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Generalized Lie bialgebroids and Jacobi structures. (English) Zbl 1001.17025

The authors introduce and investigate a deformation of the notion of a Lie algebroid and Lie bialgebroid in the sense of K. Mackenzie and P. Xu [Duke Math. J. 73, 415-452 (1994; Zbl 0844.22005)] in the presence of 1-cocyles in such a way that any Jacobi manifold induces a canonical generalized Lie bialgebroid. More precisely, let \([\cdot,\cdot]\) be a Lie algebroid bracket on a vector bundle \(A\) over \(M\), and let \(\varphi_0\in\Gamma(A^*)\) be a 1-cocycle, i.e. \(\text{d}\varphi_0=0\) for the corresponding de Rham exterior differential \(\text{d}\). Then, one can consider the deformed \(\varphi_0\)-differential \[ \text{d}_{\varphi_0}\omega=\text{d}\omega+\varphi_0\wedge \omega \] and the corresponding \(\varphi_0\)-Lie derivative \({\mathcal L}_{\varphi_0}\). One can also obtain accordingly the \(\varphi_0\)-Schouten bracket \([\cdot,\cdot]_{\varphi_0}\). The pair \(((A,\varphi_0),(A^*,X_0))\) of generalized Lie algebroids is called a generalized Lie bialgebroid if for all \(\mu\in\Gamma(\bigwedge^kA^*)\), \(\nu\in\Gamma(\bigwedge^l A^*)\) \[ \text{d}_{\varphi_0}[\mu,\nu]_{X_0}=[\text{d}_{\varphi_0}\mu ,\nu]_{X_0}-(-1)^k[\mu,\text{d}_{\varphi_0}\nu]_{X_0} \] (in fact, in the paper a different but equivalent condition is given). Now, let \((\Lambda,E)\) be a Jacobi structure on \(M\). Then, the induced canonical Lie algebroid structure on \(A=T^*M\oplus{\mathbb R}\) discovered by Y. Kerbrat and Z. Souici-Benhammadi [C. R. Acad. Sci., Paris, Sér. I 317, 81-86 (1993; Zbl 0804.58015)] with the 1-cocycle \(\varphi_0((\mu,f)) =-E(\mu)\) forms a generalized Lie bialgebroid with the generalized Lie algebroid \((TM\oplus{\mathbb R},X_0)\), where on \(A^*=TM\oplus{\mathbb R}\) we take the Lie algebroid structure of first-order differential operators \[ [(Y_1+g_1),(Y_2+g_2)]=[Y_1,Y_2]+(Y_1(g_2)-Y_2(g_1)) \] and the 1-cocycle is \(X_0((Y,g))=g\). As a kind of converse, it is proved that on the base of any generalized Lie bialgebroid a Jacobi structure is determined.
It is also shown that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid, so a duality theorem can be deduced. Finally, some special cases are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

MSC:

17B62 Lie bialgebras; Lie coalgebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D10 Contact manifolds (general theory)
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References:

[1] K.H. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Research Notes in Mathematics, Vol. 174, Pitman, London, 1988. · Zbl 0671.58001
[2] A. Coste, P. Dazord, A. Weinstein, Groupoı̈des symplectiques, Pub. Dép. Math. Lyon, Vol. 2/A, 1987, pp. 1-62.
[3] Courant, T.J., Dirac manifolds, Trans. AMS, 319, 631-661, (1990) · Zbl 0850.70212
[4] Dazord, P.; Lichnerowicz, A.; Marle, Ch.M., Structure locale des variétés de Jacobi, J. math. pures appl., 70, 101-152, (1991) · Zbl 0659.53033
[5] Drinfeld, V.G., Hamiltonian Lie groups, Lie bialgebras and the geometric meaning of the classical yang – baxter equation, Sov. math. dokl., 27, 68-71, (1983) · Zbl 0526.58017
[6] V.G. Drinfeld, Quantum groups, in: Proceedings of the International Congress on Mathematics, Vol. 1, Berkeley, 1986, pp. 789-820.
[7] Fuchssteiner, B., The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Prog. theoret. phys., 68, 1082-1104, (1982) · Zbl 1098.37540
[8] Grabowski, J.; Urbánski, P., Tangent lifts of Poisson and related structures, J. phys. A, 28, 6743-6777, (1995) · Zbl 0872.58028
[9] Guédira, F.; Lichnerowicz, A., Géométrie des algébres de Lie locales de Kirillov, J. math. pures appl., 63, 407-484, (1984) · Zbl 0562.53029
[10] Higgins, P.J.; Mackenzie, K., Algebraic constructions in the category of Lie algebroids, J. algebra, 129, 194-230, (1990) · Zbl 0696.22007
[11] Ibáñez, R.; de León, M.; Marrero, J.C.; Martı́n de Diego, D., Co-isotropic and legendre – lagrangian submanifolds and conformal Jacobi morphisms, J. phys. A, 30, 5427-5444, (1997) · Zbl 0947.53041
[12] D. Iglesias, J.C. Marrero, Some linear Jacobi structures on vector bundles, C.R. Acad. Sci., Paris I 331 (2000) 125-130. arXiv: math.DG/0007138. · Zbl 0983.53055
[13] D. Iglesias, J.C. Marrero, Generalized Lie bialgebras and Jacobi structures on Lie groups, Preprint, 2001. arXiv: math.DG/0102171. · Zbl 1059.17014
[14] Y. Kerbrat, Z. Souici-Benhammadi, Variétés de Jacobi et groupoı̈des de contact, C.R. Acad. Sci., Paris I 317 (1993) 81-86. · Zbl 0804.58015
[15] Kirillov, A., Local Lie algebras, Russian math. surveys, 31, 55-75, (1976) · Zbl 0357.58003
[16] Kosmann-Schwarzbach, Y., Exact gerstenhaber algebras and Lie bialgebroids, Acta appl. math., 41, 153-165, (1995) · Zbl 0837.17014
[17] Kosmann-Schwarzbach, Y.; Magri, F., Poisson Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations, Ann. inst. H. Poincaré phys. théoret., 49, 433-460, (1988) · Zbl 0667.16005
[18] M. de León, B. López, J.C. Marrero, E. Padrón, Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality, Preprint, 1999. arXiv: math.DG/9910079.
[19] de León, M.; Marrero, J.C.; Padrón, E., On the geometric quantization of Jacobi manifolds, J. math. phys., 38, 12, 6185-6213, (1997) · Zbl 0898.58024
[20] P. Libermann, Ch.M. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer Academic Publishers, Dordrecht, 1987. · Zbl 0643.53002
[21] Lichnerowicz, A., LES variétés de Poisson et leurs algébres de Lie associées, J. diff. geom., 12, 253-300, (1977) · Zbl 0405.53024
[22] Lichnerowicz, A., LES variétés de Jacobi et leurs algébres de Lie associées, J. math. pures appl., 57, 453-488, (1978) · Zbl 0407.53025
[23] Lu, J.-H.; Weinstein, A., Poisson Lie groups, dressing transformations and Bruhat decompositions, J. diff. geom., 31, 501-526, (1990) · Zbl 0673.58018
[24] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, Cambridge, 1987. · Zbl 0683.53029
[25] Mackenzie, K.; Xu, P., Lie bialgebroids and Poisson groupoids, Duke math. J., 73, 415-452, (1994) · Zbl 0844.22005
[26] Ngô-van-Quê, Sur l’espace de prolongement différentiable, J. Diff. Geom. 2 (1968) 33-40. · Zbl 0164.22604
[27] Pradines, J., Théorie de Lie pour LES groupoı̈des différentiables, calcul différentiel dans la catégorie des groupoı̈des infinitésimaux, C.R. acad. sci., Paris A, 264, 245-248, (1967) · Zbl 0154.21704
[28] G. Sánchez de Alvarez, Geometric methods of classical mechanics applied to control theory, Ph.D. Thesis, University of California, Berkeley, 1986.
[29] Vaisman, I., Remarkable operators and commutation formulas on locally conformal Kähler manifolds, Compositio math., 40, 287-299, (1980) · Zbl 0401.53019
[30] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathemaics, Vol. 118, Birkhauser, Basel, 1994. · Zbl 0810.53019
[31] I. Vaisman, The BV-algebra of a Jacobi manifold, Ann. Polon. Math. 73 (2000) 275-290. arXiv: math.DG/9904112. · Zbl 0989.53050
[32] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523-557; Errata and Addenda 22 (1985) 255. · Zbl 0524.58011
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