On the twisted Dorfman-Courant like brackets

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.