next up previous
Next: Levi decomposition Up: generalities in finite dimensional Previous: Weyl's theorem on complete

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''.

    $\displaystyle 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

$\displaystyle \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.

$\displaystyle [(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

    $\displaystyle [x,y]_{L_1 \ltimes L_2}=x.y.
$


next up previous
Next: Levi decomposition Up: generalities in finite dimensional Previous: Weyl's theorem on complete
2007-12-19