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.
展开▼
机译：在本文中，我们探讨了特征零积分域上有限秩的自由幂等李代数模型理论以及有限秩的自由幂等群模型理论的某些方面。在每种情况下，我们给出基本等同于这些结构的对象的代数描述。我们的方法如下。在李代数的情况下，我们研究了人们可以在多大程度上从李环结构中恢复代数的模块结构。对于一个小组，问题是人们只能在知道小组操作的情况下可以在多大程度上恢复幂运算。在这两种情况下，都证明无法完全恢复多余的结构，需要引入某些“准”对象来完成特征化。我们对有限阶自由幂立李代数的研究将AG Myasnikov对有限维代数的研究扩展到了提起。在有限阶的自由幂等群的情况下，我们扩展了O. V. Belegradek关于单斜群的结果以及A.G. Myasnikov关于在幂零域中采用指数的幂等群的工作。这一研究线源于1950年代的A. Tarski和A. Mal'cev的作品。
展开▼
