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

Abstract

For symmetrizable Kac-Moody Lie algebra $\text{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\text{U}}(\text{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\text{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\text{U}}(\text{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave the PBW-basis of ${\text{U}}^{+}(\text{g})$.

23 pages