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In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra $\mathbb{C}[W]$ and the universal enveloping algebra $U(sl_n)$ have been realized as the cohomology ring $H^\bullet(Z)$ for some certain variety $Z$. Nevertheless, in the paper the author says: "Of course, given an algebra $A$, there is no a-priori recipe helping to find a relevant geometric data $(M,Z)$; ...It is fair to say, however, that it is still quite a mystery why a geometric realization of the algebras $A$ that we are interested in is possible at all."

In the recent years, is there any progress in this area? For example is there any theory that tells us when this $Z$ exists and how to find it? For the universal enveloping algebra $U(g)$ for $g$ other than $sl_n$, does the geometric realization exist?

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  • $\begingroup$ I would suggest to retitle the question as "What algebras can be realized geometrically" ? Still in this form the question might be vague and even may be the answer can be that "any algebra" - I am not sure, may be I am wrong. Probably the actual point is in some 'nice realization' where 'nice' is not clearly understood at the moment. There is general construction "Hall algebras" surveyed in arxiv.org/abs/0910.4460 Lectures on canonical and crystal bases of Hall algebras Olivier Schiffmann $\endgroup$ – Alexander Chervov Jul 9 '12 at 7:42
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To answer your last question, a geometric realization of $U(\mathfrak{g})$ and its (irreducible highest weight) representations is given for any symmetric Kac-Moody algebra by the work of Nakajima (see his 1998 paper in Duke Math Journal).

I doubt that a complete answer to your general question exists. However, in recent years, people have been looking at connections between geometric realizations and categorifications. One thing that both of these tend to have in common is that they give rise to "nice" bases with integrality and positivity properties. For the case of $U(\mathfrak{g})$, this is the canonical (or global) basis. For the Hecke algebra, this is the Kazhdan-Lusztig basis. Thus, something that could be viewed as a condition on an algebra in order for one to expect nice geometric realizations and categorifications would be the existence of such a basis.

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