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.

[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.

[3]

J. Carinena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202.

[4]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[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.

[6]

Irene Dorfman, "Dirac Structures and Integrability of Nonlinear Evolution Equations,'' Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.

[7]

H. Geiges, Symplectic couples on $4$-manifolds, Duke Math. J., 85 (1996), 701-711. doi: 10.1215/S0012-7094-96-08527-0.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 35-81.

[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.

[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.

[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.

[18]

Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.

[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.

[20]

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137.

[21]

Y. Terashima, On Poisson functions, J. Sympl. Geom., 6 (2008), 1-7.

[22]

A. Weinstein, A note on the Wehrheim-Woodward category, J. Geom. Mechanics, 3 (2011), 507-515.

[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.

show all references

References:
[1]

P. Antunes, Poisson quasi-Nijenhuis structures with background, Lett. Math. Phys., 86 (2008), 33-45. doi: 10.1007/s11005-008-0272-5.

[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.

[3]

J. Carinena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202.

[4]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[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.

[6]

Irene Dorfman, "Dirac Structures and Integrability of Nonlinear Evolution Equations,'' Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester, 1993.

[7]

H. Geiges, Symplectic couples on $4$-manifolds, Duke Math. J., 85 (1996), 701-711. doi: 10.1215/S0012-7094-96-08527-0.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 35-81.

[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.

[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.

[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.

[18]

Zhang-Ju Liu, A. Weinstein and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.

[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.

[20]

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137.

[21]

Y. Terashima, On Poisson functions, J. Sympl. Geom., 6 (2008), 1-7.

[22]

A. Weinstein, A note on the Wehrheim-Woodward category, J. Geom. Mechanics, 3 (2011), 507-515.

[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.

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