@Article{Mikulski2021,
author = "Mikulski, Włodzimierz M.",
title = "On the gauge-natural operators similar to the twisted Dorfman-Courant bracket",
journal = "Opuscula Math.",
year = "2021",
volume = "41",
number = "2",
pages = "205--226",
doi = "https://doi.org/10.7494/OpMath.2021.41.2.205",
url = "https://doi.org/10.7494/OpMath.2021.41.2.205",
abstract = "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.",
keywords = "natural operator, linear vector field, linear form, twisted Dorfman-Courant bracket, the Jacobi identity in Leibniz form",
issn = "1232-9274"
}