Abstract
The spaces of tensor densities over a manifold M are modules over the Lie algebra Vect(M) of vector fields over the manifold. When M is a contact manifold, one can consider the algebra C(M) of vector fields which preserves the contact structure. If the manifold is endowed with a contact projective structure, there is an embedding of the linear symplectic algebra sp(2n+2, ℝ) in C(M). In this Letter, we determine the C(M)- and the sp(2n+2, ℝ)-invariant bidifferential operators on tensor densities.
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References
Arnol'd, V. I.: Matematicheskie metody klassicheskoi mekhaniki, Nauka, Moscow 1974; English transl. Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, 1978.
Arnol'd, V. I. and Givental, A. B.: Symplectic geometry, in: Current Problems in Mathematics, Fundamental Directions,Vol. 4, VINITI, Moscow, 1986, pp. 5–139; English transl. Encyclopaedia of Mathematical Sciences, Vol. 4, Springer-Verlag, Berlin, 1989.
Boniver, F. and Lecomte, P. B. A.: A remark about the Lie algebra of infinitesimal transformations of the Euclidean space, Preprint, University of Liège, 1999.
Gordan, P.: Vorlesungen öber Invariantentheorie (G. Kerschensteiner, ed.) Vol. 2, Teubner, Leipzig, 1887.
Grozman, P. Ya.: Classification of bilinear invariant operators over tensor fields, Funct. Anal. Appl. 14(2) (1980).
Lecomte, P. B. A. and Ovsienko, V. Yu.: Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization, Preprint, CPT-CNRS Luminy (1996).
Ovsienko, V. and Roger, C.: Deformations of Poisson brackets and extensions of Lie algebras of contact vector fields, UspekhiMat. Nauk 47(6) (1992); English transl. in Russian Math. Surveys 47 (1992), 6.
Ovsienko, V. Yu. and Ovsienko, O. D.: Projective structures and infinite-dimensional Lie algebras associated with a contact manifold, Adv. Soviet Math. 17 (1993).
Weyl, H.: The Classical Groups, their Invariants and Representations, Princeton Math. Series, Princeton Univ. Press, 1946.
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Mathonet, P. Invariant Bidifferential Operators on Tensor Densities over a Contact Manifold. Letters in Mathematical Physics 48, 251–261 (1999). https://doi.org/10.1023/A:1007505731725
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DOI: https://doi.org/10.1023/A:1007505731725