The lattice of ideals of a Lie algebra
Let \(L\) be a finite dimensional Lie algebra over a field \(F\) and \({\mathfrak L} (L)\) be the lattice of ideals of \(L\). The author classifies the Lie algebras in which the lattice \({\mathfrak L} (L)\) is complemented, complemented and distributive, or abelian and show that in case \(\dim L/L^ 2 >1\), the derived algebra and the solvable radical of \(L\) (for fields of characteristic 0) can be characterized in terms of the ideal lattice structure of \(L\). If the intersection of all maximal subalgebras of \(L\) is trivial, that is, if the Frattini subalgebra of \(L\) is zero, let us say that \(L\) has property \({\mathfrak P}\). Let \(F\) have characteristic zero, \(M\) be another Lie algebra over \(F\) such that \({\mathfrak L} (L)\) and \({\mathfrak L} (M)\) are isomorphic and suppose that \({\mathfrak P}\) holds in \(L\). Then the author shows that either \({\mathfrak P}\) holds in \(M\) or \(F\) is not algebraically closed and \(L\) and \(M\) both have a severely constrained structure.
scientific article; zbMATH DE number 735304