Conformal Killing symmetric tensors on Lie groups

TL;DR

This paper introduces metric Lie algebras of Killing type, characterizing conformal Killing symmetric tensors as sums of Killing tensors and metric multiples, and classifies such algebras in various low-dimensional cases.

Contribution

It defines metric Lie algebras of Killing type and provides a classification for several classes of Lie algebras regarding this property.

Findings

2-step nilpotent Lie algebras are of Killing type

Certain low-dimensional Lie algebras are of Killing type

Classification extends to Lie algebras with abelian factors

Abstract

We introduce the notion of metric Lie algebras of Killing type, which are characterized by the fact that all conformal Killing symmetric tensors are sums of Killing tensors and multiples of the metric tensor. We show that if a Lie algebra is either 2-step nilpotent, or 2- or 3-dimensional, or 4-dimensional non-solvable, or 4-dimensional solvable with 1-dimensional derived ideal, or has an abelian factor, then it is of Killing type with respect to any positive definite metric.