Poisson brackets and automorphisms of Poisson algebras in two variables

Let K be a field of characteristic zero and P_{n}=< x_{1},\ x_{2},..., x_{n}> be free Poisson algebra generated by the set { x_{1},\ x_{2},..., x_{n}} over K. In this thesis, it is investigated whether or not Bergman Centralizer Theorem is valid for free Poisson algebra P_{n}. It is shown that in free Poisson algebras over a field of characteristic zero the centralizer of every nonconstant element is a polynomial algebra on a single variable. Afterwards, it is shown that the locally nilpotent derivations of P_{2} are triangulable. Finally, it is proved that the automorphisms of P_{2} are tame.