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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

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The title is not very explanatory; you might want to change it. – Qiaochu Yuan Dec 3 '09 at 15:19
I have changed the title, maybe it looks better now – Shizhuo Zhang Dec 3 '09 at 17:15
up vote 5 down vote accepted

I tend to think of the geometric content of Hall algebra construction as reflecting the categorification of quantum groups as studied by Lusztig, Rouquier, Khovanov-Lauda, etc. You can read Lusztig's original paper, though I feel like a sadist even suggesting that. I wrote a bit about this in nLab recently; you might find that useful.

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I would not say that the Hall algebra construction is some sort of "reconstruction" (to me, reconstruction means that you find an algebraic object so that you can reconstruct the category as the representation category of this object).

I agree with Ben Webster that it is better related to categorification. For a more geometric approach to Hall algebras I would recommend to look at the work of Olivier Schiffmann.

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