Fixed points of endomorphisms of free Lie algebras and fixed point subalgebras

In this thesis we show the existence of non-trivial fixed points of an IA-endomorphism of a free metabelian Lie algebra L and develop an algorithm detecting them. We also prove that the fixed point subalgebra Fix? of an IA-endomorphism ? of L is not finitely generated. Then, we show that IA-endomorphisms of a free metabelian Lie algebra L of rank two has no non-trivial fixed points under certain conditions. Morover, in situation rank(L)?3, we construct an IA-endomorphism and we show that this IA-endomorphism is not an automorphism and has no non-trivial fixed points. Indeed, we show that there is an IA-automorphism without fixed points when rank(L)=4 and rank(L)?5. In addition, we improve a technics for showing that the fixed point subalgebra of some automorphisms of a free Lie algebra and a free metabelian Lie algebra of rank two is not finitely generated. By using this technics, we determined whether the rank of the fixed point subalgebra of some tame and non-tame automorphisms is finite or not. Besides, in a free nilpotent Lie algebra, we determined the necessary condition for some automorphisms to have non-trivial fixed points and we show that the rank of the fixed point subalgebras of some automorphisms are finitely generated.