Ricci-flat and Einstein pseudoriemannian nilmanifolds

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.

This paper contains an account of our work on pseudoriemannian Einstein metrics on nilpotent Lie groups and some new results, mostly regarding the Ricci-flat case. We restrict to left-invariant metrics, corresponding to scalar products g on the corresponding Lie algebra, also called metrics; the Einstein condition is then an algebraic equation in the entries of g, though generally quite complicated. A solution to (1) is called a Ricci-flat metric when λ = 0, and when λ = 0 an Einstein metric of nonzero scalar curvature. The nilpotent Lie groups we consider often have rational structure constants, and therefore admit a lattice, i.e. a compact quotient (see [24]); thus, the solutions that we obtain typically determine compact Einstein manifolds. Examples of Ricci-flat nilpotent Lie algebras appear in the literature in particular contexts: four-dimensional ( [30]), bi-invariant ( [9,14,19]), nearly parakähler ( [5]), G * 2 -holonomy ( [13]), or 2-step ( [16]). The first example of an Einstein metric with nonzero scalar curvature on a nilpotent Lie algebra was constructed by the authors in [7].
The problem of constructing Einstein nilpotent Lie algebras has no Riemannian counterpart: by [25], every Riemannian metric on a nonabelian nilpotent Lie algebra has a direction of positive Ricci curvature and a direction of negative Ricci curvature, and cannot therefore be Einstein. Nevertheless, there is a well-established theory of Einstein Riemannian solvmanifolds (see [21] for a survey), within which some of the techniques we use originated. Indeed, the construction of Riemannian Einstein solvmanifolds is reduced to the study of the Ricci operator on a nilpotent Lie algebra, as they are characterized by the so-called nilsoliton equation ( [22]), involving the Ricci operator of the metric restricted to the nilradical.
Since [17], an effective approach used to study the Ricci operator on a nilpotent Lie algebra is to parametrize metric Lie algebras by fixing an orthonormal basis and letting the structure constants vary; the Ricci operator can then be interpreted as a moment map in the sense of geometric invariant theory ( [20]). In fact, the Ricci operator can also be viewed as a moment map in the sense of symplectic geometry (see [7]). This is true for every signature, although the convexity properties exploited in [20] appear not to hold in the indefinite case.
A considerable amount of research has been devoted to the classification of low-dimensional nilsolitons (see [23,18,32,12,27,26,2]; more references can be found in [21]); most of these results employ, directly or indirectly, the notion of a nice basis. A basis {e 1 , . . . , e n } of a Lie algebra is called nice if each [e i , e j ] is a multiple of some e h and each contraction e i de j is a multiple of some element of the dual basis e 1 , . . . , e n ; nice bases were introduced in [23] in the study of nilsolitons, with the observation that e 1 , . . . , e n are eigenvectors of the Ricci operator for any metric for which they form an orthogonal nice basis.
In [6], we defined a nice Lie algebra as a pair (g, B), with B a nice basis on the Lie algebra g, and an equivalence of nice Lie algebras as an isomorphism that maps basis elements to multiples of basis elements. This is the natural definition for classification purposes, since the nice condition is unaffected by rescaling any element of the basis. With this terminology, we have been able to classify nice nilpotent Lie algebras up to dimension 9. The striking fact is that, at least up to dimension 7, most nilpotent Lie algebras admit exactly one nice basis up to equivalence (see Theorems 1.5 and 1.6). This fact was proved in [6] using the classification of nilpotent Lie algebras of dimension ≤ 7 (see [15]).
On a nice nilpotent Lie algebra, there are two natural classes of metrics that can be considered: diagonal metrics, that correspond to diagonal matrices in a nice basis, and σ-diagonal metrics, that correspond to diagonal matrices multiplied by a suitable order two permutation matrix σ. In this paper we will only consider the case where σ is a diagram involution, meaning that [σ(e i ), σ(e j )] is a multiple of σ([e i , e j ]); this condition will be implicitly assumed for the rest of this introduction. It was shown in [4] that, for any diagram involution σ, σdiagonal metrics have a diagonal Ricci operator, like diagonal metrics. For both classes, the Einstein equation (1) reduces to a system of n polynomial equations in n unknowns. As n increases, finding a solution (e.g. with a computer algebra system) or proving directly its nonexistence becomes harder. In fact, for λ = 0, there is an algebraic obstruction to the existence of an Einstein metric ( [7]); this is only a necessary condition, though it holds for general invariant metrics on nilpotent Lie groups, nice or not. In [4, Corollary 2.6] we obtained sharper necessary conditions for the existence of an Einstein metric in the nice diagonal and σ-diagonal settings. With some computational work, this led to a classification of nice nilpotent Lie algebras of dimension ≤ 8 carrying an Einstein metric of nonzero scalar curvature (see Section 3).
In this paper we determine conditions on a nice Lie algebra that are both necessary and sufficient for the existence of an Einstein metric of diagonal or σ-diagonal type (Theorems 2.2 and 2.7). These conditions are still polynomial, but they involve a lower number of parameters and equations than (1). We apply this criterion to the case λ = 0, obtaining a classification of diagonal and σ-diagonal Ricci-flat metrics on nice nilpotent Lie algebras of dimension ≤ 8. In particular, we obtain a one-parameter family of non-isometric Ricci-flat metrics (Example 4.7). This paper is organized as follows. The first section reviews the classification of nice nilpotent Lie algebras of dimension ≤ 9 and some open problems in this context. The second section contains a characterization of nice nilpotent Lie algebras admitting an Einstein metric of diagonal or σ-diagonal type. The third section reviews our results for the case λ = 0 and some related open problems. The final section is dedicated to the Ricci-flat case; it contains a classification of diagonal and σ-diagonal Ricci-flat metrics on nice nilpotent Lie algebras of dimension ≤ 8, as well as some remarks on the 2-step case and examples related to parahermitian geometry.
Acknowledgements We thank Viviana del Barco and the referee for useful suggestions.

Nice Lie algebras
In this section we survey our work on the classification of nice Lie algebras and state some open questions; for details, we refer to [6].
The classification of nice Lie algebras is based on linear algebra and combinatorics. We define a labeled diagram as a directed acyclic graph (with no multiple arrows) endowed with a function from the set of arrows to the set of nodes; the node so associated to an arrow is called its label. Two labeled diagrams will be regarded as isomorphic if there are compatible bijections between the corresponding nodes, arrows and labels; such a map is called an isomorphism. The group of self-isomorphisms of a labeled diagram ∆ is called its group of automorphisms Aut(∆).
Given a labeled diagram, we write i j − → k to indicate an arrow from node i to node k labeled by the node j. We write i (N2) any two distinct arrows with the same destination have different labels; (N3) if i j − → k is an arrow, then i differs from j and j i − → k is also an arrow; (N4) there do not exist four different nodes i, j, k, v such that exactly one of Let g be a lie algebra; let B = {e 1 , . . . , e n } be a basis, and denote by B * = {e 1 , . . . , e n } its dual basis. We say that B is nice if • for any e i , e j ∈ B, [e i , e j ] is a multiple of some element of B; • for any e i ∈ B, e j ∈ B * , e i de j is a multiple of some element of B * .
It is clear that rescaling one or more basis elements does not affect this definition, motivating the following: Definition 1.1. A nice Lie algebra is a pair (g, B), where g is a Lie algebra and B a nice basis of g. Two nice Lie algebras (g 1 , B 1 ) and (g 2 , B 2 ) are equivalent if there exists a Lie algebra isomorphism g 1 ∼ = g 2 that maps each element of B 1 to a multiple of an element of B 2 ; such a map is called an equivalence. As a matter of notation, we will use a string of the form 62:4a (0, 0, 0, 0, e 13 + e 24 , e 12 + e 34 ) (2) to indicate a Lie algebra with a basis B = {e 1 , . . . , e 6 } such that de 1 = 0 = · · · = de 4 , de 5 = e 13 + e 24 , de 6 = e 12 + e 34 (where, as usual, we have written e ij in lieu of e i ∧ e j ); the label 62:4a is the name of this Lie algebra in the classification of [6]. A nice Lie algebra (g, B) is reducible if there exist nice Lie algebras (g 1 , B 1 ), (g 2 , B 2 ) such that g = g 1 ⊕ g 2 and B = B 1 ∪ B 2 , and irreducible otherwise. Notice that an irreducible nice Lie algebra may be reducible in the category of Lie algebras. For instance, the nice Lie algebra (2) is isomorphic, but not equivalent, to the reducible nice Lie algebra 62:2 (0, 0, e 12 , 0, 0, e 45 ).
In this paper, we will only be interested in the case where g is nilpotent. To each nice nilpotent Lie algebra (g, B) we can associate a nice diagram ∆ by the following rules: • the nodes of ∆ are the elements of the nice basis B; • there is an arrow e i ej − → e k if e k is a nonzero multiple of [e i , e j ].
It is clear that equivalent nice Lie algebras determine isomorphic diagrams.
Conversely, nice diagrams can be used to construct nice nilpotent Lie algebras; however, this requires fixing some additional data. Given a nice diagram ∆ with nodes 1, . . . , n, let {e i } be the standard basis of R n and {e i } its dual basis, and let V ∆ ⊂ Λ 2 (R n ) * ⊗ R n be the vector space spanned by e ij ⊗ e k , where i j − → k ranges among arrows of ∆. We can parametrize the e ij ⊗ e k by the index set I ∆ that contains {{i, j}, k} whenever i j − → k is an arrow. The generic element of V ∆ has the form We shall also write c = c ijk e ij ⊗ e k , where {{i, j}, k}, i < j ranges in I ∆ . Let V ∆ be the open subset of V ∆ where each coordinate c I is nonzero. . For any c in V ∆ , suppose that the derivation d of Λ(R n ) * given by de k = c ijk e ij satisfies d 2 = 0, thereby defining a Lie algebra g = (R n , d). Then (g, {e i }) is nice and nilpotent. If in addition c ∈V ∆ , the nice diagram associated to this Lie algebra is ∆.
On V ∆ , there is a natural action of Aut(∆); explicitly, if σ is a permutation of the nodes that belongs to Aut(∆), we writẽ In addition, the Lie group D n of invertible diagonal matrices of order n acts naturally on V ∆ . By construction, two elements c, c of V ∆ define equivalent Lie algebras if they are in the same Aut(∆) D n -orbit. Let g be nice with diagram ∆ and fix an ordering on the index set I ∆ . In this paper we will use lexicographic ordering, obtained by associating to each {{i, j}, k} ∈ I ∆ with i < j the triplet (k, i, j), i.e.
Notice that by (N2), the two elements of I ∆ coincide when k = h and i = l.
The action of D n on V ∆ has weight vectors e ij ⊗ e k ; we denote by M ∆ the matrix whose rows are the weights for this action, following the ordering of I ∆ . Explicitly, if E 11 , . . . , E nn is the canonical basis of d n , we have so the weight of e ij ⊗ e k is the linear map Up to a sign convention, M ∆ is known as the root matrix in the literature. Example 1.3. In the case of (2), we have The first row, for instance, corresponds to the weight vector e 13 ⊗ e 5 ; in other words, to the fact that [e 1 , e 3 ] is a nonzero multiple of e 5 . The other rows are obtained in the same way.
The root matrix is a useful tool in the construction of Einstein metrics, since the existence of a Riemannian nilsoliton metric on a nice Lie algebra only depends on the root matrix ( [28]). It is also useful in the classification of nice Lie algebras; for instance, [18] used the root matrix to classify nice nilpotent Lie algebras with invertible root matrix and simple pre-Einstein derivation in dimensions ≤ 8. In this case, any two elements ofV ∆ define equivalent Lie algebras and the group Aut(∆) is trivial.
In the general case, one needs to study the set of Aut(∆) D n -orbits in V ∆ ; since Aut(∆) is discrete, it is natural to break the problem in two and describe a section for the action of D n first. One possible way to do so is described in [29], by taking a subspace ∆ p 0 that corresponds to a linear subspace in logarithmic coordinates. Our approach is to consider a linear subspace in the coordinates c I .
By way of notation, since the entries of M ∆ are integers, we can define its reduction mod 2, that will be indicated by M ∆,2 . Proposition 1.4 ([6, Proposition 2.2]). Choose J ∆,2 ⊂ J ∆ ⊂ I ∆ so that J ∆,2 parametrizes a maximal set of Z 2 -linearly independent rows of M ∆,2 and J ∆ parametrizes a maximal set of R-linearly independent rows of M ∆ . Set ThenW = W ∩V ∆ is a fundamental domain inV ∆ for the action of D n .
Considering now the full group Aut(∆) D n , it is not difficult to compute the action of Aut(∆) on the set of connected components ofW , and choose connected components W 1 , . . . , W k , one in each orbit. By construction, elements of different families are in different Aut(∆) D n -orbits (so they cannot define equivalent nice Lie algebras), although a single W i may contain two points in the same orbit. Finally imposing the quadratic equations corresponding to the Jacobi equality, one obtains (at most) k inequivalent families of nice Lie algebras with diagram ∆.
Implementing this strategy with a computer (see https://github.com/ diego-conti/DEMONbLAST), we obtained: • one does not admit any nice bases; • 3 admit exactly two inequivalent nice bases; • the remaining 30 admit exactly one nice basis up to equivalence. The analogous statement in dimension 7 is complicated by the fact that continuous families of Lie algebras appear. • 34 does not admit any nice bases; • 11 admit exactly two inequivalent nice bases; • the remaining 130 admit exactly one nice basis up to equivalence.
Extending these results to higher dimensions is made difficult by the fact that nilpotent Lie algebras are not classified, although with the same methods we have been able to classify nice nilpotent Lie algebras of dimension 8 and 9 (see [6]). Nevertheless, it is natural to ask whether the low-dimensional behaviour generalizes.
Question 1.7. For fixed n ∈ N, are there nilpotent Lie algebras of dimension n with an infinite number of inequivalent nice bases? What is the largest number f (n) ∈ N ∪ {+∞} of inequivalent nice bases that can be found on a nilpotent Lie algebra of dimension n?
By the above-mentioned result, we know that f (n) = 1 for n ≤ 5 and f (6) = 2 = f (7). In addition, in [6, Corollary 3.7], we proved that the number of inequivalent nice bases on a fixed nilpotent Lie algebra is at most countable. Proof. We first show that f is nondecreasing. Given a Lie algebra g with center Z and derived Lie algebra g , we can decompose each nice basis as where B 0 = B ∩ (Z \ g ) and B + is its complement; this corresponds to decomposing the nice diagram into the union of the subgraph of isolated vertices and the subgraph of vertices of positive degree. It is clear that two bases B and B on g are equivalent if and only if B + is equivalent to B + . Therefore, if B and B are inequivalent nice bases on a nilpotent Lie algebra g of dimension n, B ∪ {e n+1 } and B ∪ {e n+1 } are inequivalent nice bases on g ⊕ R. This shows that f (n + 1) ≥ f (n).
To see that f is unbounded, let g be the nilpotent Lie algebra denoted by N 6,2,5 in [15]. As shown in [6], g has two inequivalent nice bases; the associated nice diagrams ∆ 1 and ∆ 2 (see Figure 1) are connected and not isomorphic.
Given h, k ∈ N, we can define a nice diagram ∆ hk by adjoining h copies of ∆ 1 and k copies of ∆ 2 ; this determines a nice basis on the nilpotent Lie algebra Two such nice bases can only be equivalent if the underlying nice diagrams ∆ hk and ∆ h k are isomorphic; in turn, this implies h = h and k = k , because diagram isomorphisms map connected components to connected components.
This shows that f (6n) ≥ n + 1; since f is nondecreasing, the statement follows.
Another striking consequence of the classification is that two nice bases on a fixed nilpotent Lie algebra of dimension ≤ 7 with isomorphic diagrams are always equivalent; thus, a nilpotent Lie algebra of dimension ≤ 7 has as many nice diagrams as inequivalent nice bases. It is then natural to ask: Question 1.9. How many nonisomorphic nice diagrams can a nilpotent Lie algebra have? Question 1.10. How many inequivalent nice bases with the same nice diagram can a nilpotent Lie algebra have?

Einstein metrics on nice Lie algebras
Nice Lie algebras lend themselves to the construction of Einstein metrics. In this section we review the method of ([4, Section 2]) to construct Einstein pseudoriemannian metrics on a nice nilpotent Lie algebra and provide a new condition on a fixed Lie algebra to determine whether the method can be applied.
We are interested in left-invariant metrics on a Lie group G, which can be identified with scalar products on its Lie algebra g; such a scalar product will be called a metric on g. We shall consider two distinct classes of metrics, namely diagonal and σ-diagonal metrics.
By our definition, nice Lie algebras are endowed with a nice basis B; a metric on a nice Lie algebra is diagonal if its basis is orthogonal. Fixing an order in the basis, we define the signature of a diagonal metric g i e i ⊗ e i as the vector the notation being justified by the identity (−1) logsign x = sign(x). We shall also write logsign g for the vector with entries logsign g i . If a nice Lie algebra g is reducible (in the nice category), a diagonal metric on g is the direct sum of diagonal metrics on its factors; geometrically, this situation corresponds to a product metric. In particular, if the diagonal metric on g is Einstein, so is the metric on each factor; for this reason, diagonal metrics are most interesting when the nice Lie algebras are irreducible. For The root matrix determines a homomorphism of abelian Lie algebras which is the differential at the identity of the Lie group homomorphism Diagonal metrics are naturally parametrized by elements g = (g 1 , . . . , g n ) D ∈ D n . It turns out (see [23]) that the Ricci tensor of a diagonal metric is again diagonal; for an explicit formula, we will use the following: ). Let g be a diagonal metric on a nice Lie algebra with diagram ∆ and structure constants c. Define X by Then the Ricci operator is given by The Einstein equation with cosmological constant 1 2 k then reads ( t M ∆ X) D = kId; we shall denote by [k] the vector in R m with all entries equal to k, and equivalently write Given real vectors X = (x 1 , . . . , x m ) and α = (α 1 , . . . , α m ), in usual multiindex notation we shall write |X| α = m j=1 |x j | αj . The existence of a diagonal Einstein metric can be determined via the following: Theorem 2.2. Let g be a nice Lie algebra with diagram ∆ and structure constants c ∈ V ∆ . Then g has a diagonal metric of signature δ satisfying Ric = 1 2 kId, k ∈ R if and only if for some X ∈ R m : (H) X does not belong to any coordinate hyperplane; (L) logsign X = M ∆,2 δ; (P) for a basis α 1 , . . . , α k of ker t M ∆ , we have The metric g has signature δ; it solves equivalently, X is characterized by Taking componentwise logarithms, we find where the left-hand side denotes a vector with entries log which is equivalent to Condition (P).
A similar result was presented in [4, Theorem 2.3], which characterized the existence of the metric in terms of the existence of a solution to the polynomial equation e M∆ (g) = X, with X as in (K); in practice, applying this criterion amounts to solving a polynomial system of n equations in n unknowns. The improvement of Theorem 2.2 is that (P) corresponds to a polynomial system of k equations in k unknowns, and k is typically less than n; for instance when n = 8, case-by-case computations show that the maximum value of k is 5.
Remark 2.3. By construction, given X as in Theorem 2.2, the metric g is obtained by solving X D = e M∆ (g)(c D ) 2 . In particular, X determines the metric uniquely up to the kernel of e M∆ .
To understand this ambiguity in the choice of g, identify g with R n by fixing an order in the nice basis. Then g = (g 1 , . . . , g n ) D ∈ D n defines a Lie algebra automorphism if for any nonzero bracket [e i , e j ] = c ijk e k one has [ge i , ge j ] = c ijk ge k , i.e. g k gigj = 1. This holds precisely when e M∆ (g) = Id; therefore, ker e M∆ coincides with the group of diagonal automorphisms of g.
Thus, when e M∆ (g) = e M∆ (h), we obtain a Lie algebra automorphism f = gh −1 : g → g. If g and h lie in the same connected component of D n (in particular, they have the same signature), then we can write f = exp t so that exp t/2 defines an isometry between the metric Lie algebras (g, g i (e i ) 2 ) and (g, h i (e i ) 2 ). Thus, X determines the metric in an essentially unique way, at least for fixed signature; we refer to [6] for more details. In this case there are no Einstein metrics of nonzero scalar curvature because there exist derivations with nonzero trace (see Theorem 3.1); in terms of Theorem 2.2, this is reflected in the fact that M ∆ X = [1] has no solution. For k = 0, M ∆ X = 0 has solutions of the form The structure constants are c = (λ, 1, 1 − λ, 1, . . . , 1). Condition (P) gives Condition (H) implies x 8 + x 9 = 0, so the second equation in (5) is satisfied if This is sufficient to prove that this nice Lie algebra has no diagonal Ricci-flat metric, since logsign X is not in Im M ∆,2 .
For comparison, we note that applying [6, Theorem 2.3] (or Proposition 2.1) shows the Ricci-flat condition to be equivalent to the system Equation (5) is obtained by eliminating the g i from this system, and the condition on logsign X means that, after imposing (5), it is not possible to choose the signs of the g i consistently in order to satisfy (6).
Remark 2.5. In the above example, it is possible to solve separately |X| αi = |c| 2αi and logsign X = M ∆,2 (δ), with X in ker t M ∆ ; the essential fact is that the two equations cannot be solved simultaneously.
The second class of metrics that we consider is that of σ-diagonal metrics. Given a permutation of order two σ ∈ Σ n , we say that a σ-diagonal metric is a scalar product of the form where g is a σ-invariant element of (R * ) n . By construction, logsign g is always an element of ((Z 2 ) n ) σ , i.e. a σ-invariant element of (Z 2 ) n . Notice that the signature of the scalar product depends on both logsign g and σ.
We will consider the case where σ is a diagram involution, i.e. an element of Aut(∆); we therefore have a commutative diagram ). Let ∆ be a nice diagram with a diagram involution σ, and let g be a σ-diagonal metric on a nice Lie algebra with diagram ∆ and structure constants c. Define X by Then the Ricci operator is given by Theorem 2.7. Let g be a nice Lie algebra with diagram ∆ and structure constants c ∈ V ∆ . Let σ be a diagram involution. Then g has a σ-diagonal metric g such that logsign g = δ and Ric = 1 2 kId if and only if for some σ-invariant X ∈ R m : (H) X does not belong to any coordinate hyperplane; (L σ ) logsign X + logsign c + logsignc = M ∆,2 δ; (P σ ) for a basis α 1 , . . . , α k of (ker t M ∆ ) σ , we have Proof. Following the proof of Theorem 2.2, a σ-diagonal metric has the form g = (−1) δ exp v, where both δ and v are σ-invariant. The metric g has signature equivalently, X is characterized by notice that X is σ-invariant, because so are c,c and g. Taking componentwise logarithms, we find Thus, the existence of a σ-diagonal metric with signature δ and Ric = 1 2 kId is equivalent to the existence of a vector X satisfying conditions (K), (H), (L σ ) and (7) for some σ-invariant v ∈ d n . For fixed X, Equation (7) has a solution in v if and only if the left-hand side is orthogonal to ker t M ∆ ; such a solution can always be assumed to be σ-invariant up to replacing v with 1 2 (v + σ(v)). Since σ is symmetric of order two, we have an orthogonal decomposition As ker t M ∆ is σ-invariant, a vector in V + is orthogonal to ker t M ∆ if and only if it is orthogonal to ker t M ∆ ∩ V + = (ker t M ∆ ) σ . Therefore, Equation (7)  Then σ = (23)(56) is an automorphism, acting on V ∆ ∼ = R 4 as (12)(34). The kernel of t M ∆ is generated by (1, −1, −1, 1), so it does not contain any nontrivial σ-invariant element. Therefore, this nice Lie algebra does not admit any σ-diagonal Ricci-flat metric.
Of special interest is the case of permutations σ ∈ Aut(∆) with no fixed points, corresponding to linear isomorphisms σ : R n → R n that do not fix any element in the nice basis. Then n is even and a σ-diagonal metric has the form consequently, it has neutral signature.
Remark 2.9. Recall that an almost paracomplex structure on a manifold M is an endomorphism K : T M → T M such that K 2 = Id and the ±1-eigendistributions have the same rank. A neutral metric g is compatible with K if g(K·, K·) = −g (see [31] and the references therein); in this case (K, g) is called an almost parahermitian structure. A fixed-point-free permutation σ ∈ Aut(∆) defines an almost paracomplex structure which is not compatible in this sense with the σ-diagonal metric g, because the σ-invariant vectors are not null vectors for the metric g. We observe that this almost paracomplex structure is not integrable (i.e. the eigendistributions are not involutive), unless the Lie algebra is abelian. Assume that g is not abelian, and suppose that [e i , e j ] = ae k for some nonzero constant a. Since σ is an automorphism, [e σi , e σj ] = be σ k for some nonzero constant b. Write where by the nice conditions v lies in the span of the basis elements that differ from both e k and e σ k . Since the nonzero vector ae k + be σ k cannot be in both eigenspaces, at least one of the eigendistributions is not involutive.
Remark 2.10. By contrast, given an automorphism σ with no fixed point, it is possible to construct an almost parahermitian structure (K, g) for any σdiagonal metric g. In fact, partition the nice basis as B = B + ∪σ(B + ) and define an almost paracomplex structure K such that K| B + = Id and K| σ(B + ) = −Id; the σ-diagonal metric is an almost parahermitian metric compatible with K. We observe that, in general, the almost paracomplex structure is not integrable. These types of metrics fit into the framework of [5]; their Ricci tensor can therefore be related to the intrinsic torsion of the almost parahermitian structure (see Example 4.10).

The case of nonzero scalar curvature
In this section we review our results concerning Einstein metrics of nonzero scalar curvature s (namely, solutions of (1) with λ = 0); the results stated here are contained in [6,4,7]. The existence of an Einstein metric with nonzero scalar curvature puts strong constraints on the Lie algebra, even outside of the nice context. In fact, the following holds:  The difficulty of answering this question is that the Ricci tensor of a general 7-dimensional metric has 8 2 components, as opposed to the metrics considered in Section 2, that have at most 7 nonzero components. Even if we restrict to diagonal or σ-diagonal metrics on nice Lie algebras, however, determining the existence of an Einstein metric generally requires solving nonlinear equations equivalent to (P) and (P σ ).
In dimension 8, there are infinitely many inequivalent nice nilpotent Lie algebras; more precisely, there are 45 continuous families and 872 isolated nice Lie algebras (see [6]). However, only few of them admit diagonal or σ-diagonal Einstein metrics with s = 0: • exactly 6 admit a diagonal Einstein metric with s = 0; • exactly 4 admit a σ-diagonal Einstein metric with s = 0 for some diagram involution σ. With this exception, the Lie algebras of Table 1 are pairwise nonisomorphic, as one can check by taking the quotient by a line in the center and using the classification of [15].  It is striking that the Lie algebras appearing in Table 1 are precisely the 8dimensional nice Lie algebras that are characteristically nilpotent; this condition means that all derivations are nilpotent (see [1] and the references therein). In particular, characteristically nilpotent Lie algebras trivially satisfy the obstruction of Theorem 3.1. In the nice context, the set of derivations diagonalized by the nice basis corresponds to the kernel of the root matrix; thus, the characteristically nilpotent condition implies that the root matrix is injective. However, the two conditions are not equivalent, as can be seen by considering the nice nilpotent Lie algebra 9521:70a (0, 0, 0, 0, e 12 , e 14 + e 23 , e 13 + e 24 , e 15 , e 18 + e 25 + e 34 ), which has injective root matrix, but is not characteristically nilpotent.
Notice that characteristically nilpotent Lie algebras only exist in dimension 7 and greater (see [10]); nice characteristically nilpotent Lie algebras, by contrast, have dimension at least 8, as one can verify using the classification. The above observations make it natural to ask: Question 3.6. Do all characteristically nilpotent Lie algebras admit an Einstein metric with s = 0?
In dimension 9 and higher, the polynomial equations become increasingly difficult to solve. However, we can use a simpler sufficient condition for the existence of an Einstein metric that only depends on linear computations: Restricting to the case where M ∆ is surjective, we obtain the following: • exactly 48 admit a diagonal Einstein metric with s = 0; • exactly 7 admit a σ-diagonal Einstein metric with s = 0 for some diagram involution σ.
Remark 3.9. All the Lie algebras appearing in Theorems 3.4 and 3.8 have rational structure constants. Therefore, the associated Lie group G has a lattice Γ, and the left-invariant Einstein metric on G induces an Einstein metric on a nilmanifold, namely the compact quotient Γ\G.
Remark 3.10. In light of Question 3.6, we notice that among the Einstein Lie algebras appearing in Theorem 3.8 there are 44 characteristically nilpotent Lie algebras.
In the situation of Theorem 3.8, there do not appear families of nice nilpotent Lie algebras; this is a general consequence of the surjectivity of M ∆ . It would be interesting to investigate the general behaviour of families of nice nilpotent Lie algebras sharing the same diagram and root matrix. Indeed, [28] showed that the existence of a Riemannian nilsoliton metric on a nice nilpotent Lie algebra only depends on the root matrix -or, in our language, the diagram. Since nilsoliton metrics, like Einstein metrics, arise as critical points of the scalar curvature, it is natural to ask whether this property applies to Einstein metrics in the pseudoriemannian case, i.e.: Question 3.11. Are there any families of nice Lie algebras with the same diagram such that only some elements of the families admit a (diagonal) Einstein metric with s = 0?
We point out that, with the methods of this section, it is not difficult to construct families of nice Lie algebras that have a diagonal Einstein metric for all values of the parameters (see e.g. [4,Remark 4.6]).
In the case s = 0, a family of nice Lie algebras can admit a Ricci-flat metric for all values of the parameters, some or none; among the Lie algebras considered in this paper, see e.g. 741:6 (Theorem 4.3), 8542:15a and 85321:48 (Theorem 4.5).
Fixing the Lie algebra, a different question to consider is the following: The analogous question for s = 0 can be answered in the affirmative; see Example 4.7.

Ricci-flat metrics
In this section we apply Theorem 2.2 and Theorem 2.7 constructively and classify nice Lie algebras of dimension ≤ 8 admitting diagonal or σ-diagonal Ricciflat metrics, with σ a diagram involution. Unlike the situation of Section 3, some of the Lie algebras obtained in this section appear in families; for rational values of the parameter(s), the argument of Remark 3.9 yields a Ricci-flat metric on a nilmanifold.
We use the classification of nice Lie algebras contained in [6]. We start by observing that, for k = 0, conditions (K) and (H) can only be satisfied when M ∆ is nonsurjective. In dimension less than seven the only Lie algebra with nonsurjective M ∆ is the Lie algebra 631:6. We will carry out the computations in detail in the following example, where we also introduce the conventions used throughout the section. Then t M ∆ X = 0 has general solution and (P) is trivially satisfied. If nonzero, the vector X belongs to one of two orthants, according to the sign of x. By (L), the vectors δ ∈ Z 6 2 such that logsign X ∈ M ∆,2 (δ) form the possible signatures of a diagonal Ricci-flat metric g. Here and in the sequel, we will refer to these vectors δ as the admissible signatures.
Given a diagonal metric g = g i e i ⊗ e i we will identify its signature by listing the indices i for which g i is negative; for instance, 14 stands for a diagonal metric such that g 1 and g 4 are negative and the other g i are positive; in conventional language, the signature of this metric is (n − 2, 2). The set of admissible signatures S will be ordered by length and lexicographic order.
Recall that when g is a Ricci-flat metric, then so is −g: thus, assuming the dimension is 6, if 14 is an admissible signature then 2356 is also admissible. For brevity, we will only list one admissible signature in each like complementary pair, namely the one coming first in the order. On a fixed nice Lie algebra, we will denote by 1 2 S the halved set of admissible signatures obtained in this way. In particular, for 631:6 we obtain 5, 12, 13, 26, 36, 146, 156}; this shows that this Lie algebra contains a Ricci-flat metric in each indefinite signature (p, q).
To obtain the explicit metrics, we have to solve the system e M∆ (g) = X, which (normalizing to x = 1) leads to This system has solution g 4 = g 1 g 2 , g 5 = −g 1 g 3 , g 6 = g 1 g 2 g 3 , giving a 3-parameter family of Ricci-flat metrics, consistently with the fact that ker M ∆ has dimension 3 (see Remark 2.3). In this case there are no diagram involutions σ with a σ-diagonal metric because the only nontrivial automorphism is (23)(45), which does not preserve any nonzero element of ker t M ∆ .
We have obtained the following: There are no σ-diagonal Ricci-flat metrics on any nice nilpotent Lie algebra of dimension ≤ 6, for any diagram involution σ.   Table 2.
Proof. Linear computations on a case-by-case basis show that the nice Lie algebras of dimension 7 that satisfy (K) and (H) are precisely those listed in Table 2 plus the following: For 75421:4 we find that vectors in ker t M ∆ have the form X = (x 7 , x 7 + x 6 , −x 7 − x 6 , −x 7 − x 6 , x 6 , −x 7 , x 7 ), and (P) reads the only solution is x 6 = x 7 = 0, which violates (H); similarly for 74321:12.
For 75432:3, (P) is trivially satisfied. Imposing (L) on a generic vector of ker t M ∆ not contained in a coordinate hyperplane, we find 8 admissible signatures, i.e.  Table 2.
For 741:6, we get X = (x 6 , −x 5 − x 6 , x 5 , x 5 , −x 5 − x 6 , x 6 ), and (P) gives: which has a solution x 6 = ( 1 λ − 1)x 5 for all λ. Note that x 5 + x 6 has the same sign as λ, and x 5 is positive if and only if x 6 has the same sign as −1 + 1/λ. Hence, the admissible signatures are represented by: These results are collected in Table 2.
For a σ-diagonal metric g = g i e i ⊗ e σi , we will also denote by g ij the coefficient of e i ⊗ e j , and represent the signature by listing the indices i for which g iσi = g i is negative. We remark that if σ interchanges i and j, then g iσi and g jσj coincide by construction, so either both or none of i and j must appear in the list. For each signature (p, q), we will denote by S σ (p, q) the set of δ ∈ Z n 2 such that there is a diagram involution σ and a Ricci-flat σ-diagonal metric g of signature (p, q) with logsign g = δ; as usual, each δ will be represented by the indices of its nonzero entries.
Theorem 4.5. The 8-dimensional nice nilpotent Lie algebras admitting a diagonal Ricci-flat metric are listed in Table 3.
Proof. For each 8-dimensional nice nilpotent Lie algebra in the classification of [6], we compute the generic vector X in ker t M ∆ , ruling out those Lie algebras for which no choices of X satisfies (K) and (H). In the remaining cases we need to study the polynomial condition (P) and its compatibility with (H) and (L). We omit explicit computations, since they are essentially the same as in The same holds for 8542:15a and 8542:15b with a 2 < 0 (recall that parameters that appear in the structure constants of a family of nice Lie algebras are always assumed to be nonzero).
The nice nilpotent Lie algebras that satisfy all the conditions of Theorem 2.2 are listed in Table 3 together with their admissible signatures, according with the convention explained in Example 4.1.
Notice that some of the Lie algebras listed in Tables 2 and 3 are decomposable in the category of nice Lie algebras: since a diagonal metric is the product of diagonal metrics on each factor, the Ricci-flat diagonal metric can be recovered from Ricci-flat diagonal metrics of lower dimension. In particular we have that:  Finally, we apply Theorem 2.7 in the 8-dimensional case, obtaining the following classification.
Theorem 4.6. The 8-dimensional nice nilpotent Lie algebras admitting a diagram involution σ and a σ-diagonal diagonal Ricci-flat metric are listed in Table 4.
Proof. As in the proof of Theorem 4.4, we first list all the nice Lie algebras that admit a nontrivial automorphism σ of order two such that (K) has a σ-invariant solution that does not belong to any coordinate hyperplane.
Among the surviving Lie algebras, we find that the following:  Table 4.
For the remaining Lie algebras and each automorphism σ of order 2, we impose the conditions (K), (H), (L σ ) and (P σ ). The results are listed in Table 4.
The above classifications show that diagonal and σ-diagonal metrics Ricciflat metrics are quite scarce in the nice nilpotent context, although most of these Lie algebras do admit a Ricci-flat metric of more general type (see the references quoted in the introduction, as well as the forthcoming [3]). For example, all the nearly parakähler 8-dimensional examples presented in [5, Section 7] admit a nice basis; in the notation of [6], they can be written as Thus, these nice Lie algebras admit a Ricci-flat metric of neutral signature, but not a diagonal or σ-diagonal Ricci-flat metric with σ a diagram involution, as they do not appear in Tables 3 and 4. This is in sharp contrast with the case of nonzero scalar curvature, where the only known examples of Einstein nilpotent Lie algebras are nice Lie algebras with a diagonal or σ-diagonal metric.
Thus, we obtain a family of Ricci-flat metrics depending on 3 parameters. We can rescale the metric to normalize ad | g : g → g , imposing 1 = g(ad | g , ad | g ) = g(e 34 ⊗ e 7 , e 34 ⊗ e 7 ) = x; in addition, the parameter g 1 can be eliminated since it reflects the kernel of M ∆ (see Remark 2.3). We obtain the one-parameter family of Ricci-flat metrics g 1 = 1 = g 2 = g 3 , g 4 = y, g 5 = −y, g 6 = y 2 , g 7 = y.
The Riemann tensor R : Λ 2 g → End g and its projection R : Λ 2 g → End g satisfy g(R, R) = 1 2 y + y 2 + 1, g(R , R ) = −y 2 − y + 13 8 ; this proves that the metrics (8) are pairwise nonisometric. The generic element of ker t M ∆ is X = (− 1 2 x, x, − 1 2 x, x, − 1 2 x, − 1 2 x, − 1 2 x, − 1 2 x, x); (H) is trivially satisfied for x = 0, and (P) gives 64a 2 = 1. Thus, there are two nice Lie algebras in this family that admit a diagonal Ricci-flat metric, with admissible signatures determined by (L), i.e. In particular, we do not obtain Ricci-flat metrics of Lorentzian signature. In fact, it was proved in [16] that Ricci-flat Lorentzian metrics on 2-step nilpotent Lie algebras have degenerate center; diagonal metrics on a nice Lie algebra never have this property. We note that these metrics are not ad-invariant, i.e. they do not satisfy [x, y], z + y, [x, z] = 0, x, y, z ∈ g.
In fact, a diagonal metric on a 2-step nice Lie algebra cannot be ad-invariant, as one can see by taking x and y to be elements of the nice basis with z = [x, y] = 0. This is consistent with [8, Corollary 2.6].
to [5] for more details). In particular, the SL(5, R) intrinsic torsion can be split into ten invariant components: To compute explicitly the intrinsic torsion, let g be the generic σ-invariant metric that solves the equation e M∆ (g) = X for a generic σ-invariant X. Then the nonzero components of the metric tensor are: g 27 = −g 45 , g 68 = −g 13 , g 90 = −xg 13 g 45 .