TAILIEUCHUNG - Đề tài "Higher symmetries of the Laplacian "

We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 1. Introduction The space of smooth first order linear differential operators on Rn that preserve harmonic functions is closed under Lie bracket. For n ≥ 3, it is finitedimensional (of dimension (n2 + 3n + 4)/2). Its commutator subalgebra is isomorphic to so(n + 1, 1), the Lie algebra of conformal motions of Rn . Second order symmetries of the Laplacian. | Annals of Mathematics Higher symmetries of the Laplacian By Michael Eastwood Annals of Mathematics 161 2005 1645 1665 Higher symmetries of the Laplacian By Michael Eastwood Abstract We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 1. Introduction The space of smooth first order linear differential operators on R that preserve harmonic functions is closed under Lie bracket. For n 3 it is finitedimensional of dimension n2 3n 4 2 . Its commutator subalgebra is isomorphic to so n 1 1 the Lie algebra of conformal motions of R . Second order symmetries of the Laplacian on R3 were classified by Boyer Kalnins and Miller 6 . Commuting pairs of second order symmetries as observed by Win-ternitz and Fris 52 correspond to separation of variables for the Laplacian. This leads to classical coordinate systems and special functions 6 41 . General symmetries of the Laplacian on R give rise to an algebra filtered by degree see Definition 2 below . For n 3 the filtering subspaces are finite-dimensional and closely related to the space of conformal Killing tensors as in Theorems 1 and 2 below. The main result of this article is an explicit algebraic description of this symmetry algebra namely Theorem 3 and its Corollary 1 . Most of this article is concerned with the Laplacian on Rn. Its symmetries however admit conformally invariant analogues on a general Riemannian manifold. They are constructed in 5 and further discussed in 6. The motivation for this article comes from physics especially the recent theory of higher spin fields and their symmetries see 40 45 48 and references therein. In particular conformal Killing tensors arise explicitly in 40 and implicitly in 48 for similar reasons. Underlying this progress is the AdS CFT correspondence 25 38 53 . Indeed we shall use a version of .

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