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#### Quantum AlgebraRepresentation Theorymathscidoc:1702.29004

Acta Mathematica Sinica, English Series, 29, (10), 1833-1856, 2013
Let $\mathbf{U}$ be the quantized enveloping algebra and $\dot{\mathbf{U}}$ its modified form. Lusztig gives some symmetries on $\mathbf{U}$ and $\dot{\mathbf{U}}$. Since the realization of $\mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on $\mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $\dot{\mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztig's symmetries on $\dot{\mathbf{U}}$ by applying the BGP-reflection functors to $\dot{\mathcal{H}}$.
BGP-reflection functors, Lusztig's symmetries, Ringel-Hall algebras
@inproceedings{jie2013bgp-reflection,
title={BGP-Reflection Functors and Lusztig's Symmetries of Modified Quantized Enveloping Algebras},
author={Jie Xiao, and Minghui Zhao},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219193202506910475},
booktitle={Acta Mathematica Sinica, English Series},
volume={29},
number={10},
pages={1833-1856},
year={2013},
}
Jie Xiao, and Minghui Zhao. BGP-Reflection Functors and Lusztig's Symmetries of Modified Quantized Enveloping Algebras. 2013. Vol. 29. In Acta Mathematica Sinica, English Series. pp.1833-1856. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219193202506910475.