The authors investigate solvable Lie groups admitting a left invariant Einstein metric. Let be a connected solvable Lie group with a left invariant metric and its Lie algebra. Then, is nilpotent and if is an orthogonal decomposition then, if is Einstein, we have , i.e., Einstein solvmanifolds are of standard type. The investigation of standard Einstein solvmanifolds can be further reduced to the rank one case . A nilpotent Lie algebra is said to be an Einstein nilradical if it admits an inner product such that there is a standard metric solvable extension of which is Einstein. Let and . For a nilpotent Lie algebra , there is a unique derivation such that, for the Lie algebra , with , , can admit an Einstein solvmanifold modelled on , where , . The group acts on and is a -invariant algebraic subset.
The authors study the moment map of the action of the group and use geometric invariant theory to find Einstein nilradicals . They define a -invariant stratification for 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.