Lie algebras in which every 1-dimensional weak subideal is an ideal (Q752157)

A Lie algebra L in which every one-dimensional subideal is an ideal is called a \({\mathfrak C}\)-algebra. If every nilpotent inner derivation of a Lie algebra L is zero, L is called a \((A_{\infty})\)-algebra. The authors study \({\mathfrak C}(wsi)\)-algebras, Lie algebras in which every one- dimensional weak subideal is an ideal. Let \(\sigma\) be an ordinal and \({\mathfrak A}\) denote the class of abelian Lie algebras. Denote by \(E'_{\sigma}(wsi){\mathfrak A}\) the class of Lie algebras having an ascending series \((L_{\alpha})_{\alpha \leq \sigma}\) of weakly ascending subalgebras such that \(L_ 0=0\), \(L_{\sigma}=L\), \(L_{\alpha}\triangleleft L_{\alpha +1}\) and \(L_{\alpha +1}/L_{\alpha}\in {\mathfrak A}\) for \(\alpha <\sigma\) and \(L=\cup_{\alpha >\lambda}L_{\alpha}\) for \(\lambda <\sigma\). Define E(wsi)\({\mathfrak A}=\cup_{\sigma >0}E(wsi){\mathfrak A}\). Let L be over a field of characteristic zero. It is proved that if either (a) L is a serially finite Lie algebra whose locally soluble radical belongs to E(wsi)\({\mathfrak A}\) or (b) L is a subideally finite Lie algebra, then L is a \({\mathfrak C}(wsi)\)-algebra if and only if \(L=R\oplus S\), where R is an ideal which is abelian or almost abelian and S is a semisimple \((A_{\infty})\)-ideal of L. If the field is algebraically closed and L satisfies either (a) or it is a weak subideally finite Lie algebra, then L is a \({\mathfrak C}(wsi)\)-algebra if and only if L is either abelian or weakly abelian.