the Schouten–Nijenhuis bracket is the usual Lie bracket of vector fields. The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading is changed to the one of opposite parity (so that the even and odd subspaces are switched), though with this new grading it is no longer a supercommutative ring. Accordingly, the Jacobi identity may also be expressed in the symmetrical form Generalizations There is a common generalization of the Schouten–Nijenhuis bracket for alternating multivector fields and the Frölicher–Nijenhuis bracket due to Vinogradov (1990). A version of the Schouten–Nijenhuis bracket can also be defined for symmetric multivector fields in a similar way. The symmetric multivector fields can be identified with functions on the cotangent space T*(M) of M that are polynomial in the fiber, and under this identification the symmetric Schouten–Nijenhuis bracket corresponds to the Poisson bracket of functions on the symplectic manifold T*(M). There is a common generalization of the Schouten–Nijenhuis bracket for symmetric multivector fields and the Frölicher–Nijenhuis bracket due to Dubois-Violette and Peter W. Michor (1995). References External links Nicola Ciccoli Schouten–Nijenhuis bracket in notes on From Poisson to Quantum Geometry Bilinear maps Differential geometry