Einstein nilpotent Lie groups.

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $\mathrm{GL}(n,\mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature under gauge transformations of the metric. Using the notion of nice Lie algebra, we give the first example of a nilpotent Lie group with a left-invariant pseudoriemannian Einstein metric of nonzero scalar curvature. We show that nilpotent Lie groups of dimension $\leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.