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

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For symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the modified quantized enveloping algebra U(g) and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra 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 U(g), where the root category is the T²-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 U+(g).