中文摘要：

设k_（ij）（1≤i〈j≤n）是给定的正整数,分别记G={ （1 k12a12…k1na1n 0 1…k2na2n…… 0 0…1 ）｜aij∈Z},R={ （0 k12a12…k1na1n……0 0…k2na2n 0 0…1 ）｜aij∈Z},本文证明：当G成群且G的上、下中心群列重合时,其相伴Lie环L（G）与Lie环R同构,其中R的Lie积定义为[A,B]=AB-BA.即得到了此时L（G）的矩阵表示.

英文摘要：

Let Tr1（n,Z） be the group of all n x n（upper） unitriangular matrices over the integral ring.Let kij（1≤i≤j≤n） be given positive integers and G={ （1 k12a12…k1na1n 0 1…k2na2n…… 0 0…1 ）｜aij∈Z},R={ （0 k12a12…k1na1n……0 0…k2na2n 0 0…1 ）｜aij∈Z}, If G is a subgroup of Tr1（n,Z）,an d its upper central series coincides with its lower central series,then the associated Lie ring L（G） of G is isomorphic to the Lie ring R,where the Lie product of A,B is defined as[A,B]= AB - BA for arbitrary two elements A and B of R.That is to say,we obtain the matrix representation of L（G）.