Sur les suites d'algèbres de Lie de dérivations

Abstract

If ${\frak g}$ is a Lie algebra of finite dimension, let $ ({\frak g}_n) $ be the sequence of Lie algebras of derivations associated to $ {\frak g}:{\frak g}_0={\frak g}, {\frak g}_{n+1}=\hbox {Der}\,{\frak g}_n $ . If the center of $ {\frak g} $ is null, E. Schenkman showed that this sequence ends finitely for a complete Lie algebra $ {\frak g}_{\infty } $ (i.e. a Lie algebra with null center and inner derivations). In the first part of this work we construct $ {\frak g}_{\infty } $ . In the second part are studied the sequences for which the center of each $ {\frak g}_n $ is different from zero. One shows, if the dimensions of $ {\frak g}_n $ are bounded, that a limit $ {\frak g}_{\infty } $ exists too: it is the direct product of the ground field by a complete Lie algebra equal to its derived ideal. Note that the proofs used here are valid for a field of characteristic 0 only.