SchoutenNijenhuis
The Schouten–Nijenhuis bracket is a graded Lie bracket defined on the exterior algebra of the tangent bundle of a smooth manifold M, that is, on the space of multivector fields ∑k Γ(∧k TM). It extends the ordinary Lie bracket of vector fields to multivector fields and provides a natural algebraic framework for Poisson geometry and related structures. If P is a p-vector and Q is a q-vector, the bracket [P, Q] lies in ∧^{p+q−1} TM and lowers degree by one, so the bracket has degree −1.
Key properties include graded antisymmetry [P, Q] = −(−1)^{(p−1)(q−1)}[Q, P], and the graded Jacobi identity, making the
Applications and significance include its central role in Poisson geometry: a bivector π defines a Poisson structure
Historically, the bracket is named after Jan Arnoldus Schouten, who introduced the construction in the 1950s,