A Parameterization of the Canonical Bases of Affine Modified Quantized Enveloping Algebras

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 $\textbf{U}^+$.