Title:

Subideals of Lie algebras

We study infinitedimensional 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 torsionfree 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 nstep sub ideal and shows that such algebras are nilpotent of class bounded in terms of n. This is the Lietheoretic 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 2step subideals implies Minsi, the minimal condition for all subideals, and that any Lie algebra satisfying Minsi is an extension of a Jalgebra by a finitedimensional algebra (a Jalgebra is one in which every subideal is an ideal.) We show that algebras satisfy1ng Minsi have an ascending series of ideals with factors simple or finitedimensional 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 Minsi. As a byproduct 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 Jalgebras, 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 torsionfree 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 infinitedimensional 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.
