On the elementary theories of free nilpotent Lie algebras and free nilpotent groups

In this thesis we explore certain aspects of the model theory of a free nilpotent Lie algebra of finite rank over a characteristic zero integral domain, and the model theory of a free nilpotent group of finite rank. In each case we give an algebraic description of objects elementarily equivalent to these structures. Our approach is as follows. In the case of a Lie algebra, we investigate to what extent one can recover the module structure of the algebra from the Lie ring structure. In the case of a group the question is to what extent one can recover exponentiation from knowing only the group operations. In both cases it turns out that one can not recover the extra structures completely and needs to introduce certain "quasi" objects to complete the characterization.;Our work on free nilpotent Lie algebras of finite rank extends A. G. Myasnikov's work on finite dimensional algebras over a filed. In the case of free nilpotent groups of finite rank we extend O. V. Belegradek's results on unitriangular groups and A.G. Myasnikov's work on nilpotent groups taking exponents in a characteristic zero field. This line of research takes its roots in the works of A. Tarski and A. Mal'cev from 1950's.