September  2015, 7(3): 255-280. doi: 10.3934/jgm.2015.7.255

Hypersymplectic structures on Courant algebroids

1. 

CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal, Portugal

Received  January 2015 Revised  June 2015 Published  July 2015

We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simple way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hypersymplectic structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated.
Citation: Paulo Antunes, Joana M. Nunes da Costa. Hypersymplectic structures on Courant algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 255-280. doi: 10.3934/jgm.2015.7.255
References:
[1]

P. Antunes, Crochets de Poisson Gradués et Applications: Structures Compatibles et Généralisations des Structures Hyperkählériennes, Ph.D thesis, École Polytechnique, Palaiseau, France, 2010. Google Scholar

[2]

P. Antunes, C. Laurent-Gengoux and J. M. Nunes da Costa, Hierarchies and compatibility on Courant algebroids, Pac. J. Math., 261 (2013), 1-32. doi: 10.2140/pjm.2013.261.1.  Google Scholar

[3]

P. Antunes and J. M. Nunes da Costa, Hyperstructures on Lie algebroids, Rev. in Math. Phys., 25 (2013), 1343003, 19pp. doi: 10.1142/S0129055X13430034.  Google Scholar

[4]

P. Antunes and J. M. Nunes da Costa, Induced hypersymplectic and hyperkähler structures on the dual of a Lie algebroid, Int. J. Geom. Meth. Mod. Phys., 11 (2014), 1460030 (9 pages). doi: 10.1142/S0219887814600305.  Google Scholar

[5]

P. Antunes and J. M. Nunes da Costa, Hyperstructures with torsion on Lie algebroids, preprint,, , ().   Google Scholar

[6]

H. Bursztyn, G. Cavalcanti and M. Gualtieri, Generalized Kähler and hyper-Kähler quotients, in Poisson geometry in mathematics and physics, Contemp. Math., Amer. Math. Soc., Providence, RI, 450 (2008), 61-77. doi: 10.1090/conm/450/08734.  Google Scholar

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P. S. Howe and G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B, 379 (1996), 80-86. doi: 10.1016/0370-2693(96)00393-0.  Google Scholar

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J. Grabowski, Courant-Nijenhuis tensors and generalized geometries, in Groups, geometry and physics, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 29 (2006), 101-112.  Google Scholar

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N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., 54 (2003), 281-308. doi: 10.1093/qmath/hag025.  Google Scholar

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Y. Kosmann-Schwarzbach, Quasi, twisted, and all that ... in Poisson geometry and Lie algebroid theory, in The Breadth of symplectic and Poisson geometry, J.E. Marsden and T. Ratiu, eds., Progr. Math., 232, Birkhäuser, Boston, MA, 2005, pp. 363-389. doi: 10.1007/0-8176-4419-9_12.  Google Scholar

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Y. Kosmann-Schwarzbach, Poisson and symplectic functions in Lie algebroid theory, in Higher Structures in Geometry and Physics, A. Cattaneo, A. Giaquinto and Ping Xu, eds., Progr. Math. 287, Birkhäuser, Boston, 2011, pp. 243-268. doi: 10.1007/978-0-8176-4735-3_12.  Google Scholar

[12]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond, T. Voronov, ed., Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185. doi: 10.1090/conm/315/05479.  Google Scholar

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D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137. doi: 10.1023/A:1020708131005.  Google Scholar

[14]

M. Stiénon, Hypercomplex structures on Courant algebroids, C. R. Acad. Sci. Paris, 347 (2009), 545-550. doi: 10.1016/j.crma.2009.02.020.  Google Scholar

[15]

T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond, T. Voronov, ed., Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168. doi: 10.1090/conm/315/05478.  Google Scholar

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P. Xu, Hyper-Lie Poisson structures, Ann. Sci. École Norm. Sup., 30 (1997), 279-302. doi: 10.1016/S0012-9593(97)89921-1.  Google Scholar

show all references

References:
[1]

P. Antunes, Crochets de Poisson Gradués et Applications: Structures Compatibles et Généralisations des Structures Hyperkählériennes, Ph.D thesis, École Polytechnique, Palaiseau, France, 2010. Google Scholar

[2]

P. Antunes, C. Laurent-Gengoux and J. M. Nunes da Costa, Hierarchies and compatibility on Courant algebroids, Pac. J. Math., 261 (2013), 1-32. doi: 10.2140/pjm.2013.261.1.  Google Scholar

[3]

P. Antunes and J. M. Nunes da Costa, Hyperstructures on Lie algebroids, Rev. in Math. Phys., 25 (2013), 1343003, 19pp. doi: 10.1142/S0129055X13430034.  Google Scholar

[4]

P. Antunes and J. M. Nunes da Costa, Induced hypersymplectic and hyperkähler structures on the dual of a Lie algebroid, Int. J. Geom. Meth. Mod. Phys., 11 (2014), 1460030 (9 pages). doi: 10.1142/S0219887814600305.  Google Scholar

[5]

P. Antunes and J. M. Nunes da Costa, Hyperstructures with torsion on Lie algebroids, preprint,, , ().   Google Scholar

[6]

H. Bursztyn, G. Cavalcanti and M. Gualtieri, Generalized Kähler and hyper-Kähler quotients, in Poisson geometry in mathematics and physics, Contemp. Math., Amer. Math. Soc., Providence, RI, 450 (2008), 61-77. doi: 10.1090/conm/450/08734.  Google Scholar

[7]

P. S. Howe and G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B, 379 (1996), 80-86. doi: 10.1016/0370-2693(96)00393-0.  Google Scholar

[8]

J. Grabowski, Courant-Nijenhuis tensors and generalized geometries, in Groups, geometry and physics, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 29 (2006), 101-112.  Google Scholar

[9]

N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., 54 (2003), 281-308. doi: 10.1093/qmath/hag025.  Google Scholar

[10]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that ... in Poisson geometry and Lie algebroid theory, in The Breadth of symplectic and Poisson geometry, J.E. Marsden and T. Ratiu, eds., Progr. Math., 232, Birkhäuser, Boston, MA, 2005, pp. 363-389. doi: 10.1007/0-8176-4419-9_12.  Google Scholar

[11]

Y. Kosmann-Schwarzbach, Poisson and symplectic functions in Lie algebroid theory, in Higher Structures in Geometry and Physics, A. Cattaneo, A. Giaquinto and Ping Xu, eds., Progr. Math. 287, Birkhäuser, Boston, 2011, pp. 243-268. doi: 10.1007/978-0-8176-4735-3_12.  Google Scholar

[12]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond, T. Voronov, ed., Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185. doi: 10.1090/conm/315/05479.  Google Scholar

[13]

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137. doi: 10.1023/A:1020708131005.  Google Scholar

[14]

M. Stiénon, Hypercomplex structures on Courant algebroids, C. R. Acad. Sci. Paris, 347 (2009), 545-550. doi: 10.1016/j.crma.2009.02.020.  Google Scholar

[15]

T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond, T. Voronov, ed., Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168. doi: 10.1090/conm/315/05478.  Google Scholar

[16]

P. Xu, Hyper-Lie Poisson structures, Ann. Sci. École Norm. Sup., 30 (1997), 279-302. doi: 10.1016/S0012-9593(97)89921-1.  Google Scholar

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