Semi-modular subalgebras of a Lie algebra (Q1076781)

Let \(L\) be a Lie algebra, the subalgebra \(U\) of \(L\) is called upper semimodular if, whenever \(B\) is a subalgebra of \(L\) such that \(U\cap B\) is a maximal subalgebra of \(B\), then \(U\) is a maximal subalgebra of \(<U,B>\) (the subalgebra generated by \(U\) and \(B\)). The subalgebra \(U\) of \(L\) is called lower semimodular if, whenever \(B\) is a subalgebra of \(L\) such that \(U\) is maximal in \(<U,B>\), then \(U\cap B\) is maximal in \(B\). In spite of its appearance these are not dual concepts in lattice theory. The subalgebra \(U\) of \(L\) is called semimodular if it is both upper and lower semimodular. Ideals are obvious examples of these three kinds of subalgebras. The first two conditions are shown to be genuinely weaker than modularity, studied by \textit{R. K. Amayo} and \textit{J. Schwarz} [Hiroshima Math. J. 10, 311--322 (1980; Zbl 0441.17002)], and for the third it is proved that, for a finite dimensional Lie algebra over a field of characteristic zero, a subalgebra is semimodular if and only if it is modular.