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

Abstract

For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebraU and its canonical basisḂ given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set M which depends only on the root category R and prove that there is a bijection between M andḂ, where 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 U + .