On the development of minimality conditions in lie algebras

This article discusses the main stages of the development of Lie algebras satisfying minimality conditions. The minimality condition implies stabilization of descending chains and can be applied to various substructures of the Lie algebra: ideals, subideals, subalgebras, as well as inner ideals. All these conditions were transferred to Lie algebras from ring theory and group theory. Such Lie algebras are called "Artinian"by analogy with rings of the same name, since the minimality condition for rings was first considered by Emil Artin. In order to generalize the minimality condition to ideals, quasi-Artinian Lie algebras arose in the study of Artinian Lie algebras. The article presents the main research results of Russian and foreign scientists whose works are devoted to Artinian and quasi-Artinian Lie algebras. The first papers on Lie algebras with minimality conditions belong to the British mathematician Ian Nicholas Stewart. It was he who began the transfer of all the necessary concepts and conditions from group theory to Lie algebras. Stewart considered minimality conditions in relation to subideals and n-step Lie algebra subideals. A little later, Ralph Amayo became his co-author. Together they continued their work on transferring the concepts for ascending and descending chains from solvable groups to Lie algebras and proving new results about them. The main results of their research were included in the book "Infinite-dimensional Lie Algebras the most famous among mathematicians. Their joint work did not end there, the research of minimality conditions was continued. In their works, Amayo and Stewart formulated open-ended questions, the answers to which were later found by their followers. Among them, the following scientists can be distinguished: S. Togo, F. Kubo, T. Ikeda, F.A.M. Aldosrey. Their works are devoted to the study of the relations between various conditions of minimality. The minimality condition can also be applied to inner ideals, since inner ideals are considered an analogue of the one-sided ideal of rings in Lie algebras. Georgia Benkart can be considered the founder of this approach. Also, this direction belongs to the works of M. Osborne, F. Lopez, E. Garcia, G. Lozano. As noted above, in order to generalize the minimality condition to ideals in the study of Artinian Lie algebras, F.A.M. Aldosrey in 1983 considered quasi-Artinian Lie algebras. After studying this work, F. Kubo and M. Honda made up a number of questions that were not answered in it, and thus continued the study of quasi-Artinian Lie algebras. In 2021, a joint work by F. Aldosrey and I. Stewart was published, in which they conducted two essentially different studies of the structure of infinite-dimensional Lie algebras, connected by the fact that special attention was paid to chain conditions, radicals, and generalizations of the “ideal” relation. Among domestic scientists, Yu.A. Bakhturin, M.V. Zaitsev, S.A. Pihtilkov, V.M. Polyakov were engaged in studies of minimality conditions for ideals. The object of their study were special Lie algebras. S.A. Pihtilkov and E.V. Meshcherina also considered the minimality conditions for subalgebras and inner ideals and conducted a study of the relationship between Lie algebras with these minimality conditions. All the results of research in this area were applied by S.A. Pihtilkov, E.V. Meshcherina and A.N. Blagovisnaya to solve the problem of solvability of the primary radical of the Artinian Lie algebra. When writing this paper, the author tried to present the results in chronological order.