If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of quantized enveloping algebra. Some others generalized to triangulated(derived category of coherent sheaves on projective spaces,so on so forth)
I wonder know whether there exists some explicit geometric explanation to these works. I think we can consider the quantized enveloping algebra as noncommutative affine scheme. Therefore, it seems these work are doing some sort of reconstruction of schemes in abelian category or derived category.
I wonder whether Bondal-Orlov's work has any relationship(explicit)to these stuff. Therefore, what I am really interested in is what is the real meaning of Ringel-Hall algebra
At last, it is well known that Belinson-Bernstein's theorem established the equivalence of category of U(g)-module and category of D-modules on flag variety of Lie algebra. And some others, say Bezrukavnikov,Frenkel,gaitsgory generalized these results to Kac-Moody algebra.On the other hand, Van den Berg used generalized Ringel Hall algebra to realize quantum group of Kac-Moody algebra. I wonder whether anybody here can say something about these stuff.
All the other comments are welcomed