Browsing by Subject "twisted Dorfman-Courant bracket"

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Access status: Open Access ,
On the gauge-natural operators similar to the twisted Dorfman-Courant bracket
(Wydawnictwa AGH, 2021) Mikulski, Włodzimierz M.
All $\mathcal{VB}_{m,n}$-gauge-natural operators $C$ sending linear $3$-forms $H \in \Gamma^{l}_E(\bigwedge^3T^*E)$ 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 $TE \oplus T^*E \to E$ are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets $C$ (i.e. $C$ as above such that $C_0$ is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear $3$-forms $H$. An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.
Access status: Open Access ,
On the twisted Dorfman-Courant like brackets
(Wydawnictwa AGH, 2020) Mikulski, Włodzimierz M.
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.