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 , 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 ,,
(ii) a bar involution of $U^+$ and
(iii) a basis at infinity of $U^+$ coming from any PBW basis, see .
(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.