On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

• Main Author: Włodzimierz M. Mikulski
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2021-03-01
• Series: Opuscula Mathematica
• Subjects: natural operator; linear vector field; linear form; twisted dorfman-courant bracket; the jacobi identity in leibniz form
• Online Access: https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdf

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.