置换上的一类Hopf代数结构  

A HOPF ALGEBRA STRUCTURE ON PERMUTATIONS

作　　者：赵明泽 李慧兰 Zhao Mingze;Li Huilan(School of Mathematics and Statistics, Shandong Normal University, 250358, Jinan, China)

机构地区：[1]山东师范大学数学与统计学院,济南250358

出　　处：《山东师范大学学报（自然科学版）》2020年第4期395-401,共7页Journal of Shandong Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(11701339).

摘　　要：Hopf代数具有兼容的代数结构和余代数结构,且拥有对极映射,因而具有很强的稳定性,是代数组合学的重要研究内容之一.从1979年开始,已有多类置换上的Hopf代数结构被研究,且它们与代数表示理论、代数几何等都有紧密的联系.本文证明了如果在置换集合上定义洗牌积和切牌余积两种运算,则在置换集合上构造了代数结构和余代数结构,且这两种结构满足兼容性,因而是一种双代数结构.由于分级连通的双代数都是Hopf代数,因而置换集合在这两种运算下构成一个Hopf代数.进一步,根据对极映射的定义给出了这个Hopf代数的对极映射公式及其证明.A Hopf algebra has a compatible algebra and coalgebra structure,and it has an antipode,so its structure is very stable.It is one of the important research contents of algebraic combinatorics.Since 1979,various types of Hopf algebras on permutations have been studied and they are closely related to representation theory of algebras,algebraic geometry,etc.In this paper,it is proved that if two operations shuffle product and deconcatenation coproduct are defined on the permutation set,then the algebraic structure and the coalgebra structure constructed on the permutation set are compatible,so it is a bialgebra.Since all graded and connected bialgebras are Hopf algebras,so the permutation set with these two operations becomes a Hopf algebra.Furthermore,according to the definition of the antipode,we find and prove the formula for the antipode of this Hopf algebra.

关 键 词：HOPF代数 置换 洗牌积 切牌余积 对极映射 

分 类 号：G633.62[文化科学—教育学]