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Einstein solvmanifolds: Existence and non-existence questions. (English) Zbl 1222.53048

The authors investigate solvable Lie groups admitting a left invariant Einstein metric. Let $\left(S,〈,〉\right)$ be a connected solvable Lie group with a left invariant metric and $𝔰$ its Lie algebra. Then, $𝔫=\left[𝔰,𝔰\right]$ is nilpotent and if $𝔰=𝔞\oplus 𝔫$ is an orthogonal decomposition then, if $\left(S,〈,〉\right)$ is Einstein, we have $\left[𝔞,𝔞\right]=0$, i.e., Einstein solvmanifolds are of standard type. The investigation of standard Einstein solvmanifolds can be further reduced to the rank one case $dim𝔞=1$. A nilpotent Lie algebra $𝔫$ is said to be an Einstein nilradical if it admits an inner product $〈·,·〉$ such that there is a standard metric solvable extension of $\left(𝔫,〈·,·〉\right)$ which is Einstein. Let $V={\bigwedge }^{2}{ℝ}^{n}\otimes {ℝ}^{n}$ and $𝒩=\left\{\mu \in V:\mu \phantom{\rule{4.pt}{0ex}}\text{satisfies}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{Jacobi}\phantom{\rule{4.pt}{0ex}}\text{identity}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{nilpotent}\right\}$. For a nilpotent Lie algebra $𝔫=\left({ℝ}^{n},\mu \right)$, there is a unique derivation $D\in \text{Der}\left(\mu \right)$ such that, for the Lie algebra $𝔰=ℝH\oplus {ℝ}^{n}$, $\left[·,·\right]$ with $\left[H,X\right]=DX$, $\left[X,Y\right]=\mu \left(X,Y\right)$, $X,Y\in 𝔫$ can admit an Einstein solvmanifold ${S}_{\mu }$ modelled on $\left(𝔰,〈·,·〉\right)$, where $〈H,X〉=0$, $〈H,H〉=1$. The group $G{L}_{n}$ acts on $V$ and $𝒩$ is a $G{L}_{n}$-invariant algebraic subset.

The authors study the moment map of the action of the group $G{L}_{n}$ and use geometric invariant theory to find Einstein nilradicals $\mu \in 𝒩$. They define a $G{L}_{n}$-invariant stratification for $𝒩$ and use it to determine whether a given nilpotent Lie algebra can be the nilradical of a rank one Einstein solvmanifold. They present examples of $ℕ$-graded (2-step) nilpotent algebras which are not Einstein nilradicals. They give a classification of 7-dimensional 6-step nilpotent algebras which are Einstein nilradicals. Using graphs, the authors study 2-step nilpotent Lie algebras giving criteria for which graphs the associated Lie algebra is an Einstein nilradical.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Homogeneous manifolds (differential geometry) 22E25 Nilpotent and solvable Lie groups
##### References:
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