Semi direct products of Lie algebras.

DEFINITION 5.54   Let $L$ be a Lie algebra over a commutative ring $k$ . Then:

1. A ($k$ -linear) derivation of $L$ is a $k$ -linear map $D: L\to L$ such that it obeys the following ``Leibniz rule''.
   $$D([x,y])=[D x,y]+ [x, D y] \qquad (\forall x,y\in L).$$

2. We denote by $\operatorname{Der}_k(L)$ the set of all derivations of $L$ .

LEMMA 5.55  

1. Any derivation $D$ of a Lie algebra $L$ is lifted to a derivation on the universal enveloping algebra $U(L)$ .
2. $\operatorname{Der}_k(L)$ forms a Lie algebra under the usual $k$ -linear structure and the usual commutator as the bracket product.

DEFINITION 5.56   Let $L_1,L_2$ be Lie algebras over a commutative ring $k$ . we say ``$L_1$ acts on $L_2$ as a derivation'' if there is given a Lie algebra homomorphism

$$\pi: L_1 \to \operatorname{Der}_k(L_2).$$

If the action $\pi$ is obvious in context, we shall simply denote $x.y$ instead of $\pi(x).y$ .

DEFINITION 5.57   Let $L_1,L_2$ be Lie algebras over a commutative ring $k$ . Assume there is given an action $\pi$ of $L_1$ on $L_2$ . Then we define a semi direct product $L_1\ltimes_\pi L_2$ of $L_1$ and $L_2$ by introducing the $k$ -module $L_1 \oplus L_2$ with the following bracket product.

$$[(x_1,x_2),(y_1,y_2)]=([x_1,y_1], [x_2,y_2]+x_1.y_2-y_1 . x_2).$$

Note that

1. $L_1$ and $L_2$ are (identified with) subalgebras of $L_1\ltimes_\pi L_2$ .
2. Further more, $L_2$ is an ideal of $L_1\ltimes_\pi L_2$ .
3. For $x \in L_1$ and $y \in L_2$ , we have
   $$[x,y]_{L_1 \ltimes L_2}=x.y.$$