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

TL;DR

This paper provides a new parameterization of the canonical bases of affine modified quantized enveloping algebras for symmetric Lie algebras, linking them to root categories and building on PBW-basis results.

Contribution

It introduces a set depending only on the root category and establishes a bijection with the canonical basis for affine symmetric Lie algebras.

Findings

Established a bijection between a root category-based set and the canonical basis.

Connected the parameterization to the PBW-basis of +( extbf{g}).

Extended the understanding of canonical bases in affine types.

Abstract

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ 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 $\dot{\textbf{U}}(\textbf{g})$, 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 $\textbf{U}^+(\textbf{g})$.