ad-nilpotent elements, quasi-nilpotent elements and invariant filtrations of infinite dimensional Lie algebras of Cartan type
The author investigates various types of nilpotency of the infinite- dimensional Lie algebras of Cartan type (e.g., the ad-nilpotence and the quasi-nilpotence) and determines the sets of nilpotent elements \(N\) which are invariant. It is shown that the natural and the noncontractible filtrations can be defined by means of \(N\) and so are invariant. He also proves the invariance of these filtrations of the (finite-dimensional) Lie algebras of generalized Cartan type by a similar approach. The invariance of the filtrations is an important problem in the study of the Lie algebras of Cartan type and has been discussed by many authors. In the article under review, the proof is from a more unifying point of view and the results are more general that cover some cases which have not been treated before. His approach has also the advantage that it can be used to discuss the invariance of filtrations (slightly modified) of the Lie algebras of Cartan type over a commutative ring [\textit{N. Jin} and \textit{G.-Y. Shen}, Invariant filtrations of the Lie algebras of Cartan type over a commutative ring (in Chinese) (to appear)] which in turn has applications in the determination of the forms of the Lie algebras of generalized Cartan type over an algebraically non-closed field of prime characteristic [\textit{B. Shu}, Forms of Lie algebras of Cartan type, Doctoral dissertation, East China Normal University (1993)].
infinite-dimensional Lie algebras of Cartan type
scientific article; zbMATH DE number 148453