A parameterization of the canonical bases of affine modified quantized enveloping algebras

Jie Xiao Tsinghua University Minghui Zhao Beijing Forestry University

Quantum Algebra Representation Theory mathscidoc:1702.29005

Chinese Annals of Mathematics, Series B, 37, (2), 235-258, 2016
For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, Lusztig introduced the corresponding modified quantized enveloping algebra $\dot{\textbf{U}}$ and its canonical basis $\dot{\textbf{B}}$ in [13]. In this paper, in case $\mathfrak{g}$ is a symmetric Kac-Moody Lie algebra of finite or affine type, we define a set $\tilde{\mathcal{M}}$ which depends only on the root category $\mathcal{R}$ and prove that there is a bijection between $\tilde{\mathcal{M}}$ and $\dot{\textbf{B}}$, where $\mathcal{R}$ is the $T^2$-orbit category of the bounded derived category of corresponding Dynkin or tame quiver. Our method is based on a result of Lin, Xiao and Zhang in [10], which gives a PBW-type basis of $\textbf{U}^+$.
Ringel-Hall algebras, Root categories, Modified quantized enveloping algebras, Canonical bases
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@inproceedings{jie2016a,
  title={A parameterization of the canonical bases of affine modified quantized enveloping algebras},
  author={Jie Xiao, and Minghui Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219195047263029476},
  booktitle={Chinese Annals of Mathematics, Series B},
  volume={37},
  number={2},
  pages={235-258},
  year={2016},
}
Jie Xiao, and Minghui Zhao. A parameterization of the canonical bases of affine modified quantized enveloping algebras. 2016. Vol. 37. In Chinese Annals of Mathematics, Series B. pp.235-258. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219195047263029476.
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