# American Institute of Mathematical Sciences

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. [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. [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. [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. [5] P. Antunes and J. M. Nunes da Costa, Hyperstructures with torsion on Lie algebroids, preprint, arXiv:1501.00887. [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. [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. [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. [9] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., 54 (2003), 281-308. doi: 10.1093/qmath/hag025. [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. [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. [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. [13] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137. doi: 10.1023/A:1020708131005. [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. [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. [16] P. Xu, Hyper-Lie Poisson structures, Ann. Sci. École Norm. Sup., 30 (1997), 279-302. doi: 10.1016/S0012-9593(97)89921-1.

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##### 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. [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. [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. [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. [5] P. Antunes and J. M. Nunes da Costa, Hyperstructures with torsion on Lie algebroids, preprint, arXiv:1501.00887. [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. [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. [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. [9] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., 54 (2003), 281-308. doi: 10.1093/qmath/hag025. [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. [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. [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. [13] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61 (2002), 123-137. doi: 10.1023/A:1020708131005. [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. [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. [16] P. Xu, Hyper-Lie Poisson structures, Ann. Sci. École Norm. Sup., 30 (1997), 279-302. doi: 10.1016/S0012-9593(97)89921-1.
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