The Elements in Crystal Bases Corresponding to Exceptional Modules  

作　　者：Yong JIANG Jie SHENG Jie XIAO 

机构地区：[1]Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China,

出　　处：《Chinese Annals of Mathematics,Series B》2010年第1期1-20,共20页数学年刊（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China (No. 10631010) ;the NationalKey Basic Research Programme of China (No. 2006CB805905)

摘　　要：According to the Ringel-Green theorem,the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group.Furthermore,its Drinfeld double can be identified with the whole quantum group,in which the BGP-reflection functors coincide with Lusztig's symmetries.It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra,hence in the integral form of the quantum group.Then it is proved that these elements lie in the crystal basis up to a sign.Eventually,it is shown that the sign can be removed by the geometric method.The results hold for any type of Cartan datum.

关 键 词：Crystal basis Hall algebra Exceptional module 

分 类 号：TP311.13[自动化与计算机技术—计算机软件与理论] P619.281[自动化与计算机技术—计算机科学与技术]