几何代数基础新视角下的初步探讨  

PRELIMINARY STUDY ON THE BASIS OF GEOMETRIC ALGEBRA FROM A NEW PERSPECTIVE

作　　者：邹宇[1] 付云皓[1] 张景中[1,2] 

机构地区：[1]广州大学计算机科学与教育软件学院,广州510006 [2]中国科学院成都计算机应用研究所,成都610041

出　　处：《系统科学与数学》2010年第1期1-11,共11页Journal of Systems Science and Mathematical Sciences

基　　金：国家重点基础研究发展规划(2004CB318003);中国科学院知识创新重要方向基金(KZCX2-YW-S02)资助项目

摘　　要：提出有关几何代数基础的一个问题:在给定了变换群的几何上,可能建立哪些代数结构?首先证明,不可能在欧氏平面上的点之间定义一种在保距变换下不变的运算,使之在此运算下形成阿贝尔群.进一步的讨论证明,只有将欧氏几何扩大为质点几何,才能在其上建立在保距变换下不变的可交换可结合的运算,而且这种运算只能是质点几何中的加法.如果希望在此运算下构成阿贝尔群,就必须引入向量.最后讨论了所获结果的意义,并提出若干问题.A problem about the basis of geometric algebra in this paper is posed： what kind of algebraic structures may be set up on the geometry given transformation groups？ Firstly it is proven that it is impossible to define an operation which is invariant under distance-preserving transformations among points of Euclidean plane so that Euclidean planar point set can form an abelian group under the defined operation. Further discussions show that only by expanding Euclidean geometry into particle geometry could the commutative and associative operation which is invariant under distance-preserving transformations be established, and this operation can only be the addition operation in the particle geometry. If we hope to make it an abelian group under the operation we have to introduce vectors. At last, we conclude with a discussion of the significance of the results and put forward a number of issues.

关 键 词：几何代数 欧氏平面点集 加法运算 阿贝尔群 

分 类 号：O187[理学—数学]