Schouten bracket

Overview

The Schouten bracket is a generalization of the Lie bracket for multi-vector fields.

Definition

For multi-vector fields and , the Schouten bracket is the unique multivector field such that

where

Alternate definition

Another way to write the Schouten bracket is with the formula for given by

Properties

The Schouten bracket is the unique -biliear operation

satisfying the following properties:

1. When it is the usual Lie bracket of vector fields
2. When and it is the Lie derivative

3. Graded skew-symmetry

4. Graded Leibniz identity:

for , , and .

Moreover, the graded Jacobi identity holds.

References

@crainic2021 - Chapter 2 @michor2008