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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 $(S,\langle ,\rangle)$ be a connected solvable Lie group with a left invariant metric and $\frak s$ its Lie algebra. Then, $\frak n=[\frak s,\frak s]$ is nilpotent and if $\frak s=\frak a \oplus\frak n$ is an orthogonal decomposition then, if $(S,\langle,\rangle)$ is Einstein, we have $[\frak a, \frak a]=0$, i.e., Einstein solvmanifolds are of standard type. The investigation of standard Einstein solvmanifolds can be further reduced to the rank one case $\dim \frak a =1$. A nilpotent Lie algebra $\frak n$ is said to be an Einstein nilradical if it admits an inner product $\langle \cdot,\cdot\rangle$ such that there is a standard metric solvable extension of $(\frak n,\langle \cdot,\cdot\rangle)$ which is Einstein. Let $V=\bigwedge^2\Bbb R^n\otimes \Bbb R^n$ and $\Cal N=\{\mu\in V:\mu \text{ satisfies the Jacobi identity and is nilpotent}\}$. For a nilpotent Lie algebra $\frak n=(\Bbb R^n,\mu)$, there is a unique derivation $D\in \text{Der}(\mu)$ such that, for the Lie algebra $\frak s=\Bbb R H\oplus\Bbb R^n$, $[\cdot,\cdot]$ with $[H,X]=DX$, $[X,Y]=\mu(X,Y)$, $X,Y\in \frak n$ can admit an Einstein solvmanifold $S_{\mu}$ modelled on $(\frak s, \langle \cdot,\cdot\rangle)$, where $\langle H, X\rangle=0$, $\langle H,H\rangle=1$. The group $GL_n$ acts on $V$ and $\Cal N$ is a $GL_n$-invariant algebraic subset. The authors study the moment map of the action of the group $GL_n$ and use geometric invariant theory to find Einstein nilradicals $\mu\in\Cal N$. They define a $GL_n$-invariant stratification for $\Cal N$ 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 $\Bbb N$-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.

53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30Homogeneous manifolds (differential geometry)
22E25Nilpotent and solvable Lie groups
Full Text: DOI arXiv
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