Symmetry via Lie algebra cohomology

TL;DR

This paper explores the connection between the Killing operator's prolongation on Riemannian manifolds and Lie algebra cohomology, offering a model for understanding symmetries in differential operators.

Contribution

It demonstrates how simple tensor identities related to the Killing operator can be interpreted through Lie algebra cohomology, providing a framework for more complex symmetry operators.

Findings

Identifies Lie algebra cohomology as a tool for understanding Killing operator prolongation

Provides a model for analyzing symmetries in differential operators

Links tensor identities to cohomological structures

Abstract

The Killing operator on a Riemannian manifold is a linear differential operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as arising from the identification of certain Lie algebra cohomologies. The point is that this case provides a model for more complicated operators similarly concerned with symmetry.