Poisson-Jacobi reduction of homogeneous tensors

Details Poisson-Jacobi reduction of homogeneous tensors Poisson-Jacobi reduction of homogeneous tensors

2003-10-16

arxiv.org/abs/math/0310246v4

J. Phys. A 37: 5383-5399,2004

Mathematics

Differential Geometry

19 pages, minor corrections, final version to appear in J. Phys. A: Math. Gen

Scientific paper

10.1088/0305-4470/37/20/010

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $\Delta$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\Delta$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case -- a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\Delta$-homogeneous symplectic structures on $M$ and contact structures on $N$.

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