On the Derivation Algebras of Lie Algebras

Source abstract

Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L , that is, the Lie algebra of all derivations of L . Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L ? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases. Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L) . However, there is still some intimate connection between the structure of D(L) and that of L .