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Higher symmetries of the Laplacian. (English) Zbl 1091.53020

Symmetries of the Laplacian \(\Delta\) on a Riemannian manifold are linear differential operators \(D\) so that \(\Delta D = \delta \Delta\) for some linear differential operator \(\delta\). In particular, symmetries preserve harmonic functions.
First order linear differential operators on \(\mathbb R^n\) preserving harmonic functions form a closed set under Lie bracket which is finite-dimensional for \(n \geq 3\). Its commutator subalgebra is isomorphic to the Lie algebra of conformal motions of \(\mathbb R^n\). The author studies general symmetries of the Laplacian on \(\mathbb R^n\), which give rise to an algebra, filtered by degree. Answering to questions by E. Witten, the author shows that for \(n \geq 3\), the filtering subspaces are finite-dimensional and closely related to the space of conformal Killing tensors. In particular, any symmetry \(D\) of the Laplacian on a Riemannian manifold is canonically equivalent to one whose symbol is a conformal Killing tensor.
The motivation for such a study comes from the theory of higher spin fields and their symmetries [see e.g., A. Mikhailov, Notes on higher spin symmetries. Rep. No. NSF-ITP-01-181, ITEP-TH-66/01, arXiv:hep-th/0201019]. The main result is an explicit algebraic description of this symmetry algebra as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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