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For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebra \(\dot U\) and its canonical basis \(\dot B\) 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 \(\tilde M\) which depends only on the root category R and prove that there is a bijection between \(\tilde M\) and \(\dot B\), 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+.

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