On lie algebras with modular lattices of subalgebras

Abstract

The aim of the article is to classify Lie algebras over an arbitrary field with modular lattices of subalgebras. The study of various classes of universal algebras, all subalgebras of which have a lattice satisfying the modular identity (M-algebras), has a long history. The first works which dealt the group case appeared at the end of the 1920s and the beginning of the 1930s (A. RottEnder, 0. Ore, R. Baer). There are a great number of papers devoted to the study of modular semi-groups, associative and nonassociative rings and algebras, quasigroups, modules over commutative rings, and other algebraic structures. However, a full classification of the universal algebras with a modular lattice of subalgebras for any class of the “classical” universal algebras does not exist. In the middle of the 1960s an attempt was made to describe Lie algebras with various restrictions imposed on the lattice of subalgebras and, in particular, to describe the Lie algebras with a modular lattice of subalgebras (Kolman Cl, 23, Goto [3], Barnes [4]). A few years later, the papers of Towers [S], Amayo [6], Amayo and Schwarz [7], Gein [S] appeared, in which they studied Lie algebras with modular lattices of subalgebras. Similar problems were considered in [9, 10, 111. The above papers classified the Lie algebras with modular lattices of subalgebras, when certain restrictions are imposed on the ring of operators, on 80 0021-8693186 53.00