# American Institute of Mathematical Sciences

June  2012, 4(2): 165-180. doi: 10.3934/jgm.2012.4.165

## Dirac pairs

 1 Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France

Received  April 2011 Published  August 2012

We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally, on Lie algebroids.
Citation: Yvette Kosmann-Schwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165-180. doi: 10.3934/jgm.2012.4.165
##### References:
 [1] P. Antunes, Poisson quasi-Nijenhuis structures with background, Lett. Math. Phys., 86 (2008), 33-45. doi: 10.1007/s11005-008-0272-5.  Google Scholar [2] A. Barakat, A. De Sole and V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Japan. J. Math., 4 (2009), 141-252.  Google Scholar [3] J. Carinena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202.  Google Scholar [4] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar [5] I. Ya. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.  Google Scholar [6] Irene Dorfman, "Dirac Structures and Integrability of Nonlinear Evolution Equations,'' Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.  Google Scholar [7] H. Geiges, Symplectic couples on $4$-manifolds, Duke Math. J., 85 (1996), 701-711. doi: 10.1215/S0012-7094-96-08527-0.  Google Scholar [8] I. M. Gel'fand and I. Ja. Dorfman, Hamiltonian operators and algebraic structures associated with them, (Russian) Funktsional. Anal. i Prilozhen., 13 (1979), 13-30; English transl., Funct. Anal. Appl., 13 (1979), 248-262.  Google Scholar [9] I. M. Gel'fand and I. Ja. Dorfman, Schouten bracket and Hamiltonian operators, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 71-74; English transl., Funct. Anal. Appl., 14 (1980), 223-226. doi: 10.1007/BF01086188.  Google Scholar [10] Long-Guang He and Bao-Kang Liu, Dirac-Nijenhuis manifolds, Rep. Math. Phys., 53 (2004), 123-142. doi: 10.1016/S0034-4877(04)90008-0.  Google Scholar [11] Y. Kosmann-Schwarzbach, Jacobian quasi-bialgebras and quasi-Poisson Lie groups, in "Mathematical Aspects of Classical Field Theory'' (eds. M. Gotay, J. E. Marsden and V. Moncrief) (Seattle, WA, 1991), Contemp. Math., 132, American Mathematical Society, Providence, RI, (1992), 459-489.  Google Scholar [12] Y. Kosmann-Schwarzbach, Poisson and symplectic functions in Lie algebroid theory, in "Higher Structures in Geometry and Physics''(eds. A. Cattaneo, A. Giaquinto and Ping Xu), Progr. Math., 287, Birkhäuser/Springer, New York (2011), 243-268.  Google Scholar [13] Y. Kosmann-Schwarzbach, Nijenhuis structures on Courant algebroids, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 625-649. doi: 10.1007/s00574-011-0032-5.  Google Scholar [14] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 35-81.  Google Scholar [15] Y. Kosmann-Schwarzbach and V. Rubtsov, Compatible structures on Lie algebroids and Monge-Amp\ere operators, Acta. Appl. Math., 109 (2010), 101-135. doi: 10.1007/s10440-009-9444-2.  Google Scholar [16] A. Kushner, V. Lychagin and V. Rubtsov, "Contact Geometry and Nonlinear Differential Equations,'' Encyclopedia of Mathematics and its Applications, 101, Cambridge University Press, Cambridge, 2007.  Google Scholar [17] Zhang-Ju Liu, Some remarks on Dirac structures and Poisson reductions, in "Poisson Geometry'' (eds. J. Grabowski and P. Urbanski) (Warsaw, 1998), Banach Center Publications, 51, Polish Acad. Sci., Warsaw (2000), 165-173.  Google Scholar [18] Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  Google Scholar [19] V. V. Lychagin, V. N. Rubtsov and I. V. Chekalov, A classification of Monge-Ampère equations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 281-308.  Google Scholar [20] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137.  Google Scholar [21] Y. Terashima, On Poisson functions, J. Sympl. Geom., 6 (2008), 1-7.  Google Scholar [22] A. Weinstein, A note on the Wehrheim-Woodward category, J. Geom. Mechanics, 3 (2011), 507-515. Google Scholar [23] Yanbin Yin and Longguang He, Dirac strucures on protobialgebroids, Sci. China Ser. A, 49 (2006), 1341-1352. doi: 10.1007/s11425-006-1997-1.  Google Scholar

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##### References:
 [1] P. Antunes, Poisson quasi-Nijenhuis structures with background, Lett. Math. Phys., 86 (2008), 33-45. doi: 10.1007/s11005-008-0272-5.  Google Scholar [2] A. Barakat, A. De Sole and V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Japan. J. Math., 4 (2009), 141-252.  Google Scholar [3] J. Carinena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202.  Google Scholar [4] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.  Google Scholar [5] I. Ya. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246. doi: 10.1016/0375-9601(87)90201-5.  Google Scholar [6] Irene Dorfman, "Dirac Structures and Integrability of Nonlinear Evolution Equations,'' Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.  Google Scholar [7] H. Geiges, Symplectic couples on $4$-manifolds, Duke Math. J., 85 (1996), 701-711. doi: 10.1215/S0012-7094-96-08527-0.  Google Scholar [8] I. M. Gel'fand and I. Ja. Dorfman, Hamiltonian operators and algebraic structures associated with them, (Russian) Funktsional. Anal. i Prilozhen., 13 (1979), 13-30; English transl., Funct. Anal. Appl., 13 (1979), 248-262.  Google Scholar [9] I. M. Gel'fand and I. Ja. Dorfman, Schouten bracket and Hamiltonian operators, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 71-74; English transl., Funct. Anal. Appl., 14 (1980), 223-226. doi: 10.1007/BF01086188.  Google Scholar [10] Long-Guang He and Bao-Kang Liu, Dirac-Nijenhuis manifolds, Rep. Math. Phys., 53 (2004), 123-142. doi: 10.1016/S0034-4877(04)90008-0.  Google Scholar [11] Y. Kosmann-Schwarzbach, Jacobian quasi-bialgebras and quasi-Poisson Lie groups, in "Mathematical Aspects of Classical Field Theory'' (eds. M. Gotay, J. E. Marsden and V. Moncrief) (Seattle, WA, 1991), Contemp. Math., 132, American Mathematical Society, Providence, RI, (1992), 459-489.  Google Scholar [12] Y. Kosmann-Schwarzbach, Poisson and symplectic functions in Lie algebroid theory, in "Higher Structures in Geometry and Physics''(eds. A. Cattaneo, A. Giaquinto and Ping Xu), Progr. Math., 287, Birkhäuser/Springer, New York (2011), 243-268.  Google Scholar [13] Y. Kosmann-Schwarzbach, Nijenhuis structures on Courant algebroids, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 625-649. doi: 10.1007/s00574-011-0032-5.  Google Scholar [14] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 35-81.  Google Scholar [15] Y. Kosmann-Schwarzbach and V. Rubtsov, Compatible structures on Lie algebroids and Monge-Amp\ere operators, Acta. Appl. Math., 109 (2010), 101-135. doi: 10.1007/s10440-009-9444-2.  Google Scholar [16] A. Kushner, V. Lychagin and V. Rubtsov, "Contact Geometry and Nonlinear Differential Equations,'' Encyclopedia of Mathematics and its Applications, 101, Cambridge University Press, Cambridge, 2007.  Google Scholar [17] Zhang-Ju Liu, Some remarks on Dirac structures and Poisson reductions, in "Poisson Geometry'' (eds. J. Grabowski and P. Urbanski) (Warsaw, 1998), Banach Center Publications, 51, Polish Acad. Sci., Warsaw (2000), 165-173.  Google Scholar [18] Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  Google Scholar [19] V. V. Lychagin, V. N. Rubtsov and I. V. Chekalov, A classification of Monge-Ampère equations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 281-308.  Google Scholar [20] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137.  Google Scholar [21] Y. Terashima, On Poisson functions, J. Sympl. Geom., 6 (2008), 1-7.  Google Scholar [22] A. Weinstein, A note on the Wehrheim-Woodward category, J. Geom. Mechanics, 3 (2011), 507-515. Google Scholar [23] Yanbin Yin and Longguang He, Dirac strucures on protobialgebroids, Sci. China Ser. A, 49 (2006), 1341-1352. doi: 10.1007/s11425-006-1997-1.  Google Scholar
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