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| A Parameterization of the Canonical Bases of Affine Modified Quantized Enveloping Algebras |
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Citation： |
Jie XIAO,Minghui ZHAO.A Parameterization of the Canonical Bases of Affine Modified Quantized Enveloping Algebras[J].Chinese Annals of Mathematics B,2016,37(2):235~258 |
| Page view： 6479
Net amount： 4102 |
Authors： |
Jie XIAO; Minghui ZHAO |
Foundation： |
This work was supported by the Fundamental Research Funds for the Central Universities (No.BLX 2013014)
and the National Natural Science Foundation of China (No.11131001). |
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| Abstract： |
For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, Lusztig
introduced the corresponding modified quantized enveloping algebra
$\dot{\textbf{U}}$ and its canonical basis $\dot{\textbf{B}}$ given
by Lusztig in 1992. In this paper, in the case that $\mathfrak{g}$
is a symmetric Kac-Moody Lie algebra of finite or affine type, the
authors define a set $\wt{\mathcal{M}}$ which depends only on the
root category $\mathcal{R}$ and prove that there is a bijection
between $\wt{\mathcal{M}}$ and $\dot{\textbf{B}}$, where
$\mathcal{R}$ is the $T^2$-orbit category of the bounded derived
category of the corresponding Dynkin or tame quiver. The method in
this paper is based on a result of Lin, Xiao and Zhang in 2011,
which gives a PBW-type basis of $\textbf{U}^+$. |
Keywords： |
Ringel-Hall algebras, Root categories, Modified quantized enveloping algebras, Canonical bases |
Classification： |
Ringel-Hall algebras, Root categories, Modified quantized enveloping algebras, Canonical bases |
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