Schouten–Nijenhuis bracket

Summary

The Schouten–Nijenhuis bracket extends the Lie bracket of vector fields to multivector fields, the sections of the exterior powers of the tangent bundle. It lowers the total degree by one and obeys graded versions of antisymmetry, the Jacobi identity and the Leibniz rule. Its main use is in Poisson geometry: a bivector field $\pi$ defines a Poisson structure exactly when $[\pi, \pi] = 0$. It also describes the deformation theory of Poisson structures through Poisson cohomology.