Lie derivation

Lie derivation appears in a wide contexts. It is based on the following observation. Let $A$ be a commutative ring. Let $B$ be a commutative $A$ -algebra. Let $\epsilon$ be the dual number.

$$A_\epsilon=A[\epsilon]/(\epsilon^2), \quad B_\epsilon=B[\epsilon]/(\epsilon^2) .$$

Then for any $x\in \operatorname{Hom}_B(\Omega^1_{B/A},B)=\operatorname{Der}_A(B,B)$ , we define

$$\operatorname{id}+\epsilon x: B_\epsilon \to B_\epsilon.$$

It is an $A_\epsilon$ -algebra automorphism of $B_\epsilon$ which reduces to the identity when we put $\epsilon=0$ . So any natural construction on algebras may be transformed by this map. Among such constructions is the modules of differential forms, modules of derivations, tensor products of them, and the dual of them.

We may also introduce

$$L_x: \Omega^\bullet_{B/A} \to \Omega^\bullet_{B/A}$$

to be the unique $A$ -linear map which satisfies the following properties.

1. $L_x(f)=x.f=\langle x, d f\rangle$ for any $f \in B$ .
2. $L_x(d f)=d (\langle x, d f\rangle)$ for any $f \in B$ .
3. $L_x$ is an even derivation. That means,
   $$L_x(\alpha \wedge \beta) =(L_x\alpha)\wedge \beta +\alpha \wedge ... ...ll \alpha \in \Omega^\bullet_{B/A},\ \forall \beta \in \Omega^\bullet_{B/A}).$$

Other useful Lie derivation is that for vector fields. Namely, Let $x,y$ be two $A$ -derivations from $B$ to $B$ . Then we define

$$L_x(y)=[x,y](=xy-yx).$$

Lie derivations commute with contractions. Namely,

$$L_x \langle y, df\rangle =\langle L_x (y), df\rangle +\langle y, L_x(df)\rangle \quad (\forall x, y \in \operatorname{Der}_A(B), \forall f\in B).$$

We leave it the reader to do the detailed discussion.