The twisted gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

Jan Kurek, Włodzimierz Mikulski

Abstract


We completely describe all gauge-natural operators \(C\) which send linear \((p+2)\)-forms \(H\) on vector bundles \(E\) (with sufficiently large dimensional bases) into \(\mathbf{R}\)-bilinear operators \(C_H\) transforming pairs \((X_1\oplus\omega_1,X_2\oplus\omega_2)\) of couples of linear vector fields and linear \(p\)-forms on \(E\) into couples \(C_H(X_1\oplus\omega_1, X_2\oplus\omega_2)\) of linear vector fields and linear \(p\)-forms on \(E\). Further, we extract all \(C\) (as above) such that \(C_0\) is the restriction of the well-known Courant bracket and \(C_H\) satisfies the Jacobi identity in Leibniz form for all closed linear \((p+2)\)-forms \(H\).

Keywords


Natural operator; linear vector field; linear p-form; Jacobi identity in Leibniz form

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References


Kolar, I., Michor, P. W., Slovak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.

Kurek, J., Mikulski, W. M., The gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms, Ann. Univ. Mariae Curie-Skłodowska Sect. A 75(2) (2021), 73–92.

Mikulski, W. M., The natural operators similar to the twisted Courant bracket on couples of vector fields and p-forms, Filomat 44(12) (2020), 4071–4078.

Mikulski, W. M., On the gauge-natural operators similar to the twisted Dorfman–Courant bracket, Opuscula Math 41(2) (2021), 205–226.




DOI: http://dx.doi.org/10.17951/a.2022.76.1.31-46
Data publikacji: 2022-10-05 20:39:33
Data złożenia artykułu: 2022-10-04 21:01:49


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