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Decomposition of symmetric tensor fields in the presence of a flat contact projective structure. (English) Zbl 1165.53055

Summary: Let M be an odd-dimensional Euclidean space endowed with a contact 1-form \(\alpha\). We investigate the space of symmetric contravariant tensor fields over \(M\) as a module over the Lie algebra of contact vector fields, i.e., over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by \(\alpha\). If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form \(\alpha\) defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra \(\text{sp}(2n + 2)\) the algebra of vector fields which preserve both the contact structure and the pro jective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [Enseign. Math. (2) 52, No. 3–4, 215–229 (2006; Zbl 1117.53057)].

MSC:

53D10 Contact manifolds (general theory)
17B66 Lie algebras of vector fields and related (super) algebras

Citations:

Zbl 1117.53057
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