Isometries of 3-dimensional semi-Riemannian Lie groups

Abstract: Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. Using Lie theoretical methods, we present a unified approach to obtain all pairs $(G, g)$ whose full isometry group $\mathrm{Isom}(G, g)$ has dimension greater or equal to four. As a consequence, we determine, for every pair $(G, g)$ up to automorphism and scaling, the dimension of $\mathrm{Isom}(G, g)$, which can be three, four or six.