American Institute of Mathematical Sciences

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

Hypersymplectic structures on Courant algebroids

Paulo Antunes 1, and  Joana M. Nunes da Costa 1,

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.
Keywords: Lie algebroid., Courant algebroid, Hypersymplectic, hyperkähler.
Mathematics Subject Classification: Primary: 53D17; Secondary: 53D18, 53C2.
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

[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

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

[1]

David Li-Bland, Pavol Ševera. Integration of exact Courant algebroids. Electronic Research Announcements, 2012, 19: 58-76. doi: 10.3934/era.2012.19.58
[2]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
[3]

Benoît Jubin, Norbert Poncin, Kyosuke Uchino. Free Courant and derived Leibniz pseudoalgebras. Journal of Geometric Mechanics, 2016, 8 (1) : 71-97. doi: 10.3934/jgm.2016.8.71
[4]

Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701
[5]

Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani, Tatiana Toro. The Cauchy problem for Schrödinger flows into Kähler manifolds. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 389-439. doi: 10.3934/dcds.2010.27.389
[6]

Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137
[7]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[8]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[9]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020
[10]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39
[11]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
[12]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
[13]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007
[14]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[15]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[16]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124
[17]

Jundong Zhou. A class of the non-degenerate complex quotient equations on compact Kähler manifolds. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2361-2377. doi: 10.3934/cpaa.2021085
[18]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

[19]

Johannes Huebschmann. On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021009
[20]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences