On the properties of the prime radical of a weakly artinian Lie algebra

S. A. Pikhtilkov ; O. A. Pikhtilkova ; A. N. Blagovisnaya

Čebyševskij sbornik, Tome 18 (2017) no. 1, pp. 134-142

Résumé

This article deals with the issues of the structural theory of Lie algebras. The construction of the structural theory of algebraic systems implies the existence of certain structures of a special form, which are simpler than the base system. The important tool to study algebraic systems is the radical. The development of the structural theory of Lie algebras led to the emergence of various radicals. There are many radicals of Lie algebras in numerous publications. For example, the Killing radical, the Parfenov radical, the Jacobson radical and the prime radical are considered in various articles. The important area of research is the study of radicals of infinite-dimensional Lie algebras. The article is devoted to proving properties of prime radical of a weakly artinian Lie algebra. A Lie algebra is said to be a weakly artinian if the Lie algebra satisfies the descending chain condition on ideals. In the first section of the paper we introduced the concept of the prime radical in the following way. A Lie algebra $L$ is said to be prime if $[U,V]=0$ implies $U=0$ or $V=0$ for any ideals $U$ and $V$ of $L$. We say that the ideal $P$ of a Lie algebra $L$ is prime if the factor algebra $L/P$ is prime. The intersection of all prime ideals is called the prime radical $P(L)$ of a Lie algebra $L$. In the second section it is shown that any finite set of elements of the prime radical of a weakly artinian Lie algebra generates the nilpotent subalgebra. This means that the prime radical is locally nilpotent. The third section is devoted to the solvability of the prime radical of a weakly artinian Lie algebra. There is a history of solving Mikhalev’ s problem about the prime radical of a weakly artinian Lie algebra in this section also. This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik. Bibliography: 16 titles.

Détail

Keywords: weakly artinian Lie algebra, prime radical of a Lie algebra.