Opuscula Math. 40, no. 6 (2020), 703-723

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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  • 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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