Generalized Einstein metrics on Lie groups

Source abstract

We continue the systematic study of left-invariant generalized Einstein metrics on Lie groups initiated in [V. Cortés and D. Krusche, Classification of generalized Einstein metrics on 3-dimensional Lie groups, Canad. J. Math. 75(6) (2023) 2038–2095]. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra [Formula: see text], the unknowns of the system consist of a scalar product [Formula: see text] and a three-form [Formula: see text] on [Formula: see text] as well as a linear form [Formula: see text] on [Formula: see text]. As in [V. Cortés and D. Krusche, Classification of generalized Einstein metrics on 3-dimensional Lie groups, Canad. J. Math. 75(6) (2023) 2038–2095], the Lie bracket of [Formula: see text] is considered part of the unknowns. In the Riemannian case, we show that the generalized Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalized Einstein Lie groups. In the Lorentzian case, under the additional assumption [Formula: see text], we classify — up to one case — all almost Abelian generalized Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalized Einstein Riemannian Lie groups as well as generalized Einstein Lorentzian Lie groups with [Formula: see text] and non-degenerate commutator ideal.