Opuscula Math. 41, no. 2 (2021), 205-226
https://doi.org/10.7494/OpMath.2021.41.2.205
Opuscula Mathematica
On the gauge-natural operators similar to the twisted Dorfman-Courant bracket
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.
Mathematics Subject Classification: 53A55, 53A45, 53A99.
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https://orcid.org/0000-0002-2905-0461- Jagiellonian University, Department of Mathematics, S. Łojasiewicza 6, Cracow, Poland
- Communicated by P.A. Cojuhari.
- Received: 2020-11-12.
- Accepted: 2021-01-27.
- Published online: 2021-03-17.
