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| The Elements in Crystal Bases Corresponding to Exceptional Modules |
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Citation： |
Yong JIANG,Jie SHENG,Jie XIAO.The Elements in Crystal Bases Corresponding to Exceptional Modules[J].Chinese Annals of Mathematics B,2010,(1):1~20 |
| Page view： 3718
Net amount： 2951 |
Authors： |
Yong JIANG; Jie SHENG; Jie XIAO; |
Foundation： |
the National Natural Science Foundation of China (No. 10631010) and the National
Key Basic Research Programme of China (No. 2006CB805905). |
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| Abstract： |
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. |
Keywords： |
Crystal
basis, Hall algebra, Exceptional module |
Classification： |
16G20, 17B37 |
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