Isometries of 3-dimensional semi-Riemannian Lie groups

TL;DR

This paper classifies three-dimensional Lie groups with left-invariant metrics where the isometry group is larger than the group itself, identifying all such cases and their isometry group dimensions.

Contribution

It provides a unified Lie-theoretical classification of all three-dimensional Lie groups with metrics having large isometry groups, detailing when the isometry group exceeds the group dimension.

Findings

Most metrics have isometry group equal to the group itself.

Identifies specific metrics with isometry groups of dimension four or six.

Classifies all pairs (G, g) with large isometry groups in three dimensions.

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 $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$ itself, acting by left translations. It turns out that, generically, $\mathrm{Isom}(G, g)$ is actually equal to $G$, and the natural question then becomes to classify those special metrics for which this is not the case. 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 than 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.