Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

TL;DR

This paper constructs and classifies all symmetries of the conformally invariant powers of the Laplacian, known as GJMS operators, on conformally flat manifolds, including Euclidean space, revealing their algebraic structure.

Contribution

It introduces a method to generate all symmetries of GJMS operators from conformal Killing tensors, providing a complete classification on conformally flat manifolds.

Findings

Constructed symmetry operators from conformal Killing tensors.

Classified all symmetries of GJMS operators on conformally flat manifolds.

Explicitly described the algebra of symmetry operators.

Abstract

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $\Delta^k$, $k\in \mathbb{Z}>0$, on Euclidean space $\mathbb{E}^n$. The algebra formed by the symmetry operators is described explicitly.