J Grabowski et al 2004 J. Phys. A: Math. Gen. 37 5383 doi:10.1088/0305-4470/37/20/010
J Grabowski1, D Iglesias2, J C Marrero3, E Padrón3 and P Urbanski4
Show affiliationsThe 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 Δ on M, and first-order polydifferential operators on a closed submanifold N of codimension 1 such that Δ 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 also be 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 Δ-homogeneous symplectic structures on M and contact structures on N.
53D10 Contact manifolds, general
47A80 Tensor products of operators (See also 46M05)
57R25 Vector fields, frame fields
47E05 Ordinary differential operators (See also 34Bxx, 34Lxx)
Issue 20 (21 May 2004)
Received 17 October 2003, in final form 17 March 2004
Published 5 May 2004
J Grabowski et al 2004 J. Phys. A: Math. Gen. 37 5383
Michel Bauer and Denis Bernard 1999 J. Phys. A: Math. Gen. 32 5179
I D Feranchuk and A A Ivanov 2004 J. Phys. A: Math. Gen. 37 9841
Ann Merchant Boesgaard et al. 2004 ApJ 605 864
Norbert Van den Bergh 2003 Class. Quantum Grav. 20 L165
Zhao Peng-Wei et al 2009 Chinese Phys. Lett. 26 112102
L. Guzzo et al. 2007 ApJS 172 254
A. J. Green et al. 1999 ApJS 122 207
R M Gade 1999 J. Phys. A: Math. Gen. 32 7071
Mariana Guerrero and X Allen Li 2006 Phys. Med. Biol. 51 4063