On Lie algebras with modular lattices of subalgebras
An attempt is made to describe Lie algebras over an arbitrary field with a modular lattice of subalgebras. It is clear that an algebra all of whose proper subalgebras are one-dimensional has a modular lattice of subalgebras. A full classification of the simple algebras having this property does not exist. Thus all results were made in supposition that a finite-dimensional simple algebra all of whose proper subalgebras are one-dimensional is three-dimensional (this assertion has been proved by \textit{A. Premet} for algebras over perfect fields of characteristic \(>5\) [Mat. Sb., Nov. Ser. 129, No. 1, 140--153 (1986; Zbl 0595.17007)]). The main result is: Theorem. A locally finite-dimensional Lie algebra has a modular subalgebra lattice if and only if it is abelian, almost abelian or a three-dimensional simple nonsplit algebra. Remark. It is said in the paper that Proposition 4 and the above- mentioned Theorem can be proved in the same way for algebras with an upper semi-modular lattice of subalgebras. There are examples showing that this is not true.
three-dimensional simple nonsplit algebra
scientific article; zbMATH DE number 3946408