Jacobi condition and Schouten-Nijenhuis bracket

Let $\Pi$ be a bivector on $M$, so $[\Pi, \Pi]\in\gX^3 M$. Let $\om$ be a 3-form on $M$. Then $\bracket{\om}{[\Pi, \Pi]}$ is a function and

$$\bracket{\om}{[\Pi, \Pi]}=-\bracket{d(i_{\Pi}\om)}{\Pi}- \bracket{d(i_{\Pi}\om)}{\Pi} + \bracket{d\om}{\Pi\wedge\Pi}.$$

Let $\om=df\wedge dg\wedge dh$. Put $\{f,g\}=\bracket{df\wedge dg}{\Pi}$. Remark that

$$\bracket{i_{\Pi}\om}{X}=\bracket{df\wedge dg\wedge dh}{\Pi\wedge X}$$

for all $X\in\gX^1 M$. Since $d(df\wedge dg\wedge dh)=0$ we have

$$\bracket{\om}{[\Pi, \Pi]}=-2\bracket{d(i_{\Pi}\om)}{\Pi}.$$

As an application of corollary () consider a Lie algebra $\gerg$, manifold $M$, $\xi\:\gerg\to\gX(M)$ an infinitesimal map (a Lie algebra homomorphism). Then $\xi$ extends uniquely to a degree 0 map
$$\begin{align*} \wedge \xi\: \La^{\bullet}\gerg &\to \gX^{\bullet}(M) \\ x_1\wyw x_n &\mapsto \xi(x_1)\wyw \xi(x_n) \end{align*}$$
which preserves the graded brackets

$$\wedge \xi[\al,\bt]=[\wedge \xi(\al), \wedge\xi(\bt)]_{SN}$$

This is a simple consequence of the fact that both brackets are determined by their values in degree 0 and 1.

Another easy consequence of characterization $[\Pi, \Pi]=0$ is that if $\dim M=2$, then any bivector $\Pi\in\gX^2 M$ is Poisson. Indeed $[\Pi, \Pi]\in\gX^3 M=0$.

Let now $G$ be a connected Lie group such that $\Lie(G)=\gerg$ (not necessarily simply connected). Let us denote the translation operators by
$$\begin{align*} L_g\: G\to G, &\quad h\mapsto gh, \\ R_g\: G\to G, &\quad h\mapsto hg. \end{align*}$$
We will use the following notations for the tangent maps $T_eG\to T_gG$.
$$\begin{align*} L_{g,*} &\: T_h G\to T_{gh} G & R_{g,*} &\: T_h G\to T_{hg} G\\ ... ... & R_{g,*}^{\wedge} &\: \La^{\bullet}T_e G\to \La^{\bullet}T_{g} G \end{align*}$$

Let now $\al\in\La^{\bullet}\gerg$. We will denote with $\al^L$ (resp. $\al^R$) the left (resp. right) invariant multivector field on $G$ whose value at $e\in G$ (identity of $G$) is $\al$ i.e.

$$\al^L(g):=L^{\wedge}_{g, *}\al,\word{(resp.} \al^R(g):=R^{\wedge}_{g, *}\al)$$

In the same way we consider $\al^R$ to be the right invariant multivector field on $G$ whose value at $e$ is $\al$.



The condition $[\ga, \ga]=0$ is called classical Yang-Baxter equation. The above proposition can be stated as: