tailieunhanh - The theory of Jacobi systems and their Abelian representations

In this article we introduce a new generalization of the concept of Lie ring which we call Jacobi system and we investigate some elementary properties of these systems and their Abelian representations. | Turk J Math 28 (2004) , 119 – 135. ¨ ITAK ˙ c TUB The Theory of Jacobi Systems and Their Abelian Representations M. Shahryari, Y. Zamani Abstract In this article we introduce a new generalization of the concept of Lie ring which we call Jacobi system and we investigate some elementary properties of these systems and their Abelian representations. The aim of this article is to introduce a new generalization of the concept of Lie ring. The importance of Lie rings in the study of nilpotent groups as well as their role in the investigation of the Burnside problem is known. Researchers have been interested in those aspects of Lie rings which are concerned with the Burnside problem, nilpotent groups and regular automorphisms. In [3], Zamani and Shahryari introduced an algebraic system, dropping the commutativity assumption in a Lie ring. These systems are called Jacobi systems and they are analogue to near-rings, about which hundreds of papers has been written, (See [2]). The goodness of the theory of near-rings gives us the hope that we may bring the theory of Jacobi systems in the interest of doing further research in this area. In this paper we give the generalities of this theory. Topics such as J-solvable and J-nilpotent Jacobi systems, Abelian representation and some other elementary topics are included in this paper. But we do not know how much interest could be gained from this subject. We give our appreciation to the Office of Research at Tabriz University and also the Department of Research at Sahand University of Technology for their financial support during this research. 119 SHAHRYARI, ZAMANI 1. Introduction Let J be a group and suppose that there is a bi-homomorphism [ , ] : J ×J → J such that i) [ x, x ] = 1 for all x ∈ J, ii) [ [ x, y ], z ][ [ y, z ], x ][ [ z, x ], y ] = 1 for all x, y, z ∈ J. Then we say that J is a Jacobi system. Obviously any Lie ring is a Jacobi system in which the underling group is an Abelian group. Another example of a .