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Simple, semisimple, solvable, and nilpotent Lie algebras: definition

DEFINITION 5.4   For a Lie algebra $ L$ , let us define the following ideals of $ L$ .
  1. $ \operatorname{Comm}(L)=[L,L] $ , and inductively,

    $\displaystyle \operatorname{Comm}^j(L)=\operatorname{Comm}(\operatorname{Comm}^{j-1}(L)).
$

  2. $ \operatorname{ad}(L)(L)=[L,L]$ , and inductively,

    $\displaystyle \operatorname{ad}^j(L)(L)=\operatorname{ad}(\operatorname{ad}^{j-1}(L)).
$

LEMMA 5.5   For any Lie algebra $ L$ and for any positive integer $ j$ , we have

$\displaystyle \operatorname{Comm}^j(L) \subset \operatorname{ad}^j(L).
$

PROOF.. Inductively, we have

$\displaystyle \operatorname{Comm}^j(L) =[\operatorname{Comm}^{j-1}(L),\operator...
...{Comm}^{j-1}(L)]\subset [L,\operatorname{ad}^{j-1}(L)]=\operatorname{ad}^j(L).
$

$ \qedsymbol$

DEFINITION 5.6   A Lie algebra $ L$ over a field $ k$ is said to be
  1. semisimple if it has no abelian ideals.
  2. simple if it has no non trivial ideals and $ \dim(L)>1$ .
  3. solvable if $ \operatorname{Comm}^N (L)=0$ for some $ N\in \mathbb{Z}_{>0}$ .
  4. nilpotent if $ \operatorname{ad}(L)^N(L)=0$ for some $ N\in \mathbb{Z}_{>0}$ .

PROPOSITION 5.7   We have the following implications.
  1. Simple Lie algebras are semisimple.
  2. Nilpotent Lie algebras are solvable.

PROOF.. (1)is Easy. (2) follows from Lemma 5.5. $ \qedsymbol$

Semisimple algebras and solvable ones are ``orthogonal''. For now we only mention the following

LEMMA 5.8   Non zero solvable algebra $ L$ cannot be semisimple.

PROOF.. Let $ N_0$ be a positive integer such that

$\displaystyle \operatorname{Comm}^N_0 (L)\neq 0 , \qquad
\operatorname{Comm}^{N_0+1} (L)=0.
$

Then $ \operatorname{Comm}^{N_0}(L)$ is a non-zero abelian ideal of $ L$ . $ \qedsymbol$


next up previous
Next: The radicals of Lie Up: generalities in finite dimensional Previous: Ideals of Lie algebras
2007-12-19