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 space of multivector fields into a graded Lie algebra. The bracket is a derivation in each argument with respect to the wedge product: [P, Q ∧ R] = [P, Q] ∧ R + (−1)^{(p−1)q} Q ∧ [P, R]. For vector fields X, Y the bracket recovers the ordinary Lie bracket [X, Y].

Applications and significance include its central role in Poisson geometry: a bivector π defines a Poisson structure precisely when [π, π] = 0. The Schouten–Nijenhuis bracket provides a natural setting for Poisson cohomology and deformation theory, and endows the exterior algebra of vector fields with a Gerstenhaber algebra structure, combining the wedge product with the graded Lie bracket.

Historically, the bracket is named after Jan Arnoldus Schouten, who introduced the construction in the 1950s, and was later generalized and popularized in the work of Nijenhuis and collaborators, leading to the standard designation Schouten–Nijenhuis bracket in modern differential geometry.