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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• Communicated by P.A. Cojuhari.
• Accepted: 2020-10-02.
• Published online: 2020-12-01.