# Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL BASES ARISING FROM QUANTIZED ENVELOPING ALGEBRAS, 1990. He said that

the canonical basis for $U^+$, the plus part of a quantized enveloping algebra $U$ of type A,D,E. This is done by two methods (which lead to the same basis):

(1) an algebraic one based on the following three ingredients:

(i) an integer form of $U^+$ which I introduced earlier [79],[90],

(ii) a bar involution of $U^+$ and

(iii) a basis at infinity of $U^+$ coming from any PBW basis, see [91].

(2) a topological method based on the local intersection cohomology of the orbit closures in the moduli space of representations of a quiver.

I am trying to understand these constructions. The first case is $U_q(sl_2)$. What are the canonical basis of $U_q(sl_2)$? How to construct the canonical basis of $U_q(sl_2)$ using the methods of (1) and (2) respectively? Thank you very much.

• Note that Lusztig's paper is freely available online and has some discussion of low rank examples in 3.4 including this rank 1 case: ams.org/mathscinet-getitem?mr=1035415 (see also the answer to your June question on canonical bases). – Jim Humphreys Jan 20 '15 at 16:32