Group Analysis of Differential Equations and Integrable Systems − 2018

Eugen-Mihaita Cioroianu (West University of Timisoara, Romania)
Cornelia Vizman (West University of Timisoara, Romania)

Jacobi manifolds with background

Combining the definition of twisted Jacobi manifolds [10] (that involves, besides a 2-vector field and a vector field, also a 2-form $\omega$ together with its de Rham differential d$\omega$) with that of twisted Poisson structures, also called Poisson structures with a 3-form background [13], we propose and analyze a new kind of Jacobi-like manifold. This is generated through the standard vector and 2-vector fields supplemented with two arbitrary forms, namely, a 2-form $\omega$ and a 3-form $\phi$ and will be addressed as Jacobi structure with background.
Keywords: Twisted Jacobi manifold, Poisson manifold with a 3-form background.
A.M.S. classification (2000): 53C12, 53D10, 53D17

[1] A. Alekseev, Y. Kosmann-Schwarzbach, E. Meinrenken, Quasi-Poisson manifolds.
[2] P. Dazord, A. Lichnerowicz, Ch.-M. Marle, Structure locale des variétés de Jacobi, J. Math. Pures et Appl. 70 (1991) 101.
[3] I. M. Gelfand and I. Ya. Dorfman, The Schouten bracket and Hamiltonian operators, Funkt. Anal. Prilozhen. 14 (3) (1980), 71-74.
[4] J. Grabowski and G. Marmo, Jacobi structures revisited, Journal of Physics A: Mathematical and General, 34(2001) Number 49.
[5] F. Guedira, A. Lichnerowicz, Géométrie des algébres de Lie de Kirillov, J. Math. Pures et Appl. 63 (1984) 407.
[6] D. Iglesias and J.C. Marrero, Generalized Lie bialgebroids and Jacobi structures J. Geom. Phys. 40 (2001) 176-200.
[7] D. Iglesias and J.C. Marrero, Generalized Lie bialgebras and Jacobi structures on Lie groups, Israel Journal of Mathematics, 133 (2003), 285-320.
[8] A. A. Kirilov, Local Lie algebras, Russian Math. Surveys 31(4) (1976) 55.
[9] C. M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. and Phys. 23(3-4) (1997) 350.
[10] J. M. Nunes da Costa, F. Petalidou, Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen. 39(33) (2006) 10449.
[11] J. M. Nunes da Costa, F. Petalidou, On quasi-Jacobi and Jacobi-quasi bialgebroids, Lett. Math. Phys. 80(1) (2006) 155-169.
[12] J. M. Nunes da Costa, F. Petalidou, Characteristic foliation of twisted Jacobi manifolds, in: Iglesias Ponte, al., Proceedings of XV Fall Workshop on Geometry and Physics, Tenerife, 2006, Publ. de la RSME 11 (2007), 322.
[13] P. Severa, A.Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145.
[14] I. Vaisman, Jacobi manifolds, Selected topics in Geometry and Mathematical Physics 1 (2002) 81.
[15] A. Weinstein, Poisson geometry, Diff. Geom. and its Appl. 9 (1998) 213.