Title:

Subalgebras of free nilpotent and polynilpotent Lie algebras

In this thesis we study subalgebras in free nilpotent and polynilpotent Lie algebras. Chapter 1 sets up the notation and includes definitions and elementary properties of free and certain reduced free Lie algebras that we use throughout this thesis. We also describe a Hall basis of a free Lie algebra as in [4] and a basis for a free polynilpotent Lie algebra which was developed in [24]. In Chapter 2 we first consider the class of nilpotency of subalbebras of free nilpotent Lie algebras starting with twogenerator subalgebras. Then we study those subalgebras in a free nilpotent Lie algebra which, are themselves free nilpotent. We give necessary and sufficient conditions in the case of twogenerator subalgebras. Chapter 3 extends the results obtained in Chapter 2 to the polynilpotent case. First we look at twogenerator subalgebras of a free polynilpotent Lie algebra. Then we consider more general subalgebras. Finally we study those subalgebras which are themselves free polynilpotent and give necessary and sufficient conditions for twogenerator subalgebras to be free polynilpotent. In Chapter 4 we first study certain properties of ideals in free, free nilpotent and free polynilpotent Lie algebras and establish the fact that in a free polynilpotent Lie algebra a nonzero ideal which is finitelygenerated as a subalgebra must be equal to the whole algebra. Then we consider the quotient Lie algebra of a lower central term of a free Lie algebra by a term of the lower central series of an ideal. We then generalize the results to cover the free nilpotent and free polynilpotent cases. In the last section of Chapter 4 we consider ideals of free nilpotent (and later polynilpotent) Lie algebras as free nilpotent (polynilpotent) subalgebras and establish the fact that in most nontrivial cases such an ideal cannot be free nilpotent (polynilpotent). In the last chapter we consider the m+kth term of the lower central series of a free Lie algebra as a subalgebra of the mth term for m ⩽ k and generalize the results proved in [25]. We give reasons for the failure of these results in the case m > k.
