The Jordan socle and finitary Lie algebras
The goal of the paper under review is to study simple finitary Lie algebras and their algebras of derivations by using methods from the theory of Jordan systems. (We refer to the paper, which is clearly written, for the relevant definitions.) In a related paper [Commun. Algebra 32, No. 10, 3807--3824 (2004; Zbl 1088.17010)] the same authors have developed a socle theory for \(3\)-graded Lie algebras via Jordan pairs. In this paper it is shown that this socle is independent of the chosen grading, when the gradings are effective; and a definition of a Jordan socle is given for nondegenerate, but not necessarily \(3\)-graded, Lie algebras. A description of nondegenerate Lie algebras with essential Jordan socle is then obtained. In doing so, the algebras of derivations of simple finitary Lie algebras of infinite dimension over algebraically closed fields of characteristic zero are computed.
scientific article; zbMATH DE number 2136740