Tame and non-tame automorphisms and primitive elements of free Lie algebras

In this thesis we establish primitive lifting in free metabelian Lie algebras and primitive lifting of a single element in case rank 3. We also establish the systems of sets which can be lifted to a primitive systems of the free nilpotent Lie algebra free nilpotent Lie algebras. Then we present an automorphism of F=gc+1(F) which cannot be lifted to an automorphism of F=(gc+1(F))0 and we show that there are automorphisms of the free metabelian Lie algebras of rank n, which cannot be lifted to automorphisms of the free solvable Lie algebras of rank n and derived length 3, where F is a free Lie algebra. Furthermore, we give a necessary condition for an endomorphism of F to be an automorphism. This condition is explicitly based on the Dieudonn´e determinant and it yields a method to detecting non-tame automorphisms of a free metabelian Lie algebra. Finally we show that if R is an ideal of F then free Lie algebra F=R0 has automorphisms which cannot be lifted to an automorphism of F and F=R has automorphisms which cannot be lifted to an automorphism of F=R0 KeyWords: Free Lie Algebras,primitive lifting, automorphic lifting, Dieudonn´e Determinant II