On modular\(^*\) subalgebras of a Lie algebra
A subalgebra \(U\) of a Lie algebra \(L\) is called modular\(^*\) if \(\langle U,B \rangle \cap C= \langle B,U \cap C \rangle\) for all subalgebras \(B\subseteq C\) and \(\langle U \cap B,C \rangle= \langle B,C \rangle \cap U\) for all subalgebras \(C\subseteq U\). (This is a dualized version of the modular conditions.) This paper contains several results on the structure of Lie algebras containing a modular\(^*\) subalgebra. It is shown that all modular\(^*\) subalgebras of a nilpotent Lie algebra are inside the center. It is also shown that for Lie algebras \(L\) over a field of characteristic zero every proper modular\(^*\) subalgebra is inside the abelian socle of \(L\) unless \(L\) is three-dimensional nonsplit simple or almost abelian, and, finally, that if \(L\) is a Lie algebra over a field of characteristic different from 2, 3 and containing an almost abelian modular\(^*\) subalgebra, then \(L\) is almost abelian also.
scientific article; zbMATH DE number 1014006
scientific article