Classification of generalized Einstein metrics on three-dimensional Lie groups

We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by choice divergence operator gives rise to Ricci curvature tensor and we study corresponding Einstein equation. compute in terms tensors (on sum algebra its dual) encoding Courant algebroid structure, operator. The resulting expression is polynomial homogeneous degree two coefficients Dorfman bracket with respect orthonormal basis for metric. determine all three-dimensional