Opuscula Math. 40, no. 6 (2020), 703-723
https://doi.org/10.7494/OpMath.2020.40.6.703

 
Opuscula Mathematica

On the twisted Dorfman-Courant like brackets

Włodzimierz M. Mikulski

Abstract. There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted.

Keywords: natural operator, linear vector field, linear form, (twisted) Dorfman-Courant bracket, Jacobi identity in Leibniz form.

Mathematics Subject Classification: 53A55, 53A45, 53A99.

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  1. Z. Chen, Z. Liu, Omni-Lie algebroids, J. Geom. Phys. 60 (2010) 5, 799-808.
  2. T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.
  3. M. Doupovec, J. Kurek, W.M. Mikulski, The natural brackets on couples of vector fields and 1-forms, Turk. Math. J. 42 (2018), 1853-1862.
  4. M. Gualtieri, Generalized complex geometry, Ann. of Math. 174 (2011) 1, 75-123.
  5. N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308.
  6. M. Jotz Lean, C. Kirchhoff-Lukat, Natural lifts of Dorfman brackets, arXiv:1610.05986v2 [math.DG] 14 Jul 2017.
  7. I. Kolář, P.W. Michor, J. Slovák, Natural Operations in Differential Geometry, Berlin, Germany Springer-Verlag, 1993.
  8. Z.J. Liu, A. Weinstein, P. Xu, Main triples for Lie bialgebroids, J. Differ. Geom. 45 (1997), 547-574.
  9. W.M. Mikulski, The natural operators similar to the twisted Courant bracket one, Mediter. J. Math. 16 (2019), Article no. 101.
  10. W.M. Mikulski, The gauge-natural bilinear operators similar to the Dorfman-Courant bracket, 17 (2020), Article no. 40.
  11. P. Severa, A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys., Suppl. 144 (2001), 145-154.
  • Communicated by P.A. Cojuhari.
  • Received: 2020-07-01.
  • Accepted: 2020-10-02.
  • Published online: 2020-12-01.
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Cite this article as:
Włodzimierz M. Mikulski, On the twisted Dorfman-Courant like brackets, Opuscula Math. 40, no. 6 (2020), 703-723, https://doi.org/10.7494/OpMath.2020.40.6.703

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