Abstract
The aim of this paper is to construct leftinvariant Einstein pseudoRiemannian Sasaki metrics on solvable Lie groups. We consider the class of \(\mathfrak {z}\)standard Sasaki solvable Lie algebras of dimension \(2n+3\), which are in onetoone correspondence with pseudoKähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudoKähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify \(\mathfrak {z}\)standard Sasaki solvable Lie algebras of dimension \(\le 7\) and those whose pseudoKähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudoIwasawa type.
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1 Introduction
An effective method to construct Einstein metrics is by considering invariant metrics on a solvmanifold obtained by extending a suitable metric on a nilpotent Lie group of codimension one. Indeed, in the Riemannian case, Einstein solvmanifolds are described by a standard solvable Lie algebra \(\widetilde{\mathfrak {g}}\) of Iwasawa type ([22, 26]). In particular, this means that \(\widetilde{\mathfrak {g}}\) admits an orthogonal decomposition \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes \mathfrak {a}\), with \(\mathfrak {g}\) nilpotent, \(\mathfrak {a}\) abelian and \({{\,\textrm{ad}\,}}X\) selfadjoint whenever X is in \(\mathfrak {a}\). Furthermore, the restriction of the metric to \(\mathfrak {g}\) satisfies the socalled nilsoliton equation ([25]).
Things are more complicated in the indefinite case (see e.g. [14]), but it is still possible to construct Einstein solvmanifolds by extending a nilsoliton; indeed, there is a correspondence between nilsolitons and a class of Einstein solvmanifolds for which \(\widetilde{\mathfrak {g}}\) admits a decomposition as above, called a pseudoIwasawa decomposition (see [15]).
In the noninvariant setting, Einstein metrics are often studied in the presence of additional structures, such as a Killing spinor or a restriction on the holonomy (see, e.g., [2, 3]). It is then natural to ask whether Einstein metrics compatible with such special structures can be obtained in the invariant setting too.
This paper is focused on Sasaki metrics. More precisely, we consider a class of leftinvariant pseudoRiemannian Sasaki–Einstein metrics on solvable Lie groups. Sasaki–Einstein metrics admit a Killing spinor (see [20]) and may be viewed as the odddimensional counterpart of Kähler–Einstein geometry. We showed in [12] that Sasaki Lie algebras can never be of pseudoIwasawa type, regardless of whether they are Einstein; therefore, Sasaki–Einstein solvmanifolds cannot be obtained by extending a nilsoliton.
We therefore consider a different construction, represented pictorially in Fig. 1; we illustrate it in the invariant case, though it holds more generally.
Suppose that \(\widetilde{G}\) is a solvable Lie group with a leftinvariant pseudoRiemannian Sasaki structure and X is a leftinvariant, nonnull Killing field preserving the structure; one can then consider the moment map \(\widetilde{\mu }\) and the contact reduction \(\widetilde{G}//X\). Suppose that the zerolevel set G of the moment map is a nilpotent Lie subgroup, and X is in the center of its Lie algebra; then, the contact reduction \(\widetilde{G}//X\) is also a Lie group with a leftinvariant Sasaki structure. Notice that we do not assume that X is in the center of the Lie algebra of \(\widetilde{G}\); therefore, the quotient \(\widetilde{G}/\mathcal {F}_X\), which we omitted in the diagram, is not necessarily a Lie group.
Assuming that the Reeb vector field \(\xi \) is central, we have that the quotient by the Reeb foliation \(\widetilde{G}/\mathcal {F}_\xi \) is also a Lie group, with a leftinvariant pseudoKähler structure. Then, its symplectic reduction is a pseudoKähler nilpotent Lie group , which can also be described as the quotient of the contact reduction \(\widetilde{G}//X\) by the Reeb foliation. In accord with [12], we call the Kähler reduction of \(\widetilde{G}\). Notice that is Ricciflat; this is a general property of pseudoKähler nilpotent Lie groups (see [19]).
Our aim is to obtain Sasaki–Einstein solvmanifolds by inverting the diagram of Fig. 1. The straight arrows have natural inverses: one takes a circle bundle with curvature determined by the Kähler form or \(dX^\flat \) (see [21]). By contrast, the curly arrows are not bijections in general: if \(\nabla X^\flat =dX^\flat \) is known, only the metric on \(\mu ^{1}(0)\) and the second fundamental form are determined, but this does not determine the metric on the full group \(\widetilde{G}\), for general Sasaki metrics. However, the metric is determined by its restriction to the hypersurface and its second fundamental form if one requires \(\widetilde{G}\) to be Einstein, according to [24].
More explicitly, symplectic reduction can be inverted as follows. The pseudoKähler Lie group is endowed with a closed (1, 1)form \(\gamma \), corresponding to \(dX^\flat \). We can describe \(\widetilde{G}/\mathcal {F}_\xi \) as the product of a circle bundle with curvature \(\gamma =d\theta \) and a line, endowed with the complex structure for which (1, 0)forms are generated by the pullback of (1, 0)forms on along with \(dr+ie^{2r}\theta \) and the Kähler form
where \(\omega \) denotes the Kähler form of . It turns out that this can be written as a leftinvariant metric on a solvable Lie group \(\widetilde{G}/\mathcal {F}_\xi \).
In our invariant setup, the hypersurface G in \(\widetilde{G}\) corresponds to a nilpotent ideal of codimension one in a solvable Lie algebra, and the construction can be studied with the language of standard Lie algebras. However, rather than the Lie algebras of (pseudo)Iwasawa type studied in [15, 22], we need to consider a different class, that we introduced in [12] under the name of \(\mathfrak {z}\)standard Sasaki Lie algebras. This condition means that the Lie algebra \(\widetilde{\mathfrak {g}}\) is endowed with a Sasaki structure \((\phi ,\xi ,\eta ,g)\) such that \(\widetilde{\mathfrak {g}}\) takes the form of an orthogonal semidirect product \(\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), with \(\phi (e_0)\) a central element of \(\mathfrak {g}\) corresponding to the vector field X in Fig. 1 (see Sect. 1 for the precise definitions). It then becomes possible to characterize and study the construction of Fig. 1 in purely algebraic terms. In particular, we showed in [12] that the Lie algebra of the Kähler reduction comes endowed with a derivation satisfying certain conditions; in Fig. 1, determines \(\widetilde{G}/\mathcal {F}_{\xi ,X}\) as a semidirect product .
In this paper, we specialize this construction to the Sasaki–Einstein case. As a first step, we introduce a generalization of the nilsoliton condition that enables one to construct Einstein solvmanifolds, which are not of pseudoIwasawa type, by taking a semidirect product with \(\mathbb {R}\); these metrics are not necessarily Sasaki (Proposition 2.1). We then characterize \(\mathfrak {z}\)standard Sasaki–Einstein solvable Lie algebras in terms of their Kähler reduction , showing that the symmetric part of the derivation is the identity and preserves the pseudoKähler structure. This implies that lies in the Lie algebra , where \(\mathfrak {cu}(p,q)=\mathfrak {u}(p,q)\oplus {\text {Span}}\left\{ {\text {Id}}\right\} \).
In the opposite direction, as sketched in Fig. 2, we show that any pseudoKähler nilpotent Lie algebra with a derivation whose symmetric part is the identity induces:

an Einstein metric on , corresponding to \(\widetilde{G}/\mathcal {F}_{\xi ,X}\) in Fig. 1;

a pseudoKähler–Einstein metric with positive curvature on a solvable double extension \(\widetilde{\mathfrak {k}}\), corresponding to \(\widetilde{G}/\mathcal {F}_{\xi }\). Geometrically, this corresponds to setting \(\gamma =2\omega \) and applying the ansatz (1). Notice that if one flips a sign in (1) and writes \(e^{2r}\omega + e^{2r} dr\wedge \theta \), one obtains the construction of a (Riemannian) Kähler–Einstein metric with negative curvature on the bundle over a Kähler Ricciflat manifold given in [4, § 11.8] (see also [31, Equation (3.20)] and [5, Theorem 9.129]).

a Sasaki–Einstein metric on a central extension of \(\widetilde{\mathfrak {k}}\), corresponding to \(\widetilde{G}\) in the diagram. Notice that having chosen the metric with positive curvature on \(\widetilde{\mathfrak {k}}\) is essential for this step, due to the constraints on the space of leaves of a Sasaki–Einstein manifold.
We then show that any two choices of on with symmetric part equal to the identity determine isometric Einstein extensions; the isometry is at the level of solvmanifolds as pseudoRiemannian manifolds, and it does not preserve the Lie algebra structure (Theorem 4.1).
It turns out that contains elements with symmetric part equal to the identity if and only if it contains an element with nonzero trace; in that case, we show that the Lie algebra contains a canonical choice of . This canonical element is obtained by adapting a construction of Nikolayevsky. Indeed, we fix an algebraic subalgebra \(\mathfrak {h}\) of \(\mathfrak {gl}(m,\mathbb {R})\) and define the \(\mathfrak {h}\)Nikolayevsky derivation on a Lie algebra \(\mathfrak {g}\) with a Hstructure as the unique semisimple derivation N in \(\mathfrak {h}\cap {{\,\textrm{Der}\,}}\mathfrak {g}\) such that
For \(\mathfrak {h}=\mathfrak {gl}(m,\mathbb {R})\), one obtains the Nikolayevsky derivation introduced in [28], and for \(\mathfrak {h}=\mathfrak {co}(p,q)\) the metric Nikolayevsky derivation of [11]. Existence and uniqueness of the \(\mathfrak {h}\)Nikolayevsky derivation is proved similarly as in these particular cases (Proposition 2.7).
The relevant situation for this paper is the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation of a pseudoKähler Lie algebra, which turns out to have rational eigenvalues, like the ordinary Nikolayevsky derivation. Thus, we see that the element of that determines the Sasaki–Einstein extension can be assumed to be diagonalizable over \(\mathbb {Q}\) (Proposition 2.8). The existence of a nonzero \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation guarantees that there is a standard Einstein extension, corresponding to \(\widetilde{G}/\mathcal {F}_{\xi ,X}\) in Fig. 1.
The characterization of \(\mathfrak {z}\)standard Sasaki–Einstein solvmanifolds in terms of their Kähler reduction allows us to classify all \(\mathfrak {z}\)standard Sasaki–Einstein solvmanifolds of dimension 7 (Theorem 4.4). In addition, we are able to write down all \(\mathfrak {z}\)standard Sasaki–Einstein solvmanifolds that reduce to a pseudoKähler which is abelian as a Lie algebra (Corollary 4.2). This includes all Lorentzian \(\mathfrak {z}\)standard Sasaki–Einstein solvmanifolds, for which is forced to be a nilpotent Kähler Lie algebra, hence abelian.
In particular, our results give rise to explicit pseudoKählerEinstein and Sasaki–Einstein solvmanifolds in all dimensions \(\ge 4\) (Theorem 4.4, Remark 4.5). These metrics are not Ricciflat, which is a general fact for Sasaki–Einstein metrics and their KählerEinstein quotients.
We point out that our Sasaki–Einstein metrics are examples of Einstein standard solvmanifolds that are not isometric to any Einstein solvmanifold of pseudoIwasawa type. This is in sharp contrast to the Riemannian case, where [22] shows that all standard Einstein solvmanifolds are of Iwasawa type up to isometry.
We also show with an example that not all pseudoKähler Lie algebras can be extended to a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra (Example 4.6).
2 Preliminaries: structures on Lie algebras
In this section, we introduce some general language relevant to the study of Sasaki–Einstein metrics, specialized to the invariant setting, and recall some results that will be needed in the sequel.
Given a Lie algebra \(\mathfrak {g}\) of dimension m, we can think of a basis of \(\mathfrak {g}\) as a frame \(\mathbb {R}^m\cong \mathfrak {g}\). There is a natural right action of \(\textrm{GL}(m,\mathbb {R})\) on the set of frames. Given a subgroup \(H\subset \textrm{GL}(m,\mathbb {R})\), we will say that a Hstructure on \(\mathfrak {g}\) is a Horbit in the space of frames. Given any frame u, the identification \(u:\mathbb {R}^m\cong \mathfrak {g}\) induces a left action of H on \(\mathfrak {g}\). This induces an inclusion map \(H\rightarrow \textrm{GL}(\mathfrak {g})\) that depends on the frame u, but the image of the inclusion only depends on the Hstructure. Accordingly, whenever we have a Hstructure on \(\mathfrak {g}\), we will write \(H\subset \textrm{GL}(\mathfrak {g})\), \(\mathfrak {h}\subset \mathfrak {gl}(\mathfrak {g})\).
It is clear that a Hstructure on a Lie algebra \(\mathfrak {g}\) induces a leftinvariant Hstructure, in the usual sense, on any Lie group with Lie algebra \(\mathfrak {g}\).
An almost contact structure on a \((2n + 1)\)dimensional Lie algebra \(\widetilde{\mathfrak {g}}\) is a triple \((\phi , \xi , \eta )\), where \(\phi \) is a linear map from \(\widetilde{\mathfrak {g}}\) to itself, \(\xi \) is an element of \(\widetilde{\mathfrak {g}}\), \(\eta \) is in \(\widetilde{\mathfrak {g}}^*\) and
Given a nondegenerate scalar product g on \(\widetilde{\mathfrak {g}}\), the quadruple \((\phi ,\xi ,\eta ,g)\) is called an almost contact metric structure if \((\phi , \xi , \eta )\) is an almost contact structure and
for any \(X,Y\in \mathfrak {g}\). One then defines a twoform \(\Phi \) by \(\Phi (X,Y)=g(X,\phi Y)\).
Given an almost contact metric structure of signature \((2p+1,2q)\) on \(\mathfrak {g}\), one can find a frame \(e_1,\dotsc , e_{2n+1}\) with dual basis \(\{e^i\}\) such that
The common stabilizer of these tensors in \(\textrm{GL}(2p+2q+1,\mathbb {R})\) is \(\textrm{U}(p,q)\). Thus, an almost contact metric structure on a Lie algebra can be viewed as a \(\textrm{U}(p,q)\)structure.
An almost contact metric structure is called Sasaki if \(N_{\phi } + d\eta \otimes \xi = 0\) and \(d\eta = 2\Phi \), where \(N_\phi \) denotes the Nijenhuis tensor.
These definitions mimic analogous definitions for structures on manifolds (see, e.g., [7, 8]). It is clear that a Sasaki structure on \(\widetilde{\mathfrak {g}}\) defines a leftinvariant almost Sasaki structure on any Lie group \(\widetilde{G}\) with Lie algebra \(\widetilde{\mathfrak {g}}\) by left translation. In particular, g defines a pseudoRiemannian metric on \(\widetilde{G}\). We shall also refer to g as a metric on \(\widetilde{\mathfrak {g}}\) and define its LeviCivita connection, curvature and so on in terms of the corresponding objects on \(\widetilde{G}\).
Sasaki structures are an odddimensional analogue of (pseudo)Kähler structures. In our invariant setting, a pseudoKähler structure on a Lie algebra \(\mathfrak {g}\) is a triple \((g,J,\omega )\), where g is a pseudoRiemannian metric, \(J:\mathfrak {g}\rightarrow \mathfrak {g}\) is an almost complex structure satisfying the compatibility condition \(g(JX,JY)=g(X,Y)\), and \(\omega (X,Y)=g(X,JY)\), where one further imposes that \(N_J=0\) and \(d\omega =0\).
Having fixed the metric, for any endomorphism \(f:\widetilde{\mathfrak {g}}\rightarrow \widetilde{\mathfrak {g}}\) we write \(f=f^s+f^a\), where \(f^s\) is symmetric and \(f^a\) is skewsymmetric relative to the metric, i.e.
With this notation, the LeviCivita connection is given by
The Ricci tensor can be written as:
see, e.g., [13, Lemma 1.1]. If \(\widetilde{\mathfrak {g}}\) is unimodular with Killing form equal to zero, this formula simplifies to
We will need a result originally proved in [1] for Riemannian metrics and later adapted to standard indefinite metrics in [15], though the standard condition turned out not to be necessary (see [12]). The precise statement we are going to need is the following:
Proposition 1.1
([1, 12]) Let H be a subgroup of \(\textrm{SO}(r,s)\) with Lie algebra \(\mathfrak {h}\) and \(\widetilde{\mathfrak {g}}\) a Lie algebra of the form \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes \mathfrak {a}\) endowed with a Hstructure. Let \(\chi :\mathfrak {a}\rightarrow {{\,\textrm{Der}\,}}(\mathfrak {g})\) be a Lie algebra homomorphism such that, extending \(\chi (X)\) to \(\widetilde{\mathfrak {g}}\) by declaring it to be zero on \(\mathfrak {a}\),
Let \(\widetilde{\mathfrak {g}}^*\) be the Lie algebra \(\mathfrak {g}\rtimes _\chi \mathfrak {a}\). If \(\widetilde{G}\) and \(\widetilde{G}^*\) denote the connected, simply connected Lie groups with Lie algebras \(\widetilde{\mathfrak {g}}\) and \(\widetilde{\mathfrak {g}}^*\), with the corresponding leftinvariant Hstructures, there is an isometry from \(\widetilde{G}\) to \(\widetilde{G}^*\), whose differential at e is the identity of \(\mathfrak {g}\oplus \mathfrak {a}\) as a vector space, mapping the Hstructure on \(\widetilde{G}\) into the Hstructure on \(\widetilde{G}^*\).
Proof
For \(\mathfrak {h}=\mathfrak {so}(r,s)\), \(\chi (X){{\,\textrm{ad}\,}}X\) is skewsymmetric and the proof is identical to [12], Proposition 2.2]. In general, one uses that \(\chi (X){{\,\textrm{ad}\,}}X\) is in \(\mathfrak {h}\) to conclude that the action of \(G^*\) on \(\widetilde{G}\) preserves the Hstructure.
We will say that two Lie algebras endowed with a Hstructure are equivalent if there is an isometry between the corresponding simply connected Lie groups mapping one Hstructure into the other.
Recall from [15] that a standard decomposition of the Lie algebra with a fixed metric is an orthogonal splitting
where \(\mathfrak {g}\) is a nilpotent ideal and \(\mathfrak {a}\) is an abelian subalgebra. This definition generalizes the definition given in [22] for positivedefinite metrics.
In [12], we introduced a special class of standard Sasaki Lie algebras: if \(\widetilde{\mathfrak {g}}\) has both a Sasaki structure \((\phi , \xi , \eta ,g)\) and a standard decomposition of the form \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), it is called \(\mathfrak {z}\)standard if \(\phi (e_0)\) lies in the center \(\mathfrak {z}\) of \(\mathfrak {g}\). This means that the oneparameter group \(\{\exp tb\}\), with \(b=\phi (e_0)\), acts on the corresponding group in such a way that the contact quotient is still a Lie group; this implies that the pseudoKähler Lie group obtained by quotienting by the Reeb direction has a symplectic reduction which is a pseudoKähler Lie group, motivating the following:
Definition 1.2
Given a \(\mathfrak {z}\)standard Lie algebra \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) with Sasaki structure \((\phi , \xi , \eta ,g)\), the quotient Lie algebra with the metric induced by g and complex structure induced by \(\phi \) is called the Kähler reduction of \(\widetilde{\mathfrak {g}}\).
In fact, \(\mathfrak {z}\)standard Sasaki Lie algebras can be characterized in terms of their Kähler reduction as follows (Corollary 4.4 and Proposition 5.1 in [12]):
Proposition 1.3
([12], Proposition 5.1 and Corollary 4.4]) Let be a pseudoKähler nilpotent Lie algebra. Let be a derivation of , \(\tau =\pm 1\), and a central extension of \(\mathfrak {g}\) with a metric of the form:
where . Assume furthermore

\(d\xi ^\flat =2\omega \), where the righthand side is implicitly pulled back to \(\widetilde{\mathfrak {g}}\);

, where the righthand side is implicitly pulled back to \(\widetilde{\mathfrak {g}}\);

;

for some constant h.
Let \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), where
then \(\widetilde{\mathfrak {g}}\) has a \(\mathfrak {z}\)standard Sasaki structure \((\phi ,\eta ,\xi ,\widetilde{g})\) given by
Conversely, every \(\mathfrak {z}\)standard Sasaki Lie algebra arises in this way.
Figure 2, which should be compared with Fig. 1, summarizes the Lie algebras appearing in Proposition 1.3 and their relations, alongside other related Lie algebras which will appear in the sequel of the paper, namely
 :

a pseudoKähler nilpotent Lie algebra of dimension 2n;
 \(\mathfrak {g}_{\textrm{std}}\):

a solvable standard extension of by the derivation , also a quotient of \(\widetilde{\mathfrak {k}}\) by the noncentral onedimensional ideal \({\text {Span}}\left\{ b\right\} \);
 \(\mathfrak {k}\):

a nilpotent central extension of by the cocycle ;
 \(\mathfrak {g}^\circlearrowright \):

a nilpotent Sasaki central extension of by the cocycle \(d\xi ^\flat =2\omega \);
 \(\widetilde{\mathfrak {k}}\):

a solvable Kähler Lie algebra, which can be obtained as a standard extension of \(\mathfrak {k}\) by the derivation ;
 \(\mathfrak {g}\):

a nilpotent central extension of by and \(d\xi ^\flat =2\omega \);
 \(\widetilde{\mathfrak {g}}\):

a \(\mathfrak {z}\)standard Sasaki Lie algebra of dimension \(2n+3\).
Remark 1.4
Composing a semidirect product with a central extension in such a way that the two new directions span an indefinite twoplane is a procedure known as double extension (see [27]). Our construction is different because the semidirect products and central extensions corresponding to vertical arrows in Fig. 2 have the effect of adding a definite twoplane.
It is well known that, given a Sasaki manifold M, the space of leaves of the Reeb foliation has a pseudoKähler structure (see [29]), and the Ricci tensor of the latter is determined by the Ricci tensor of M (see [7, Theorem 7.3.12]). In our invariant setting, this fact takes the following form:
Proposition 1.5
Let \(\mathfrak {g}\) be a Lie algebra with a Sasaki structure \((\phi ,\xi ,\eta , g)\). Suppose \(\mathfrak {g}\) has nonzero center. Then, \(\mathfrak {z}(\mathfrak {g})={\text {Span}}\left\{ \xi \right\} \) and the quotient has an induced pseudoKähler structure with
where \({{\,\textrm{ric}\,}}\) is restricted to \(\xi ^\perp \) implicitly.
Proof
Any element of the center satisfies \(v\lrcorner \,d\eta =0\), so it is a multiple of \(\xi \). Thus, the kernel coincides with \({\text {Span}}\left\{ \xi \right\} \).
As a vector space, we identify with \(\xi ^\perp \), so that the metric is the restriction of g. The Lie algebra structure of is given by a projection, i.e.
Therefore,
Then, from equation (2), the LeviCivita connections \(\nabla \), are related by
If \(\alpha \in {\text {Ann}}(\xi )\), we have
The Sasaki condition implies
so . This implies that \(d\eta \) defines a pseudoKähler structure on .
The exterior derivative on can be identified with the restriction of d. By (3), we obtain
Remark 1.6
Every Sasaki metric on a manifold of dimension \(2n+1\) satisfies \({{\,\textrm{Ric}\,}}(\xi )=2n\xi \). Accordingly, the space of leaves of a Sasaki–Einstein manifolds satisfies
Conversely, every Kähler–Einstein manifold with positive scalar curvature, suitably normalized so as to satisfy (6), gives rise to a Sasaki–Einstein manifold in one dimension higher.
We recall the following fact from [19]:
Lemma 1.7
[19, Lemma 6.3] PseudoKähler metrics on nilpotent Lie algebras are Ricciflat.
Proof
On a simply connected manifold, it is well known that pseudoKähler metrics have holonomy contained in \(\textrm{U}(p,q)\); Ricciflatness is equivalent to holonomy being contained in \(\textrm{SU}(p,q)\), i.e., the existence of a closed complex volume form.
In the case of a nilpotent Lie algebra \(\mathfrak {g}\), the complex volume form is unique up to multiple; the fact that it is closed can be proved using the methods of [33], or directly as follows.
Let \(\theta _1,\dotsc , \theta _n\) be a complex frame of vectors of type (1, 0), and let \(\theta ^1,\dotsc , \theta ^n\) be the dual coframe of (1, 0)forms. Relative to the splitting \(\mathfrak {g}^{\mathbb {C}}=\mathfrak {g}^{1,0}\oplus \mathfrak {g}^{0,1}\), we have
where \(f_k:\mathfrak {g}^{1,0}\rightarrow \mathfrak {g}^{1,0}\) is nilpotent and hence tracefree. Therefore, for all k we have
implying that the complex volume form \(\theta ^1\wedge \dots \wedge \theta ^n\) is closed.
Proposition 1.8
There is no nilpotent Sasaki–Einstein Lie algebra.
Proof
Let \(\mathfrak {g}\) be a nilpotent Lie algebra with a Sasaki–Einstein structure. We know that its center is spanned by \(\xi \). By Proposition 1.5, the quotient \(\mathfrak {g}/{\text {Span}}\left\{ \xi \right\} \) is pseudoKähler and Einstein with positive scalar curvature. Since it is also nilpotent, it must be Ricciflat by Lemma 1.7, which is absurd.
Another link between Sasaki–Einstein and Kähler–Einstein geometry is given by the following:
Proposition 1.9
([7, Corollary 11.1.8]) Let \((\phi ,\xi ,\eta ,g)\) be an almost contact pseudoRiemannian metric structure on a manifold M of dimension \(2n+1\). The following are equivalent:

1.
\((\phi ,\xi ,\eta ,g)\) is Sasaki–Einstein;

2.
the cone \((\mathbb {R}^+\times M,J,\omega )\) is pseudoKähler and Ricciflat.
3 Einstein standard Lie algebras
In this section, we study the Einstein condition on standard Lie algebras \(\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), without assuming the pseudoIwasawa condition (see [12], Proposition 2.6]). We write down the conditions that the induced metric g and the derivation \(D={{\,\textrm{ad}\,}}e_0\) must satisfy, generalizing the nilsoliton equation. In particular, the conditions are satisfied if g is Ricciflat and the symmetric part of D is an appropriate multiple of the identity.
We then recall and generalize the construction of the Nikolayevsky and metric Nikolayevsky derivation ([11, 28]). We show that a nilpotent Lie algebra admits a standard Einstein extension with the symmetric part of D equal to a multiple of the identity if and only if it is Ricciflat and the metric Nikolayevsky derivation is nonzero. In this case, the extension is unique up to isometry.
Recall that given endomorphisms f, g of \(\mathfrak {g}\), we have
Proposition 2.1
Let \(\mathfrak {g}\) be a nilpotent Lie algebra with a pseudoRiemannian metric g, D a derivation and \(\tau =\pm 1\). Then, the metric \(\widetilde{g}=g+\tau e^0\otimes e^0\) on \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \) is Einstein if and only if
in this case, \(\widetilde{{{\,\textrm{ric}\,}}}=  \tau {{\,\textrm{tr}\,}}((D^s)^2)\widetilde{g}\).
Proof
By [15, Proposition 1.10], we have
Thus, the Einstein condition \(\widetilde{{{\,\textrm{Ric}\,}}}=\lambda {\text {Id}}\) holds if and only if
Remark 2.2
If \(h=g\), then h, g have the same Ricci tensor and opposite Ricci operators; the operators \(D\mapsto D^*\) and \(D\mapsto D^s\) are identical. Therefore, if g satisfies
then
This amounts to the fact that \(g+\tau e^0\otimes e^0\) is Einstein if and only if so is \(h\tau e^0\otimes e^0\).
Remark 2.3
We can write
It turns out that the condition that \({{\,\textrm{tr}\,}}({{\,\textrm{ad}\,}}v\circ D^*)\) vanish can be eschewed under a suitable assumption on the eigenvalues of D:
Corollary 2.4
Let \(\mathfrak {g}\) be a nilpotent Lie algebra with a pseudoRiemannian metric g, D a derivation such that \({{\,\textrm{tr}\,}}D\) is not an eigenvalue of D and \(\tau =\pm 1\). Then, the metric \(\widetilde{g}=g+\tau e^0\otimes e^0\) on \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \) is Einstein if and only if
in this case, \(\widetilde{{{\,\textrm{ric}\,}}}=  \tau {{\,\textrm{tr}\,}}((D^s)^2)\widetilde{g}\).
Proof
One direction follows from Proposition 2.1. For the other direction, assume that \(f=({{\,\textrm{tr}\,}}D){\text {Id}}+D\) is invertible and (7) holds. Since \({{\,\textrm{ad}\,}}v\) is a derivation,
where we have used \({{\,\textrm{tr}\,}}({{\,\textrm{ad}\,}}v\circ D)=0\) (see, e.g., [6, Chapter 1, Section 5.5]). Since f is invertible, this implies that \({{\,\textrm{tr}\,}}({{\,\textrm{ad}\,}}w\circ D^*)=0\) for all w, so \(\widetilde{g}\) is Einstein by Proposition 2.1.
Proposition 2.1 and Corollary 2.4 generalize a similar result of [15], where the derivation D was assumed to be symmetric. The resulting standard extensions take the form \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \), with D symmetric; such a standard decomposition is said to be of pseudoIwasawa type.
Example 2.5
Fix the Lie algebra \(\mathfrak {g}=(0,0,e^{12},0)\), which is the direct sum of the Heisenberg Lie algebra and \(\mathbb {R}\); the notation, inspired by [33], means that \(\mathfrak {g}^*\) has a fixed basis of 1forms \(e^1,e^2,e^3,e^4\) with \(de^3=e^1\wedge e^2\) and the other forms closed. The metric \(g= e^1\odot e^2+ e^3\odot e^4\) has Ricci operator equal to:
Consider the derivation
where \(\lambda \) and \(\mu \) are nonzero parameters. Then, equation (7) is satisfied for any choice of \(\tau =\pm 1\). In this case,
hence \({{\,\textrm{tr}\,}}(D^s)=0\) and \({{\,\textrm{tr}\,}}((D^s)^2)=0\).
In order to obtain a standard Einstein metric, it is sufficient, thanks to Corollary 2.4, to show that \({{\,\textrm{tr}\,}}D=0\) is not an eigenvalue. Since \(\mu \) is assumed not to be zero, D is not singular and 0 cannot be an eigenvalue. For \(\tau =1\), we obtain a twoparameter family of Ricciflat solvmanifolds of signature (3, 2); for \(\tau =1\), we obtain another family which corresponds to reversing the overall sign of the metric and applying the isomorphism \( e_2\mapsto e_2\), \(e_3\mapsto e_3\).
Notice that the resulting standard Lie algebra \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) has derived algebra equal to \(\mathfrak {g}\), because D is surjective. Therefore, the standard decomposition is unique. In addition, it is not possible to use Proposition 1.1 to obtain an isometric standard Lie algebra of pseudoIwasawa type because D and \(D^s\) do not commute.
Example 2.6
In this example, we apply Corollary 2.4 to a nilpotent Lie algebra of step greater than 2 and obtain an Einstein solvmanifold with nonzero scalar curvature. Fix the 4dimensional 3step nilpotent Lie algebra \(\mathfrak {g}=(0,0,e^{12},e^{13})\) with metric \(g= e^1\odot e^3+\frac{1}{2}e^2\otimes e^2e^{4}\otimes e^4\). Its Ricci operator equals
Consider the oneparameter family of derivations
where \(\mu \in \mathbb {R}\). Then, equation (7) is satisfied with \(\tau =1\). In this case,
hence \({{\,\textrm{tr}\,}}D={{\,\textrm{tr}\,}}(D^s)=\sqrt{\frac{3}{2}}\) and \({{\,\textrm{tr}\,}}((D^s)^2)=\frac{1}{2}\). Note that \({{\,\textrm{tr}\,}}D\) is not an eigenvalue, then by Corollary 2.4 for any choice of \(\mu \) the Lie algebra \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) has a standard Einstein metric \(\widetilde{g}= g+e^0\otimes e^0\). Even if D is not surjective, the resulting standard Einstein Lie algebra has derived algebra equal to \(\mathfrak {g}\), so the standard decomposition is unique. In addition, it is not possible to obtain an isometric standard Lie algebra of pseudoIwasawa type using either [15, Proposition 1.19] (because \(D^*\) is not a derivation) or Proposition 1.1 (because D and \(D^s\) do not commute).
As a particular case of Proposition 2.1, consider solutions of (7) such that \(D^s=a{\text {Id}}\). The case \(a=0\) corresponds to a standard extension by a skewsymmetric derivation of a Ricciflat metric, which by Proposition 1.1 yields a Ricciflat metric isometric to a product with a line.
In the case \(a\ne 0\), we have that D is a derivation in the Lie algebra
where (r, s) is the signature of g, and the inclusion \(\mathfrak {co}(r,s)\subset \mathfrak {gl}(\mathfrak {g})\) is determined by fixing an orthonormal frame. Additionally, D has nonzero trace. This implies that the metric Nikolayevsky derivation N is nonzero. We proceed to recall the construction of N, giving a slight generalization for use in later sections. For the proof, we refer to [28] and [11, Theorem 4.9].
Proposition 2.7
Let \(\mathfrak {h}\) be an algebraic subalgebra of \(\mathfrak {gl}(m,\mathbb {R})\). There exists a semisimple derivation N in \(\mathfrak {h}\cap {{\,\textrm{Der}\,}}\mathfrak {g}\) such that
The derivation N is unique up to automorphisms of \(\mathfrak {h}\).
For \(\mathfrak {h}=\mathfrak {gl}(m,\mathbb {R})\), the derivation N of Proposition 2.7 corresponds to the preEinstein or Nikolayevsky derivation introduced in [28]; accordingly, we will refer to the derivation N of Proposition 2.7 as the \(\mathfrak {h}\)Nikolayevsky derivation. For \(\mathfrak {h}=\mathfrak {co}(r,s)\), the \(\mathfrak {h}\)Nikolayevsky derivation is the metric Nikolayevsky derivation introduced in [11].
Notice that the \(\mathfrak {h}\)Nikolayevsky derivation is zero if and only if all derivations in \(\mathfrak {h}\) are traceless (i.e., \(\mathfrak {h}\) is contained in \(\mathfrak {sl}(m,\mathbb {R})\)). In particular, we see that that there is derivation with \(D^s={\text {Id}}\) if and only if the metric Nikolayevsky derivation is nonzero.
In later sections, we will consider Lie algebras with an almost pseudoHermitian structure and use the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation, where
Like the Nikolayevsky and the metric Nikolayevsky derivation, the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation turns out to have rational eigenvalues:
Proposition 2.8
Let \(\mathfrak {g}\) be a Lie algebra with an almost pseudoHermitian structure. Then, the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation of \(\mathfrak {g}\) has rational eigenvalues.
Proof
The proof follows [28] and [11, Theorem 4.9]. We can characterize elements of \(\mathfrak {cu}(p,q)\) as elements of \(\mathfrak {co}(2p,2q)\) that commute with the complex structure J.
If N is the \(\mathfrak {cu}(p,q)\)Nikolayevsky, let \(\mathfrak {g}^\mathbb {C}=\bigoplus \mathfrak {b}_t\) be the decomposition into eigenspaces and let \(\pi _t:\mathfrak {g}^\mathbb {C}\rightarrow \mathfrak {b}_t\) denote the projections. Define
Since N commutes with J, each \(\mathfrak {b}_t\) is Jinvariant. Therefore, J commutes with projections, and we can write
One can now proceed as in [11, Theorem 4.9] and show that N is the unique element of \(\mathfrak {n}\) such that \({{\,\textrm{tr}\,}}(N\psi )={{\,\textrm{tr}\,}}(\psi )\) for all \(\psi \in \mathfrak {n}\), and its coefficients \(\nu _t\) are rational numbers.
Lemma 2.9
Let H be an algebraic subgroup of \(\textrm{SO}(r,s)\) with Lie algebra \(\mathfrak {h}\) and let \(\mathfrak {g}\) be a nilpotent Lie algebra with a Hstructure. If \(D,D'\) are two elements of \((\mathfrak {h}\oplus {\text {Span}}\left\{ {\text {Id}}\right\} )\cap {{\,\textrm{Der}\,}}\mathfrak {g}\) with the same trace, then the Hstructures on \(\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \) and \(\mathfrak {g}\rtimes _{D'}{\text {Span}}\left\{ e_0\right\} \) are equivalent.
Proof
The Lie algebra \(\mathfrak {f} =(\mathfrak {h}\oplus {\text {Span}}\left\{ {\text {Id}}\right\} )\cap {{\,\textrm{Der}\,}}\mathfrak {g}\) is algebraic. Observe that that two commuting derivations of \(\mathfrak {f}\) with the same trace determine equivalent extensions by Proposition 1.1, as their difference is in \(\mathfrak {h}\cap \mathfrak {so}(r,s)\). We will use this fact repeatedly.
Denote by \(\mathfrak {r}\) the radical of \(\mathfrak {f}\). By [10], the fact that \(\mathfrak {f}\) is algebraic implies that \(\mathfrak {r}\) is also algebraic, and we can write \(\mathfrak {r}=\mathfrak {n}\rtimes \mathfrak {a}\), where \(\mathfrak {a}\) is an abelian Lie algebra consisting of semisimple elements and \(\mathfrak {n}\) is the nilradical. Since \(\mathfrak {a}\) is abelian, any two derivations in \(\mathfrak {a}\) with the same trace determine isometric extensions. Thus, we only need to show that for any \(D\in \mathfrak {f}\) there is an element of \(\mathfrak {a}\) determining an equivalent extension.
Since \(\mathfrak {f}\) is algebraic, we can write \(D=D_{ss}+D_n\), where \(D_{ss}\) is semisimple, \(D_n\) is nilpotent, and \([D_{ss},D_n]=0\). Since \(D_n\) has trace zero, D and \(D_{ss}\) determine isometric extensions. Since \(D_{ss}\) is semisimple, so are
where we have set \(\mathfrak {f}_0=\mathfrak {f}\cap \mathfrak {so}(r,s)\). We can choose a decomposition
where W is contained in \(\mathfrak {h}\) and \({{\,\textrm{ad}\,}}D_{ss}\)invariant. Indeed, it suffices to choose for W an \({{\,\textrm{ad}\,}}D_{ss}\)invariant complement of \(\mathfrak {f}_0\cap \mathfrak {r}\) in \(\mathfrak {f}_0\).
Accordingly, write \(D_{ss}=D_{\mathfrak {r}}+D_{W}\). Then,
the lefthand side belongs to the \({{\,\textrm{ad}\,}}D_{ss}\)invariant space W, and the righthand side to the ideal \(\mathfrak {r}\), so both must vanish.
Therefore, \(D_{ss}\) and \(D_{\mathfrak {r}}\) are commuting derivations with the same trace, and they determine equivalent extensions.
Using the Jordan decomposition in the algebraic Lie algebra \(\mathfrak {r}\), we see that \(D_{\mathfrak {r}}\) determines, up to equivalence, the same standard extension as its semisimple part. On the other hand, the latter is conjugate in \(\mathfrak {r}\) to an element of \(\mathfrak {a}\) by [23, Section 19.3]. The conjugation is realized by an element of the Lie group with Lie algebra \(\mathfrak {f}\), which can be assumed to have determinant one, and therefore by an element of H.
Theorem 2.10
Let \(\mathfrak {g}\) be a nilpotent Lie algebra with a pseudoRiemannian metric g such that the metric Nikolayevsky derivation N is nonzero. Then, g is Ricciflat and \(\mathfrak {g}\) has an Einstein standard extension \(\mathfrak {g}\rtimes _N{\text {Span}}\left\{ e_0\right\} \).
Conversely, suppose \(\mathfrak {g}\) is a nilpotent Lie algebra with a pseudoRiemannian metric g and an Einstein standard extension with \(D^s=a{\text {Id}}\). Then, g is Ricciflat and, up to a scaling factor, the extension is isometric to either \(\mathfrak {g}\oplus \mathbb {R}\) or \(\mathfrak {g}\rtimes _N{\text {Span}}\left\{ e_0\right\} \) according to whether a is zero or not.
Proof
Let D be a multiple of N such that \(D^s={\text {Id}}\). Every metric of the form \(e^tg\) can be written as \(g(\exp (tD)\cdot , \exp (tD)\cdot )\), i.e., it is related to g by an isomorphism. The Ricci tensor transforms accordingly; however, the Ricci tensor of \(e^tg\) coincides with that of g, and this forces it to be zero. Then, \([D,D^*]=[D,2{\text {Id}}D]=0\) and (7) holds. In addition,
where \({{\,\textrm{ad}\,}}v\) and \({{\,\textrm{ad}\,}}v\circ D\) are traceless because \(\mathfrak {g}\) is nilpotent and D is a multiple of the Nikolayevsky derivation. Thus, Proposition 2.1 implies that \(\mathfrak {g}\rtimes _D {\text {Span}}\left\{ e_0\right\} \) is Einstein.
We claim that replacing D with a nonzero multiple, say \(D'=kD\), has the effect of giving the same standard extension up to isometry and rescaling. Indeed, observe that \(\{\exp tD\}\) acts on the metric g by rescaling while leaving D unchanged. This means that the \(\widetilde{g}=g+e^0\otimes e^0\) and \(\widetilde{g}'=k^2\,g+e^0\otimes e^0\) are isometric metrics on \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \). Setting \(e_0'=ke_0\), we can write \(\widetilde{g}'=k^2(g+(e^0)'\otimes (e^0)')\), and \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _{D'}{\text {Span}}\left\{ e_0\right\} \).
Now suppose that \(\mathfrak {g}\) has a standard Einstein extension with \(D^s=a{\text {Id}}\). In this case, if \(\mathfrak {g}\) has dimension m and \(D^s=a{\text {Id}}\), then \([D,D^*]=2[D^a,D^s]=0\) and (7) becomes
If \(a=0\), D is skewsymmetric; by Proposition 1.1, we can assume \(D=0\) up to isometry, obtaining a direct product \(\mathfrak {g}\times \mathbb {R}\).
If \(a\ne 0\), D has nonzero trace and the metric Nikolayevsky derivation N is nonzero, so it too has nonzero trace. We already observed that rescaling N yields an isometric extension up to isometry. Therefore, we can assume that D and N have the same trace and conclude by Lemma 2.9.
Remark 2.11
Geometrically, we can describe the metric of Theorem 2.10 as follows. Let G be the simply connected Lie group with Lie algebra \(\mathfrak {g}\). We can exponentiate N to a oneparameter group of automorphisms \(\{f_t\}=\{\exp tN\}\subset {{\,\textrm{Aut}\,}}\mathfrak {g}\), which determines a oneparameter group of automorphisms \(\{\phi _t\}\) in \({{\,\textrm{Aut}\,}}G\). The semidirect product \(G\rtimes _{\phi _t}\mathbb {R}\) has Lie algebra \(\mathfrak {g}\rtimes _N\mathbb {R}\), and a leftinvariant metric g on G induces a leftinvariant metric \((\exp tN)g+dt^2\). Since skewsymmetric elements act trivially on the metric and \(N^s\) is a multiple of the identity, the metric takes the form of a warped product,
The fact that a metric of this form is Einstein follows directly from g being Ricciflat (see [5, § 9.109]).
4 \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras
In this section, we study the Ricci curvature of \(\mathfrak {z}\)standard Sasaki Lie algebras, characterizing the Einstein metrics in terms of their Kähler reduction. Recall from Sect. 1 that a \(\mathfrak {z}\)standard Sasaki Lie algebra is a Lie algebra \(\widetilde{\mathfrak {g}}\) carrying both a Sasaki structure \((\phi , \xi , \eta ,g)\) and a standard decomposition of the form \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) such that \(\phi (e_0)\) lies in the center \(\mathfrak {z}\) of \(\mathfrak {g}\).
Let \(\mathfrak {g}\) be a central extension of a nilpotent Lie algebra , i.e.,
As vector spaces, . Let \(\{e_s\}\) be a basis of \(\mathbb {R}^k\); the elements \(\{e^s\}\) of the dual basis can be viewed as elements of \(\mathfrak {g}^*\), and the Lie algebra structure of \(\mathfrak {g}\) is entirely determined by and the exterior derivatives \(\{de^s\}\). Explicitly,
Lemma 3.1
Let be a nilpotent Lie algebra with a metric ; on the central extension , fix a metric of the form
Then, for , the Ricci tensors of g and are related by
Proof
By construction, . For oneforms \(\alpha \) on , zeroextended to \(\mathfrak {g}\), we have . We use the fact that the musical isomorphisms relative to g and are compatible, so using (4) we obtain
Lemma 3.2
The Ricci tensor of the metric on \(\mathfrak {g}\) constructed in Proposition 1.3 is:
where \(\dim \mathfrak {g}=2n\).
Proof
Since is pseudoKähler and nilpotent, is zero by Lemma 1.7. By Lemma 3.1, we have
Then,
We can simplify these formulae by observing that
so we can view as a (1, 1) tensor . Similarly, we have \(\omega ^\sharp = J\). Then
Finally, observe that \(\omega \) and are \(d^*\)closed, so (since is unimodular),
Summing up,
Lemma 3.3
With the hypothesis of Proposition 1.3, the metric \(\widetilde{g}=g+\tau e^0\otimes e^0\) on \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes _D{\text {Span}}\left\{ e_0\right\} \) is Einstein if and only if
Proof
By Proposition 2.1, \(\widetilde{g}\) is Einstein if and only if
We have
So
Multiplying by \(\tau \) each side of (7) and using Lemma 3.2, we get
i.e.,
If this system of equations holds, is a multiple of the identity; setting , so that , we get
So the system holds if and only if and \(h=\pm 2\). This condition also implies \({{\,\textrm{tr}\,}}({{\,\textrm{ad}\,}}v\circ D^*)=0\) because \(\mathfrak {g}\) is unimodular and \({{\,\textrm{tr}\,}}({{\,\textrm{ad}\,}}v\circ D)=0\) by [6, Chapter 1, Section 5.5], proving the equivalence in the statement.
Remark 3.4
As observed in [12], Remark 5.2], changing the sign of h, , \(e_0\) and b yields an isometric metric. Therefore, we will only consider the case \(h=2\) and .
The construction of Proposition 1.3 can be specialized to the Sasaki–Einstein case as follows:
Proposition 3.5
Let be a pseudoKähler nilpotent Lie algebra, and let be a derivation of with and commuting with J. If is the central extension of characterized by \(d\xi ^*=2\omega =db^*\), where \(\{b^*,\xi ^*\}\) is the basis dual to \({\text {Span}}\left\{ b,\xi \right\} \), with the metric , then the semidirect product \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), where
has a Sasaki–Einstein structure \((\phi ,\eta ,\xi ,\widetilde{g})\) given by
Proof
We have ; applying Proposition 1.3 with \(h=2\) and \(\tau =1\) we obtain a Sasaki extension as in the statement, which is Einstein by Lemma 3.3.
Proposition 3.5 has a Kähler analogue:
Corollary 3.6
Let be a pseudoKähler nilpotent Lie algebra with nonzero metric Nikolayevsky derivation, and let be a derivation of with . If is the central extension of characterized by \(db^*=2\omega \), where \(\{b^*\}\) is the basis dual to \({\text {Span}}\left\{ b\right\} \), with the metric , then the semidirect product \(\widetilde{\mathfrak {k}}=\mathfrak {k}\rtimes {\text {Span}}\left\{ e_0\right\} \), where
has a pseudoKähler–Einstein structure \((\widetilde{\mathfrak {k}}, \widetilde{J}, \widetilde{\omega })\) given by
with \(\widetilde{{{\,\textrm{ric}\,}}}=(2n+2)\widetilde{g}\), with 2n the dimension of \(\widetilde{\mathfrak {k}}\).
Proof
Take the Lie algebra constructed in Proposition 3.5 and take the quotient by \(\xi \). Then, by Proposition 1.5 it is Kähler–Einstein with \(\widetilde{{{\,\textrm{ric}\,}}}=(2n+2)\widetilde{g}\).
Remark 3.7
Arguing as in Remark 2.11, it follows that the pseudoKähler–Einstein metric constructed in Corollary 3.6 has the form (1).
Remark 3.8
If the Lie algebra is not abelian, then Corollary 3.6 produces pseudoKähler–Einstein rankone extension which are not pseudoIwasawa, unlike the method presented in [32], where one constructs pseudoKähler–Einstein rankone extensions of pseudoIwasawatype.
Indeed, the derivation of Corollary 3.6 is selfadjoint with respect to the metric if and only if is a derivation, but since , this happens only if the identity is a derivation, i.e., if is an abelian Lie algebra.
Example 3.9
Let , with
and set \(D={\text {Id}}\). We get
Applying Corollary 3.6, one obtains the Lie algebra \(\mathfrak {k}\)
with the pseudoKähler–Einstein metric
The resulting solvmanifold can be identified with the symmetric space \(\textrm{SU}(p,q+1)/\textrm{U}(p,q)\). Indeed, fix the diagonal matrices
We can identify \(\mathfrak {su}(p,q+1)\) with the Lie algebra
The involution \(\theta ={{\,\textrm{Ad}\,}}X\) makes \((\textrm{SU}(p,q+1),\textrm{U}(p,q))\) into a symmetric pair, determining a splitting \(\mathfrak {su}(p,q+1)= \mathfrak {u}(p,q)\oplus \mathfrak {p}\). Let \(\mathfrak {a}\) be the maximal abelian subalgebra of \(\mathfrak {p}\) spanned by
The positive eigenspaces of \({{\,\textrm{ad}\,}}E_0\) generate the nilpotent Lie algebra \(\mathfrak {n}\) spanned by
where j ranges between 1 and n. Explicitly, we have
where \(\epsilon _j\) is the jth element in the diagonal of \(I_{p,q}\). The semidirect product \(\mathfrak {n}\rtimes \mathfrak {a}\) is therefore isomorphic to the Lie algebra \(\mathfrak {k}\) of (8). By [35], the symmetric metric can be expressed in terms of the Killing form B as
A straightforward computation shows that this is indeed a multiple of the metric (9). For \(q=0\), we obtain the positivedefinite symmetric metric on the Iwasawa subgroup of \(\textrm{SU}(n+1,1)\). Suggestively, this Lie group and metric appear as the fibre of quaternionKähler manifolds obtained via the cmap (see [18]).
Notice that the metric (9) is of pseudoIwasawa type; in fact, Einstein solvmanifolds arising from symmetric spaces as above are the motivating example for the notion of Iwasawa type. On the other hand, by Proposition 3.5 can also be extended to a Sasaki–Einstein Lie algebra \(\widetilde{\mathfrak {g}}\), which is not of pseudoIwasawa type. Explicitly, \(\widetilde{\mathfrak {g}}\) has a basis \(\{e_0,e_1,\dotsc , e_{2n+2}\}\) such that
and the metric is
Remark 3.10
The pseudoKähler–Einstein quotient constructed in Example 3.9 is precisely the family of [32, Example 7.6], and since is abelian, this is consistent with Remark 3.8.
5 Classification results
In this section, we characterize \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras in terms of their Kähler reduction using the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation introduced in Sect. 2. We also classify \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras of dimension \(\le 7\).
Theorem 4.1
If \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) is a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra, the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation of its Kähler reduction is nonzero.
Conversely, if is a pseudoKähler Lie algebra with nonzero \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation, it extends to a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \), uniquely determined up to equivalence.
Proof
If \(\widetilde{\mathfrak {g}}=\mathfrak {g}\rtimes {\text {Span}}\left\{ e_0\right\} \) is a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra, Proposition 1.3 asserts that \(\widetilde{\mathfrak {g}}\) can be realized as an extension of its Kähler reduction . By Proposition 3.5, is a derivation commuting with J such that . This implies that is an element of
with nonzero trace; if such a exists, the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation is nonzero.
Now assume is pseudoKähler and \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation is nonzero. By rescaling, we obtain a derivation whose symmetric part is the identity; this yields a Sasaki–Einstein extension by Proposition 3.5.
To prove uniqueness, fix two derivations , commuting with J, . The Lie algebras and have a natural \(\textrm{U}(p,q)\)structure. By Lemma 2.9, they are equivalent.
We can view \(\widetilde{\mathfrak {g}}\) as an extension of by the ideal \({\text {Span}}\left\{ b,\xi \right\} \), where \({{\,\textrm{ad}\,}}b=e^0\otimes (2b+2\xi )\) and \(db^*\) and \(d\xi ^*\) are determined by the \(\textrm{U}(p,q)\)invariant form \(\omega \). Therefore, \(\widetilde{\mathfrak {g}}\) and its counterpart obtained using are equivalent.
In the case that is abelian, we obtain:
Corollary 4.2
Every \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra such that the Kähler reduction is an abelian Lie algebra is equivalent to one of those constructed in Example 3.9.
Proof
If is an abelian Lie algebra, we can assume , with
the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation is \({\text {Id}}\), so by Theorem 4.1 the extension is equivalent to one of those constructed in Example 3.9.
In dimension 3, \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras take the form \(\mathbb {R}^2\rtimes {\text {Span}}\left\{ e_3\right\} \), with \({{\,\textrm{ad}\,}}e_3\) acting on \(\mathbb {R}^2\) as the identity. In dimension 5, \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras determine a reduction of dimension 2, which is abelian. Therefore, these metrics have the form given in Example 3.9, and we obtain:
Proposition 4.3
Let \(\widetilde{\mathfrak {g}}\) be a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra of dimension \(\le 5\). Then, \(\widetilde{\mathfrak {g}}\) is equivalent to one of
Note that the 5dimensional solvable Lie algebras appearing in Proposition 4.3 are isomorphic; up to a sign, the metric of signature (1, 4) is isometric to [16, Example 5.6].
In dimension 7, we can classify \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebras by using the classification of fourdimensional Lie algebras with a pseudoKähler metric in [30]:
Theorem 4.4
Let \(\widetilde{\mathfrak {g}}\) be a \(\mathfrak {z}\)standard Sasaki–Einstein Lie algebra of dimension 7. Then, \(\widetilde{\mathfrak {g}}\) is equivalent to one of the following:

1.
\(\widetilde{\mathfrak {g}}\) is the solvable Lie algebra
$$\begin{aligned} (e^{17},e^{27},e^{37},e^{47},2\epsilon _1e^{12}+2\epsilon _2 e^{34}+2e^{57},2\epsilon _1e^{12}+2\epsilon _2 e^{34}+2e^{57},0)\end{aligned}$$with metric
$$\begin{aligned} \widetilde{g}=\epsilon _1(e^1\otimes e^1+e^2\otimes e^2)+\epsilon _2 (e^3\otimes e^3+e^4\otimes e^4)+\gamma ,\quad \epsilon _1,\epsilon _2\in \{+1,1\}; \end{aligned}$$ 
2.
\(\widetilde{\mathfrak {g}}\) is the solvable Lie algebra
$$\begin{aligned}{} & {} \Big (\frac{2}{3}e^{17},\frac{2}{3}e^{27},\frac{a}{3} e^{27}+\frac{4}{3}e^{37}+e^{12},\frac{a}{3} e^{17}+\frac{4}{3} e^{47},\\{} & {} 2(e^{13}+e^{24}+ae^{12}+e^{57}),2(e^{13}+e^{24}+ae^{12}+e^{57}),0\Big ) \end{aligned}$$with metric
$$\begin{aligned} \widetilde{g}= a(e^1\otimes e^1+e^2\otimes e^2)+e^1\odot e^4e^2\odot e^3+\gamma ,\quad a\in \mathbb {R}; \end{aligned}$$ 
3.
\(\widetilde{\mathfrak {g}}\) is the solvable Lie algebra
$$\begin{aligned}{} & {} \Big (\frac{2}{3}e^{17},\frac{2}{3}e^{27},\frac{b}{3} e^{17}+\frac{4}{3}e^{37}+e^{12},\frac{b}{3} e^{27}+\frac{4}{3} e^{47},\\{} & {} 2a(e^{13}+e^{24})+2(e^{14}e^{23}+be^{12}+e^{57}),2a(e^{13}+e^{24})+2(e^{14}e^{23}+be^{12}+e^{57}),0\Big ) \end{aligned}$$with metric
$$\begin{aligned} \widetilde{g}=b(e^1\otimes e^1+e^2\otimes e^2)+a(e^1\odot e^4e^2\odot e^3)e^1\odot e^3e^2\odot e^4+\gamma ,\quad a,b\in \mathbb {R}; \end{aligned}$$
where we have set \(\gamma = e^5\otimes e^5+e^6\otimes e^6 e^7\otimes e^7\).
Proof
By Proposition 1.3, every \(\mathfrak {z}\)standard Sasaki Lie algebra can be obtained by extending a fourdimensional pseudoKähler Lie algebra . By the classification of [30], we have the following possibilities:
1. is abelian; we can assume that the metric is either positivedefinite or neutral. Then, we obtain the Lie algebras of Example 3.9, i.e.,
with metric
where \(\epsilon _1,\epsilon _2=\pm 1\).
2. , with \(Je_1=e_2, Je_3=e_4\), and \(\omega =e^{13}+e^{24}+ae^{12}\) for \(a\in \mathbb {R}\). Then,
The generic satisfying the hypothesis of Proposition 3.5 is:
By Theorem 4.1, we can assume \(\lambda =0\). Therefore, we obtain the extension
with the metric
3. with \(Je_1=e_2, Je_3=e_4\), and \(\omega =a(e^{13}+e^{24})+e^{14}e^{23}+be^{12}\) for \(a,b\in \mathbb {R}\). Then,
The generic satisfying the hypothesis of Proposition 3.5 is:
Again, we may assume \(\lambda =0\) and obtain
with the metric
Remark 4.5
For each of the Sasaki–Einstein Lie algebras of Theorem 4.4, the center is spanned by \(e_6\); taking the quotient gives explicit pseudoKähler–Einstein Lie algebras.
Example 4.6
Consider the 6dimensional Lie algebra , denoted by \(\mathfrak {h}_{11}\) in [17]; by [9, 33], it admits a oneparameter family of complex structures. By the work of [17], we know that it has a fourdimensional space of compatible pseudoKähler metrics.
Instead of fixing the complex structure, we use the explicit form of the two families of pseudoKähler structures given in [34].
The first one is \(\omega _1=e^{16}\lambda e^{25}(\lambda 1)e^{34}\), with the compatible complex structure
and metric
while the second one is \(\omega _2=e^{16}+e^{24}\frac{1}{2}(e^{25}e^{34})\) with the compatible complex structure
and metric .
The first case, imposing , gives
and imposing gives \(\mu _1=\frac{3}{2}\) and \(\mu _i=0\) for \(i=2,\dots ,5\), that is
Writing \(e_7,e_8,e_9\) instead of \(b,\xi ,e_0\), the metric on \(\widetilde{\mathfrak {g}}\) is then , whilst the Lie algebra is
On the other hand, gives
but imposing does not yield any solution for the \(\mu _i\).
Example 4.7
The following example shows a \(\mathfrak {z}\)standard Sasaki–Einstein \(\widetilde{\mathfrak {g}}\) obtained by extending a 6dimensional pseudoKähler Lie algebra with a derivation , which is not a multiple of the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation. Consider the Lie algebra with symplectic form \(\omega =e^{13}+e^{24}+e^{56}\) and complex structure \(J(e_1)=e_2\), \(J(e_3)=e_4\) and \(J(e_5)=e_6\). Then,
satisfies the hypothesis of Proposition 3.5 and therefore determines a \(\mathfrak {z}\)standard Sasaki–Einstein \(\widetilde{\mathfrak {g}}\) Lie algebra of dimension 9. The derivation is not diagonalizable over \(\mathbb {R}\), but has eigenvalues \((\frac{2}{3},\frac{2}{3},1i\rho ,1+i\rho ,\frac{4}{3},\frac{4}{3})\); therefore, is only a multiple of the \(\mathfrak {cu}(p,q)\)Nikolayevsky derivation when \(\rho \) is zero. Note, however, that all the resulting extensions are isometric by Theorem 4.1.
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This paper was written as part of the PhD thesis of the third author, written under the supervision of the first author, for the joint PhD programme in Mathematics Università di Milano Bicocca – University of Surrey. The authors acknowledge GNSAGA of INdAM.
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Conti, D., Rossi, F.A. & Segnan Dalmasso, R. PseudoKähler and pseudoSasaki structures on Einstein solvmanifolds. Ann Glob Anal Geom 63, 25 (2023). https://doi.org/10.1007/s10455023098940
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DOI: https://doi.org/10.1007/s10455023098940