Schouten bracket

Overview

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

Definition

For multi-vector fields $X\in {\mathfrak{X}}^{k+1}(M)$ and $Y\in {\mathfrak{X}}^{l+1}(M)$, the Schouten bracket $[X,Y]$ is the unique multivector field such that

${\mathcal{L}}_{[X,Y]}={\mathcal{L}}_X\circ {\mathcal{L}}_Y-(-1{)}^{kl}{\mathcal{L}}_Y\circ {\mathcal{L}}_X$ where ${\mathcal{L}}_X\circ {\mathcal{L}}_Y(f_1,\dots ,f_{k+l+1})={\sum }_{\sigma \in S_{k,l+1}}(-1{)}^{\sigma }{\mathcal{L}}_X(f_{\sigma (1)},\dots ,f_{\sigma (k)},{\mathcal{L}}_Y(f_{\sigma (k+1)},\dots ,f_{\sigma (k+l+1)}))$

Alternate definition

Another way to write the Schouten bracket is with the formula for $[-,-]:{\mathfrak{X}}^k(M)\times {\mathfrak{X}}^l\to {\mathfrak{X}}^{k+l-1}$ given by $[X_1\wedge \cdots \wedge X_k,Y_1\wedge \cdots \wedge Y_l]={\sum }_{i,j}(-1{)}^{i+j}[X_i,Y_j]\wedge X_1\wedge \dots \widehat{X_i}\cdots \wedge X_p\wedge Y_1\wedge \dots \widehat{Y_j}\cdots \wedge Y_l$

Properties

The Schouten bracket is the unique $\mathbb{R}$-biliear operation

$[-,-]:{\mathfrak{X}}^{k+1}(M)\times {\mathfrak{X}}^{l+1}(M)⟶{\mathfrak{X}}^{k+l+1}(M),(k,l\ge 1)$

satisfying the following properties:

1. When $k=l=0$ it is the usual Lie bracket of vector fields
2. When $k=0$ and $l=-1$ it is the Lie derivative

$[X,f]={\mathcal{L}}_{X}f=X(f)$

3. Graded skew-symmetry

$[X,Y]=-(-1{)}^{kl}[Y,X]\text{for }X\in {\mathfrak{X}}^{k+1}(M),Y\in {\mathfrak{X}}^{l+1}(M)$

4. Graded Leibniz identity:

$[X,Y\wedge Z]=[X,Y]\wedge Z+(-1{)}^{k(l+1)}Y\wedge [X,Z]$

for $X\in {\mathfrak{X}}^{k+1}(M)$, $Y\in {\mathfrak{X}}^{l+1}(M)$, and $Z\in {\mathfrak{X}}^{m+1}(M)$.

Moreover, the graded Jacobi identity holds.

$(-1{)}^{km}[X,[Y,Z]]+(-1{)}^{lk}[Y,[Z,X]]+(-1{)}^{lm}[Z,[X,Y]]=0$

References

@crainic2021 - Chapter 2 @michor2008