On isometries of some Riemannian Lie groups

Abstract

We study isometry groups of Lie groups endowed with left-invariant Riemannian metrics. We mainly consider triangular Lie groups. By the familiar Gordon-Wilson theorem, calculating the isometry groups of left-invariant metrics on such groups is reduced to calculating the automorphism groups of the corresponding Lie algebras and to distinguishing compact subgroups of these groups. We consider nilpotent Lie groups in more detail, with special attention to filiform Lie groups and their relatives (prefiliform, quasifiliform). As a rule, we state the main results in terms of the automorphism groups of Lie algebras and then give their geometric interpretation. Special attention is paid to finding the group of connected components of the isometry group (in particular, it is calculated for all filiform Lie groups) and to conditions guaranteeing that the group of rotations (that is, isometries preserving a given point) is finite for certain classes of Riemannian Lie groups.