Subideals of Lie algebras

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Abstract

We study infinite-dimensional Lie algebras, with particular regard to their subideal structure.
Chapter 1 sets up notation.
Chapter 2 gives an algebraic treatment of Mal'cev's correspondence between complete locally nilpotent torsion-free groups and locally nilpotent Lie algebras over the rational field. This enables us to translate certain of our later results into theorems about groups. As an application we prove a theorem about bracket varieties.
Chapter 3 considers Lie algebras in which every subalgebra is an n-step sub ideal and shows that such algebras are nilpotent of class bounded in terms of n. This is the Lie-theoretic analogue of a theorem of J.E. Roseblade about groups.
Chapter 4 considers Lie algebras satisfying certain minimal conditions on subideals. We show that the minimal condition for 2-step subideals implies Min-si, the minimal condition for all subideals, and that any Lie algebra satisfying Min-si is an extension of a J-algebra by a finite-dimensional algebra (a J-algebra is one in which every subideal is an ideal.) We show that algebras satisfy1ng Min-si have an ascending series of ideals with factors simple or finite-dimensional abelian, and that the type of such a series may be made any given ordinal number by suitable choice of Lie algebra. We show that the Lie algebra of all endomorphisms of a vector space satisfies Min-si. As a by-product we show that every Lie algebra can be embedded in a simple Lie algebra. Not every Lie algebra can be embedded as a subideal of a perfect Lie algebra.
Chapter 5 considers chain conditions in more specialised classes of Lie algebras. The results are applied to groups.
Chapter 6 develops the theory of J-algebras, and in particular classifies such algebras under conditions of solubility (over any fteld) or finitedimensionality (characteristic zero). We also classify locally finite Lie algebras, every subalgebra of which lies in J ,over algebraically closed fields of characteristic zero.
Chapter 7 concerns various radicals in Lie algebras. We show that not every Baer algebra is Fitting answering a question of B.Hartley. As a consequence we can exhibit a torsion-free Baer group which is not a Fitting group (previous examples have all been periodic). We show that under certain circumstances Baer implies Fitting (for groups or Lie algebras). The last section considers Gruenberg algebras.
Chapter 8 is an investigation parallelling those of Hall and Kulatilaka for groups. We ask: when does an infinite-dimensional Lie algebra have an infinitedimensional abelian subalgebra? The answer is: not always. Under certain conditions of generalised solubility the answer is 'yes' and we can make the abelian subalgebra in question have additional properties (e.g. be a subideal). The answer is also shown to be 'yes' if the algebra is locally finite (over a field of characteristic zero). This enables us to prove a theorem concerning the minimal condition for subalgebras.

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