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Serially finite Lie algebras. (English) Zbl 0598.17012

The author presents several characterizations of serially finite Lie algebras. He proves that over a field of characteristic zero, \[ L(ser){\mathfrak F}=L{\mathfrak F}\cap J(ser){\mathfrak F}=L(lsi){\mathfrak F}=J(lsi){\mathfrak F}=L(lasc){\mathfrak F}=J(lasc){\mathfrak F} \] where L\({\mathfrak F}\) is the class of locally finite Lie algebras, L(\(\Delta)\)\({\mathfrak F}\) is the class of Lie algebras L such that any finite subset of L is contained in a finite-dimensional \(\Delta\)-subalgebra, J(\(\Delta)\)\({\mathfrak X}\) is the class of Lie algebras generated by finite-dimensional \(\Delta\)-subalgebras, \(\Delta\) is any one of the relations serial, local subideal, local ascendant.
Similar characterizations have been given for the subclasses L(ser)(EA\(\cap {\mathfrak F})\), L(ser)(\({\mathfrak M}\cap {\mathfrak F})\) of serially finite Lie algebras, where \({\mathfrak F}\), \({\mathfrak M}\), EA be the classes of finite-dimensional, nilpotent and soluble Lie algebras. Furthermore defining L\({\mathfrak M}\) to be the class of locally nilpotent Lie algebras, he shows that L\({\mathfrak M}=L(ser)({\mathfrak M}\cap {\mathfrak F})\) and L\({\mathfrak M}\) coincides with the class of locally finite Lie algebras each of whose one-dimensional subalgebras is weakly serial (resp. \(\omega\)-step weakly ascendant).
Reviewer: F.A.M.Aldosray

MSC:

17B65 Infinite-dimensional Lie (super)algebras
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