The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M and first-order polydifferential operators on a closed submanifold N of codimension 1 to which ∆ is transversal. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on M to th...

The Schouten-Nijenhuis bracket on the module of K\"ahler differentials is introduced. We recover Lichnerowicz's notion Poisson manifold by using universal property derivations. prove that a nondegenerate structure corresponds exactly to symplectic structure. Finally, we explore Hamiltonian vector fields in terms

We find two one-parametric families of recursion operators and use them to construct higher symmetries for the Calogero--Bogoyavlenskii--Schiff breaking soliton equation. Then we prove that from first family pair-wise commute with respect Nijenhuis bracket (are compatible).

Killing-Yano tensors are natural generalizations of Killing vectors. We investigate whether Killing-Yano tensors form a graded Lie algebra with respect to the Schouten-Nijenhuis bracket. We find that this proposition does not hold in general, but that it does hold for constant curvature spacetimes. We also show that Minkowski and (anti)-deSitter spacetimes have the maximal number of Killing-Yan...

We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any ...

recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. using this machinery, we have defined the concept of symmetric curvature. this concept is natural and is related to the notions divergence and laplacian of vector fields. this concept is also related to the derivations on the algebra of symmetric forms which has been discus...