY tensors of dimension 1.
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