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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,,) be a connected solvable Lie group with a left invariant metric and 𝔰 its Lie algebra. Then, 𝔫=[𝔰,𝔰] is nilpotent and if 𝔰=𝔞𝔫 is an orthogonal decomposition then, if (S,,) is Einstein, we have [𝔞,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 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 (𝔫,·,·) which is Einstein. Let V= 2 n n and 𝒩={μV:μsatisfiestheJacobiidentityandisnilpotent}. For a nilpotent Lie algebra 𝔫=( n ,μ), there is a unique derivation DDer(μ) such that, for the Lie algebra 𝔰=H n , [·,·] with [H,X]=DX, [X,Y]=μ(X,Y), X,Y𝔫 can admit an Einstein solvmanifold S μ modelled on (𝔰,·,·), where H,X=0, H,H=1. The group GL n acts on V and 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 μ𝒩. They define a GL 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 -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:
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30Homogeneous manifolds (differential geometry)
22E25Nilpotent and solvable Lie groups
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