Which nilpotent Lie algebras are Einstein nilradicals? (Lauret–Will Problem 3 and the positive-graph question)

Lauret and Will (Math. Ann. 350, 2011) reduced the classification of Einstein solvmanifolds — expected, via the Alekseevskii conjecture, to exhaust all noncompact homogeneous Einstein manifolds — to a single question about nilpotent Lie algebras: which of them are Einstein nilradicals, i.e. admit a nilsoliton metric whose Ricci operator is a multiple of the identity plus a derivation. The question is completely settled through dimension 7 — all nilpotent algebras of dimension at most 6 qualify, and Fernández-Culma classified dimension 7 — and for a few special classes, and the topological form of the Alekseevskii conjecture was proved by Böhm–Lafuente in 2023. But no structural characterization and no terminating decision procedure is known for an arbitrary nilpotent Lie algebra, and the associated combinatorial problem of characterizing the positive graphs, which decides the question for the two-step algebras attached to graphs, is open beyond Lafuente's classification of graphs with at most three coherent components. The core problem remains open.

Background

A Riemannian manifold is Einstein when its Ricci tensor is a constant multiple of the metric. For a homogeneous space $G/K$ carrying a $G$-invariant Einstein metric, the compact and noncompact cases behave very differently, and in the noncompact case every known example is a solvmanifold: a simply connected solvable Lie group with a left-invariant metric. The long-standing Alekseevskii conjecture predicts that these are all the examples; note that it has two common forms. The stronger, modern isometric form says every noncompact homogeneous Einstein manifold is isometric to a solvmanifold and is still open, while the weaker topological form — the manifold is diffeomorphic to $\mathbb{R}^n$ — was proved by Böhm and Lafuente (J. Amer. Math. Soc. 36, 2023). This background conjecture frames, but is not itself, the problem posed here.

The structure of Einstein solvmanifolds is well understood. Write the Lie algebra as $\mathfrak{s}=\mathfrak{a}\oplus\mathfrak{n}$ with $\mathfrak{n}=[\mathfrak{s},\mathfrak{s}]$ the nilradical (the maximal nilpotent ideal) and $\mathfrak{a}$ the metric orthogonal complement. The solvmanifold is standard when $\mathfrak{a}$ is abelian, and Lauret proved that every Einstein solvmanifold is standard. Heber's theory then reduces the study to the rank-one case $\dim\mathfrak{a}=1$, where everything is controlled by $\mathfrak{n}$: a nilpotent Lie algebra $\mathfrak{n}$ is the nilradical of a rank-one Einstein solvmanifold if and only if $\mathfrak{n}$ admits a nilsoliton metric (also called a minimal metric), meaning an inner product whose Ricci operator satisfies $\operatorname{Ric}=\lambda\operatorname{Id}+D$ with $\lambda<0$ and $D$ a derivation of $\mathfrak{n}$. Such a metric is unique up to isometry and scaling. One calls $\mathfrak{n}$ an Einstein nilradical when it has this property, and Lauret and Will reformulate the classification of Einstein solvmanifolds as their Problem 3: which nilpotent Lie algebras are Einstein nilradicals?

Two facts delineate the difficulty. First, by Heber's theorem every Einstein nilradical admits an $\mathbb{N}$-gradation $\mathfrak{n}=\mathfrak{n}_1\oplus\cdots\oplus\mathfrak{n}_r$ with $[\mathfrak{n}_i,\mathfrak{n}_j]\subseteq\mathfrak{n}_{i+j}$, arising as the eigenspace decomposition of the derivation induced by $\mathfrak{a}$ (the eigenvalue type). Second, this necessary condition is not sufficient: Lauret and Will answered their own Problem 4 negatively, producing $\mathbb{N}$-graded, even 2-step, nilpotent Lie algebras that are not Einstein nilradicals. Their method rephrases the Einstein condition algebraically: parametrizing nilpotent brackets $\mu$ on $\mathbb{R}^n$ by a variety on which $GL(n)$ acts by change of basis, with moment map $m(\mu)=\frac{4}{\|\mu\|^2}R_\mu$ (where $R_\mu$ is the Ricci operator of $\mu$, so $m$ is the $GL(n)$-equivariant map encoding the Ricci curvature of $\mu$), the associated rank-one solvmanifold is Einstein precisely when $\mu$ is a critical point of the square norm $F=\|m\|^2$ — equivalently, when $F$ restricted to the $GL(n)$-orbit of $\mu$ attains its minimum at $\mu$ — and the stratification of the bracket variety from geometric invariant theory interlocks with the existence of Einstein points in orbit closures. In low dimensions the problem is closed: every nilpotent Lie algebra of dimension at most 6 is an Einstein nilradical, and Fernández-Culma completed the classification in dimension 7.

For 2-step algebras the problem becomes combinatorial. To a finite graph $G$ with vertices $v_1,\dots,v_p$ and edges $\ell_1,\dots,\ell_q$ one attaches the 2-step nilpotent Lie algebra $\mu_G$ with basis $e_1,\dots,e_{p+q}$ and bracket $[e_i,e_j]=e_{p+k}$ exactly when $\ell_k=v_iv_j$ with $i<j$. Lauret and Will prove that $\mu_G$ is an Einstein nilradical if and only if $G$ is positive, where positivity means the following: let $L(G)$ be the line graph of $G$ (the graph whose vertices are the edges of $G$, two being adjacent when they share a vertex) and $\operatorname{Adj} L(G)$ its adjacency matrix; then $3I+\operatorname{Adj} L(G)$ is positive definite, and $G$ is positive when the vector $(3I+\operatorname{Adj} L(G))^{-1}\mathbf{1}$ — equivalently, the edge weighting $c$ normalized by $\sum_k c_k=1$, which satisfies $3c_k+\sum_{\ell_m\sim\ell_k}c_m=\nu$ for every edge $\ell_k$, with the scalar $\nu$ thereby determined — has all entries strictly positive. Every graph with at most five vertices is positive with a single exception, and the family members $G_{r,s,t}$ are nonpositive precisely under explicit conditions on $(r,s,t)$ (for instance $rs\geq4$ or $t\geq3$), with any graph faithfully containing one of these nonpositive members likewise nonpositive; but a structural, combinatorial characterization of the positive graphs was left open. Lafuente later classified positivity for graphs with at most three coherent components (equivalence classes of vertices having the same neighbors), and the construction side keeps growing — most recently, every nilpotent Lie algebra obtained from a finite acyclic quiver (a directed graph in which loops and multiple arrows are allowed but directed cycles are not) is an Einstein nilradical (Mizoguchi–Tamaru 2025). What is missing, in both the general and the graph case, is a decision procedure or structural classification for the unrestricted question.

Problem Statement

Determine which nilpotent Lie algebras are Einstein nilradicals, in the unrestricted sense of Lauret–Will's Problem 3: given an arbitrary finite-dimensional real nilpotent Lie algebra $\mathfrak{n}$, decide whether $\mathfrak{n}$ is the nilradical of a (rank-one, standard) Einstein solvmanifold — equivalently, whether $\mathfrak{n}$ admits a nilsoliton metric, an inner product with Ricci operator $\lambda\operatorname{Id}+D$ for some $\lambda<0$ and some derivation $D$ of $\mathfrak{n}$. The question asks for necessary and sufficient conditions on $\mathfrak{n}$, equivalently a terminating decision procedure that takes the structure constants of any nilpotent Lie algebra and correctly decides the property. The problem's shared scientific objective is the classification of Einstein solvmanifolds, to which this question is equivalent via Heber's rank-one reduction and Lauret's standardness theorem. Two explicitly linked sub-questions from the same source delimit current knowledge rather than narrowing the target: (i) the question is already settled in dimensions at most 7, so the open core concerns general dimension and general step; and (ii) for the 2-step algebras $\mu_G$ attached to finite graphs, the problem reduces to the combinatorial question of structurally characterizing the positive graphs — those for which the vector $(3I+\operatorname{Adj}L(G))^{-1}\mathbf{1}$ is entrywise positive — beyond Lafuente's classification of graphs with at most three coherent components. This is not the Alekseevskii conjecture (whose topological form is proved and whose isometric form is a separate named conjecture): it is the concrete classification question to which the source paper reduces the solvable part of the noncompact homogeneous Einstein problem.

References

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