The left Schouten-Nijenhuis bracket on polynomial polyvector fields is the degree-minus-one Gerstenhaber bracket extending the commutator of derivations and by graded antisymmetry and the left graded Leibniz rule. For , this convention gives and . A right insertion convention reverses the degree-two-with-function formula.
Articles by others on the same topic
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.