December  2012, 4(4): 385-395. doi: 10.3934/jgm.2012.4.385

Semi-simple generalized Nijenhuis operators

1. 

Univ. Montpellier 2, I3M UMR CNRS 5149, F-34095 Montpellier, France

2. 

Univ. Perpignan Via Domitia, LAboratoire de Mathmatiques et PhySique, EA 4217, F-66860 Perpignan, France

Received  December 2011 Revised  June 2012 Published  January 2013

We study a special class of endomorphism fields of the generalized tangent bundle ${\mathcal{T}}M:=TM\oplus T^*M$ of a smooth manifold $M$. An operator of this class is defined as follows: it has a vanishing Courant-Nijenhuis torsion and is diagonalizable (after a possible extension of scalars) with constant dimensions of its eigenspaces. Such an endomorphism field is called a semi-simple generalized Nijenhuis operator. The generalized paracomplex and complex structures give examples of such operators.
    In this study, we distinguish two cases according to whether the operator has exactly two eigenvalues or has at least three elements in its spectrum. In the first case, we prove that either the operator is affinely related to a generalized complex structure, or it is equivalent to a pair of transverse Dirac structures on ${\mathcal{T}}M$. In the second case, the semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.
Citation: Hassan Boualem, Robert Brouzet. Semi-simple generalized Nijenhuis operators. Journal of Geometric Mechanics, 2012, 4 (4) : 385-395. doi: 10.3934/jgm.2012.4.385
References:
[1]

J. Cariñena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202. doi: 10.1088/0305-4470/37/19/006.

[2]

V. Cruceanu, P. Fortuny and P. M. Gadea, Survey on paracomplex geometry, The Rocky Mountain J. of Math., 26 (1996), 83-115. doi: 10.1216/rmjm/1181072105.

[3]

T. Courant, Dirac manifolds, Trans. Am. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.

[4]

T. Courant and A. Weinstein, Beyond Poisson structures, Travaux en Cours, 27, Hermann, Paris (1988), 39-49.

[5]

M. Crainic, Generalized complex structures and Lie brackets, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 559-578. doi: 10.1007/s00574-011-0029-0.

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms I, Indag. Math., 18 (1956), 338-359.

[7]

J. Grabowski, Courant-Nijenhuis tensor and generalized geometries, Monogr. Real Acad. Ci. Exact. Fis.-Quim. Nat. Zaragoza, 29 (2006), 101-112.

[8]

M. Gualtieri, "Generalized Complex Geometry," Ph.D Thesis, Oxford, UK, 2003.

[9]

N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math., 54 (2003), 281-308. doi: 10.1093/qjmath/54.3.281.

[10]

Y. Kosmann-Schwarzbach, Nijenhuis structures on Courant algebroids, Bull. Braz. Math. Soc., New Series, 42 (2011), 625-649. doi: 10.1007/s00574-011-0032-5.

[11]

P. Libermann, Sur les structures presque paracomplexes, C. R. Acad. Sci. Paris, 234 (1952), 2517-2519.

[12]

Z. Liu, A. Weinstein and P. Xu, Manin triples for Lie algebroids, J. Differential Geometry, 45 (1997), 547-574.

[13]

A. Nijenhuis, Jacobi-type identities for bilinear concomitants of certain tensor fields, Indag. Math. A, 58 (1955), 390-403.

[14]

A. Wade, Dirac structures and paracomplex manifolds, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 889-894. doi: 10.1016/j.crma.2004.03.031.

show all references

References:
[1]

J. Cariñena, J. Grabowski and G. Marmo, Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37 (2004), 5189-5202. doi: 10.1088/0305-4470/37/19/006.

[2]

V. Cruceanu, P. Fortuny and P. M. Gadea, Survey on paracomplex geometry, The Rocky Mountain J. of Math., 26 (1996), 83-115. doi: 10.1216/rmjm/1181072105.

[3]

T. Courant, Dirac manifolds, Trans. Am. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.

[4]

T. Courant and A. Weinstein, Beyond Poisson structures, Travaux en Cours, 27, Hermann, Paris (1988), 39-49.

[5]

M. Crainic, Generalized complex structures and Lie brackets, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 559-578. doi: 10.1007/s00574-011-0029-0.

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms I, Indag. Math., 18 (1956), 338-359.

[7]

J. Grabowski, Courant-Nijenhuis tensor and generalized geometries, Monogr. Real Acad. Ci. Exact. Fis.-Quim. Nat. Zaragoza, 29 (2006), 101-112.

[8]

M. Gualtieri, "Generalized Complex Geometry," Ph.D Thesis, Oxford, UK, 2003.

[9]

N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math., 54 (2003), 281-308. doi: 10.1093/qjmath/54.3.281.

[10]

Y. Kosmann-Schwarzbach, Nijenhuis structures on Courant algebroids, Bull. Braz. Math. Soc., New Series, 42 (2011), 625-649. doi: 10.1007/s00574-011-0032-5.

[11]

P. Libermann, Sur les structures presque paracomplexes, C. R. Acad. Sci. Paris, 234 (1952), 2517-2519.

[12]

Z. Liu, A. Weinstein and P. Xu, Manin triples for Lie algebroids, J. Differential Geometry, 45 (1997), 547-574.

[13]

A. Nijenhuis, Jacobi-type identities for bilinear concomitants of certain tensor fields, Indag. Math. A, 58 (1955), 390-403.

[14]

A. Wade, Dirac structures and paracomplex manifolds, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 889-894. doi: 10.1016/j.crma.2004.03.031.

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