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 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⁺.

Keywords

Ringel-Hall algebras / Root categories / Modified quantized enveloping algebras / Canonical bases

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