A common generalization of the Frölicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

Abstract

There is a canonical mapping from the space of sections of the bundle $\wedge T^\ast M\otimes ST\ M$ to $Ω(T^\ast M ; T(T^\ast M))$. It is shown that this is a homomorphism on $Ω(M;TM) for the Frölicher-Nijenhuis brackets, and also on $Γ(ST\ M)$ for the Schouten bracket of symmetric multi vector fields. But the whole image is not a subalgebra for the Frölicher-Nijenhuis bracket on $Ω(T^\ast M;T(T^\ast M))$.