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Nambu structures and associated bialgebroids. (English) Zbl 07006406
Summary: We investigate higher-order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that \(n\)-Lie algebroid structures correspond to \(n\)-ary generalization of Gerstenhaber algebras and are implied by \(n\)-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order \(n>2\)) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order \(n\)). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.
MSC:
17B62 Lie bialgebras; Lie coalgebras
17B63 Poisson algebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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References:
[1] Alekseevsky, D.; Guha, P., On decomposability of Nambu-Poisson tensor, Acta Math. Univ. Comenian. (N.S.), 65, 1-9, (1996) · Zbl 0864.70012
[2] Bayen, F.; Flato, M., Remarks concerning Nambu’s generalized mechanics, Phys. Rev. D (3), 11, 3049-3053, (1975)
[3] Ciccoli N, Nambu-Lie group actions, Acta Math. Univ. Comenian. (N.S.) 70(2) (2001) 251-263 · Zbl 1007.37029
[4] Courant, TJ, Dirac manifolds, Trans. Amer. Math. Soc., 319, 631-661, (1990) · Zbl 0850.70212
[5] Das A, Gondhali S and Mukherjee G, Nambu structures on Lie algebroids and their modular classes, to appear in Proc. Ind. Acad. Sci, (Math. Sci.)
[6] Das A, Reduction of Nambu-Poisson manifolds by regular distributions, Math. Phys. Anal. Geom.21(1) (2018) Art. 5, 21 pp. · Zbl 06925357
[7] Das A, Singular reduction of Nambu-Poisson manifolds, Int. J. Geom. Methods Mod. Phys. 14(9) (2017) 1750128, 13 pp.
[8] Dorfman I, Dirac Structures and Integrability of Nonlinear Evolution Equations (1993) (Chichester: John Wiley and Sons Ltd)
[9] Dufour J-P and Zung N T, Poisson Poisson, structures and their normal forms, Progress in Mathematics 242 (2005) (Basel: Birkhäuser Verlag) · Zbl 1082.53078
[10] Etingof, P.; Varchenko, A., Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys., 192, 77-120, (1998) · Zbl 0915.17018
[11] Filippov V T, \(n\)-Lie algebras, Sibirsk. Mat. Zh. 26 (1985) 126-140, 191
[12] Gautheron, P., Some remarks concerning Nambu mechanics, Lett. Math. Phys., 37, 103-116, (1996) · Zbl 0849.70014
[13] Grabowski, J.; Marmo, G., On Filippov algebroids and multiplicative Nambu-Poisson structures, Differential Geom. Appl., 12, 35-50, (2000) · Zbl 1026.17006
[14] Hagiwara, Y., Nambu-Dirac manifolds, J. Phys. A, 35, 1263-1281, (2002) · Zbl 1005.53021
[15] Hagiwara, Y., Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 37, 6713-6725, (2004) · Zbl 1079.53125
[16] Ibáñez, R.; León, M.; Marrero, JC; Padrón, E., Leibniz algebroid associated with a Nambu-Poisson Structure, J. Phys. A, 32, 8129-8144, (1999) · Zbl 0962.53047
[17] Ibáñez, R.; León, M.; López, B.; Marrero, JC; Padrón, E., Duality and modular class of a Nambu-Poisson structure, J. Phys. A, 34, 3623-3650, (2001) · Zbl 1021.53060
[18] Jurčo, B.; Schupp, P.; Vysoký, J., Nambu-Poisson gauge theory, Phys. Lett. B, 733, 221-225, (2014) · Zbl 1370.81164
[19] Kosmann-Schwarzbach, Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math., 41, 153-165, (1995) · Zbl 0837.17014
[20] Kosmann-Schwarzbach, Y., The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett. Math. Phys., 38, 421-428, (1996) · Zbl 1005.53060
[21] Landsman, NP, Lie groupoids and Lie algebroids in physics and noncommutative geometry, J. Geom. Phys., 56, 24-54, (2006) · Zbl 1088.58009
[22] de León M and Sardon C, Geometric Hamiltonian-Jacobi theory on Nambu-Poisson manifolds, J. Math. Phys. 58(3) (2017) 033508, 15 pp. · Zbl 1380.70044
[23] Liu, Z-J; Weinstein, A.; Xu, P., Manin triples for Lie bialgebroids, J. Differential Geom., 45, 547-574, (1997) · Zbl 0885.58030
[24] Liu, Z-J; Xu, P., The local structure of Lie bialgebroids, Lett. Math. Phys., 61, 15-28, (2002) · Zbl 1030.53080
[25] Lu J-H, Multiplicative and affine Poisson structures on Lie groups, Ph.D. thesis (1990) (UC Berkeley)
[26] Lu, J-H; Weinstein, A., Poisson Lie Groups, Dressing transformations, and Bruhat decompositions, J. Differential Geom., 31, 501-526, (1990) · Zbl 0673.58018
[27] Mackenzie K C H, General Theory of Lie Groupoids and Lie Algebroids (2005) (Cambridge: Cambridge University Press) · Zbl 1078.58011
[28] Mackenzie, KCH; Xu, P., Lie bialgebroids and Poisson groupoids, Duke Math. J, 73, 415-452, (1994) · Zbl 0844.22005
[29] Magri F and Morosi C, A Geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S19 (1984) (University of Milan)
[30] Marmo, G.; Vilasi, G.; Vinogradov, AM, The local structure of \(n\)-Poisson and \(n\)-Jacobi manifolds, J. Geom. Phys., 25, 141-182, (1998) · Zbl 0978.53126
[31] Mukunda, N.; Sudarshan, ECG, Structure of Dirac bracket in Classical mechanics, J. Math. Phys., 9, 411-417, (1968) · Zbl 0164.25402
[32] Nakanishi, N., On Nambu-Poisson manifolds, Rev. Math. Phys., 10, 499-510, (1998) · Zbl 0929.70015
[33] Nambu, Y., Generalized Hamiltonian Dynamics,, Phys. Rev. D (3), 7, 2405-2412, (1973) · Zbl 1027.70503
[34] Schupp P and Jurčo B, Nambu-sigma model and branes, in: Proc. of the Corfu Summer Institute 2011, School and Workshops on Elementary Particle Physics and Gravity, September 4-18 (2011), Corfu, Greece, pp. 45-53
[35] Takhtajan, L., On foundation of the generalized Nambu mechanics, Comm. Math. Phys., 160, 295-315, (1994) · Zbl 0808.70015
[36] Vaisman I, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118 (1994) (Basel: Birkhäuser Verlag) · Zbl 0810.53019
[37] Vaisman, I., Nambu-Lie groups, J. Lie Theory, 10, 181-194, (2000) · Zbl 0986.37057
[38] Wang, S-H, Calculation of Nambu mechanics, J. Comput. Math., 24, 444-450, (2006) · Zbl 1093.70009
[39] Wade, A., Nambu-Dirac Structures for Lie Algebroids, Lett. Math. Phys., 61, 85-99, (2002) · Zbl 1027.53106
[40] Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys., 200, 545-560, (1999) · Zbl 0941.17016
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