Nijenhuis structures on Courant algebroids. (English) Zbl 1241.53068
This paper deals with Nijenhuis structures on Courant algebroids in order to study the infinitesimal deformations of Courant algebroids. Let us recall that Nijenhuis operators were introduced by Fuchssteiner and developped by Carinera, Grabowski and Marmo. These authors have also introduced Courant algebroids for their role in the theory of deformations. In this work, the author first recalls some results on Nijenhuis structures on Leibniz algebras and Leibniz algebroids. Then it is proved that the Nijenhuis torsion of a skew-symmetric endomorphism, \(N\), has the usual properties of tensoriality and skew-symmetry when \(N^2\) is proportional to the identity. The author considers also the case of Courant algebroids that are the double of a Lie bialgebroid. It is shown that when \(N^2\) is proportional to the identity, the torsion of the corresponding skew-symmetric endomorphism of the double \(A+A*\) is, in a certain sense, the sum of the torsion of \(N\) and the torsion of its transpose. The role of the Nijenhuis tensors in the theory of Dirac pairs is finally discussed and will be the subject of further papers.
Reviewer: Angela Gammella-Mathieu (Metz)
MSC:
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 53D18 | Generalized geometries (à la Hitchin) |
Keywords:
Courant algebroids; Nijenhuis operators; skewsymmetric endomorphism; Lie bialgebroid; deformationsReferences:
| [1] | P. Antunes. Poisson quasi-Nijenhuis structures with background. Lett. Math. Phys., 86(1) (2008), 33–45. · Zbl 1176.53080 · doi:10.1007/s11005-008-0272-5 |
| [2] | P. Antunes. Crochets de Poisson gradués et applications: structures compatibles et généralisations des structures hyperk.hlériennes. Thèse de doctorat de l’école Polytechnique, March 2010. |
| [3] | 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. · Zbl 1244.17017 · doi:10.1007/s11537-009-0932-y |
| [4] | N. Bedjoui-Tebbal. Contractions d’algèbres de Lie et torsion de Nijenhuis. Bull. Belg. Math. Soc. Simon Stevin, 7(2) (2000), 303–310. · Zbl 1136.17302 |
| [5] | P. Bressler. The first Pontryagin class. Compos. Math., 143(5) (2007), 1127–1163. · Zbl 1130.14018 |
| [6] | J. Cariñena, J. Grabowski and G. Marmo. Contractions: Nijenhuis and Saletan tensors for general algebraic structures. J. Phys., A 34(18) (2001), 3769–3789. · Zbl 1009.17003 |
| [7] | J. Cariñena, J. Grabowski and G. Marmo. Courant algebroid and Lie bialgebroid contractions. J. Phys., A 37(19) (2004), 5189–5202. · Zbl 1058.53022 |
| [8] | J. Clemente-Gallardo and J.M. Nunes da Costa. Dirac-Nijenhuis structures. J. Phys., A 37(29) (2004), 7267–7296. · Zbl 1065.53064 |
| [9] | I. Ya. Dorfman. Dirac structures of integrable evolution equations. Phys. Lett., A 125(5) (1987), 240–246. · doi:10.1016/0375-9601(87)90201-5 |
| [10] | Irene Dorfman. Dirac structures and integrability of nonlinear evolution equations. Nonlinear Science: Theory and Applications, John Wiley & Sons, Ltd., Chichester (1993). |
| [11] | B. Fuchssteiner. Compatibility in abstract algebraic structures, in ”Algebraic aspects of integrable systems”, A.S. Fokas and I.M. Gelfand, eds., Progr. Nonlinear Differential Equations Appl., 26 (1997), 131–141, Birkhäuser, Boston, MA. · Zbl 0864.35099 |
| [12] | I.M. Gelfand and I. Ya. Dorfman. Schouten bracket and Hamiltonian operators (Russian). Funktsional. Anal. i Prilozhen., 14(3) (1980), 71–74, Funct. Anal. Appl., 14 (3) (1980), 223–226. |
| [13] | 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. · Zbl 1137.53021 |
| [14] | J. Grabowski, D. Khudaverdyan and N. Poncin. Loday algebroids and their supergeometric interpretation, arXiv:1103.5852. · Zbl 1294.53072 |
| [15] | M. Gualtieri. Generalized complex geometry. Ann. Math., 174(1) (2011), 75–123. · Zbl 1235.32020 · doi:10.4007/annals.2011.174.1.3 |
| [16] | Long-Guang He and Bao-Kang Liu. Dirac-Nijenhuis manifold. Rep. Math. Phys., 53(1) (2004), 123–142. · Zbl 1051.53068 · doi:10.1016/S0034-4877(04)90008-0 |
| [17] | F. Keller and S. Waldmann. Deformation theory of Courant algebroids via the Rothstein algebra, arXiv:0807.0584. · Zbl 1315.53103 |
| [18] | Y. Kosmann-Schwarzbach. Derived brackets. Lett. Math. Phys., 69 (2004), 61–87. · Zbl 1055.17016 · doi:10.1007/s11005-004-0608-8 |
| [19] | 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 (2005), 363–389, Birkhäuser, Boston, MA. · Zbl 1079.53126 |
| [20] | 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 (2011), 243–268, Birkhäuser, Boston, MA, arXiv:0711.2043. · Zbl 1216.53073 |
| [21] | Y. Kosmann-Schwarzbach and F. Magri. Poisson-Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Théor., 53(1) (1990), 35–81. · Zbl 0707.58048 |
| [22] | Y. Kosmann-Schwarzbach and V. Rubtsov. Compatible structures on Lie algebroids and Monge-Ampère operators. Acta. Appl. Math., 109(1) (2010), 101–135. · Zbl 1190.53080 · doi:10.1007/s10440-009-9444-2 |
| [23] | J.-L. Loday and T. Pirashvili. Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann., 296(1) (1993), 139–158. · Zbl 0821.17022 · doi:10.1007/BF01445099 |
| [24] | Zhang-Ju Liu, A. Weinstein and Ping Xu. Manin triples for Lie bialgebroids. J. Differential Geom., 45(3) (1997), 547–574. · Zbl 0885.58030 |
| [25] | D. Roytenberg. Quasi-Lie bialgebroids and twisted Poisson manifolds. Lett. Math. Phys., 61(2) (2002), 123–137. · Zbl 1027.53104 · doi:10.1023/A:1020708131005 |
| [26] | D. Roytenberg. On the structure of graded symplectic supermanifolds and Courant algebroids, in ”Quantization, Poisson brackets and beyond”, T. Voronov, ed., Contemp. Math., 315 (2002), 169–185, Amer. Math. Soc., Providence, RI. · Zbl 1036.53057 |
| [27] | D. Roytenberg. Courant-Dorfman algebras and their cohomology. Lett. Math. Phys., 90(1–3) (2009), 311–351. · Zbl 1233.16013 · doi:10.1007/s11005-009-0342-3 |
| [28] | M. Stiénon and Ping Xu. Poisson quasi-Nijenhuis manifolds. Commun. Math. Phys., 270 (2007), 709–725. · Zbl 1119.53056 · doi:10.1007/s00220-006-0168-0 |
| [29] | I. Vaisman. Reduction and submanifolds of generalized complex manifolds. Differential Geom. Appl., 25(2) (2007), 147–166. · Zbl 1126.53049 · doi:10.1016/j.difgeo.2006.08.007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
