Schouten–Nijenhuis bracket

The Schouten–Nijenhuis bracket is an important tool in differential geometry and algebraic topology, particularly in the study of multivector fields and their relations to differential forms and Lie algebras. It generalizes the Lie bracket of vector fields to multivector fields, which are generalized objects that can be thought of as skew-symmetric tensors of higher degree. ### Definition 1. **Multivector Fields**: Let \( V \) be a smooth manifold.