Classification of Odd Generalized Einstein Metrics on 3-Dimensional Non-unimodular Lie Groups

An odd generalized metric E − $$E_{-}$$ on a Lie group G of dimension n with Lie algebra 𝔤 $$\mathfrak {g}$$ is a left-invariant generalized metric defined on a Courant algebroid E H , F $$E_{H, F}$$ of type B n $$B_{n}$$ over G, with left-invariant twisting forms H and F. We say that E − $$E_{-}$$ is generalized Einstein with left-invariant divergence δ ∈ ( 𝔤 ⊕ 𝔤 ∗ ⊕ ℝ ) ∗ $$\delta \in (\mathfrak {g}\oplus \mathfrak {g}^{*}\oplus \mathbb {R})^{*}$$ if the Ricci tensor of the pair ( E − , δ ) $$(E_{-}, \delta )$$ vanishes. In this paper we describe all odd generalized Einstein metrics (together with their divergences and Courant algebroids E H , F $$E_{H, F}$$ on which they are defined) on 3-dimensional non-unimodular Lie groups.